@article{Math868,
	author = {Mathieu, \'{E}mile},
	journal = {J. Math. Pures Appl.},
	volume = {13},
	keywords = {membrane; plate},
	language = {fre},
	pages = {137--203},
	title = {M\'{e}moire sur le mouvement vibratoire d'une membrane de forme elliptique.},
	url = {http://eudml.org/doc/234720},
	year = {1868},
}

@article{Hill886,
	title = {On the part of the motion of lunar perigee which is a function of the mean motions of the sun and moon},
	author = {Hill, G. W.},
	abstract = {},
	journal = {Acta Math.},
	volume = {8},
	number = {},
	pages = {1 -- 36},
	year = {1886},
	month = {Dec},
	publisher = {},
	issn = {},
	e-issn = {},
	doi = {10.1007/BF02417081},
	**url = {https://doi.org/10.1007/BF02417081}**
}

@article{Brouw918,
	title = {Beweging van een materieel punt op den bodem fleener draaiende vaas onder den invloed der zwaartekracht},
	author = {Brouwer, L. E. J.},
	abstract = {},
	journal = {N. Arch. v. Wisk},
	volume = {2},
	number = {},
	pages = {407 -- 419},
	year = {1918},
	month = {},
	publisher = {},
	issn = {},
	e-issn = {},
	doi = {},
	**url = {}**
}

@article{Blan46,
	author = {Blanch, Gertrude},
	title = {On the Computation of Mathieu Functions},
	journal = {J. Mathematics and Physics},
	volume = {25},
	number = {1 -- 4},
	pages = {1 -- 20},
	year = {1946},
	month = {Apr},
	publisher = {Wiley},
	doi = {10.1002/sapm19462511},
**url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm19462511}**
}

@article{Bril48,
	title = {A practical method for solving Hill’s equation},
	author = {Brillouin, L\'{e}on},
	abstract = {},
	journal = {Quart. Appl. Math.},
	volume = {6},
	number = {},
	pages = {167 -- 178},
	year = {1948},
	publisher = {Brown University - Amer. Math. Soc.},
	issn = {0033-569X},
	e-issn = {1552-4485},
	doi = {10.1090/qam/27111},
	**url = {https://www.sciencedirect.com/science/article/pii/002199917190088X},**
}

@book{Lach64,
	title = {Theory and application of Mathieu functions},
	author = {McLachlan, Norman William}, 
	editor = {},
	series = {},
	volume = {1233},	
	edition = {},
	publisher = {Dover Publications},
	address = {New York},
	isbn = {9780486612331},
	year = {1964},
	month = {Jan},
	doi = {},
	**url = {}**	  
}

@book{Mag66,
	title = {Hill’s Equation},
	author = {Magnus, W. and Winkler, S.}, 
	editor = {},
	series = {Interscience Tracts in Pure and Applied Mathematics},
	volume = {20},	
	edition = {},
	publisher = {Wiley},
	address = {New York},
	isbn = {},
	year = {1966},
	doi = {10.1002/zamm.19680480218},
	**url = {https://doi.org/10.1002/zamm.19680480218}**	  
}

@article{Cano71,
	title = {Numerical solution of Mathieu's equation},
	author = {Canosa, Jos\'{e}},
	abstract = {A method is presented for the numerical solution of Mathieu's equation. The power of the method lies in the fact that it can be used equally for ordinary and extremely asymptotic problems, making possible the computation of Mathieu functions for large values of the parameter with an accuracy heretofore unattainable.},
	journal = {J. Computat. Phys.},
	volume = {7},
	number = {2},
	pages = {255 -- 272},
	year = {1971},
	publisher = {Elsevier},
	issn = {0021-9991},
	e-issn = {1090-2716},
	doi = {10.1016/0021-9991(71)90088-X},
	**url = {https://www.sciencedirect.com/science/article/pii/002199917190088X},**
}

@book{Abra72,
	author = {}, 
	title = {Mathieu Functions},
	booktitle = {Handbook of Mathematical Functions},
	abstract = {},
	editor = {Abramowitz, Milton and Stegun, Irene A.},
	series = {Appl. Math. Series} ,
	volume  = {55},
	chapter = {20},
	pages  = {722 -- 750},
	edition  = {},
	publisher = {US Dept. of Commerce, National Bureau of Standards},
	address = {Washington, DC},
	year  = {1972},
	month = {Dec},
	date = {},
	isbn  = {},
	e-isbn = {}, 
	doi   = {},
  **url = {}**  
}

@book{Bate81,
	author = {Bateman, Jason}, 
	title = {Higher transcendental functions},
	abstract = {},
	editor = {Erd\'{e}lyi, Arthur},
	series = {} ,
	volume  = {3},
	chapter = {16},
	pages  = {91 -- 166},
	edition  = {},
	publisher = {Robert E. Krieger Publishing},
	address = {Malabar, Florida},
	year  = {1981},
	month = {},
	date = {},
	isbn  = {},
	e-isbn = {}, 
	doi   = {},
	**url = {}**  
}


@article{Nay89,
	title = {Bifurcations in a forced softening duffing oscillator},
	author = {Ali H. Nayfeh and Nestor E. Sanchez},
	abstract = {The response of a damped Duffing oscillator of the softening type to a harmonic excitation is analyzed in a two-parameter space consisting of the frequency and amplitude of the excitation. An approximate procedure is developed for the generation of the bifurcation diagram in the parameter space of interest. It is a combination of second-order perturbation solutions of the system in the neighborhood of its non-linear resonances and Floquet analysis. The results show that the proposed scheme is capable of predicting symmetry-breaking and period-doubling bifurcations as well as Jumps to either bounded or unbounded motions. The results obtained are validated using analogand digital-computer simulations, which show chaos and unbounded motions, among other behaviors.},
	journal = {Int. J. Nonlin. Mech.},
	volume = {24},
	number = {6},
	pages = {483 -- 497},
	year = {1989},
	issn = {0020-7462},
	doi = {10.1016/0020-7462(89)90014-0},
  **url = {https://www.sciencedirect.com/science/article/pii/0020746289900140}**
}

@article{Blu89,
	title = {Chaos and order of laser-cooled ions in a Paul trap},
	author = {Bl{\"u}mel, R. and Kappler, C. and Quint, W. and Walther, H.},
	abstract = {We present recent experimental and theoretical results on the behavior of two-, three-, and four-ion crystals close to the Mathieu instability. In particular, we show that the crystals are stable until the Mathieu instability is reached, i.e., for the parameter space investigated, there is no ‘‘melting’’ of the crystals. Laser and rf heating are studied in detail. A simple model of chaotic rf heating as well as a classification of the ion dynamics into four characteristic regimes are presented.},
	journal = {Phys. Rev. A},
	volume = {40},
	number = {2},
	pages = {808 -- 823},
	numpages = {0},
	year = {1989},
	month = {Jul},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.40.808},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.40.808}**
}

@article{Edmo89,
	author = {Edmonds, A. R.},
	title = {Application of the theory of Hill's equation to the study of the stability of periodic classical orbits},
	abstract = {The theory of Hill's equation is applied to periodic orbits in a classical model of the magnetic hydrogen atom. It is shown how an infinite Hill determinant may be approximated and computed, thus giving the discriminant of the relevant Whittaker-Hill equation. This discriminant supplies information on the stability of the above periodic orbits which is much more easily obtained than by the numerical integrations of the orbits appearing in previous publications. The method is also applicable to other classical systems of current interest.},
	journal = {J. Phys. A: Math. Gen.},
	volume = {22},
	number = {14},
	pages = {L673},
	numpages = {},
	year = {1989},
	month = {jul},
	publisher = {IOP Publishing},
	doi = {10.1088/0305-4470/22/14/004},
**url = {https://dx.doi.org/10.1088/0305-4470/22/14/004}**
}

@article{Mond93,
	title = {Stability Analysis Of The Non-Linear Mathieu Equation},
	author = {M. Mond and G. Cederbaum and P. B. Khan and Zarmi, Yair},
	abstract = {The non-linear Mathieu equation is analyzed within the framework of the method of normal forms. Analytical conditions for explosive instability are obtained, and expressions for the period as well as the amplitude of the stable response are derived.},
	journal = {J. Sound Vib.},
	volume = {167},
	number = {1},
	pages = {77 -- 89},
	year = {1993},
	month = {Oct},
	date = {08},
	issn = {0022-460X},
	doi = {10.1006/jsvi.1993.1322},
  **url = {https://www.sciencedirect.com/science/article/pii/S0022460X83713222}**
	
}

@article{Moore93,
	title = {Quantum manifestations of order and chaos in the Paul trap},
	author = {Moore, M. and Bl{\"u}mel, R.},
	abstract = {In the pseudopotential approximation the type of dynamics of two charged particles confined in a Paul trap depends on the ratio $\lambda = \omega_z/omega_{\rho}$ of the frequencies of the axial and the radial pseudo-oscillators, respectively. For $\lambda =1/2, 1, 2$ the classical dynamics is integrable. The classical dynamics appears to be nonintegrable for $\lambda \neq 1/2, 1, 2$ and shows a mixed phase space with interspersed regular and chaotic regions. The type of classical dynamics manifests itself in the statistics of the quasienergy spectrum of the Paul trap. In the integrable case (\lambda = 1/2, 1, 2$) we find level clustering, whereas level repulsion is obtained in the chaotic regime.},
	journal = {Phys. Rev. A},
	volume = {48},
	number = {4},
	pages = {3082 -- 3091},
	numpages = {0},
	year = {1993},
	month = {Oct},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.48.3082},
  **url = {https://link.aps.org/doi/10.1103/PhysRevA.48.3082}**
}

@article{Fren01,
	author = {Frenkel, D. and Portugal, R.},
	title = {Algebraic methods to compute Mathieu functions},
	abstract = {The standard form of the Mathieu differential equation is $y''(z) + (a - 2 q\cos{2z}) y(z) = 0$, where $a$ is the characteristic number and $q$ is a real parameter. The most useful solution forms are given in terms of expansions for either small or large values of $q$. In this paper we obtain closed formulae for the generic term of expansions of Mathieu functions in the following cases: 
		(1) standard series expansion for small $q$; 
		(2) Fourier series expansion for small $q$; 
		(3) asymptotic expansion in terms of trigonometric functions for large $q$; and 
		(4) asymptotic expansion in terms of parabolic cylinder functions for large $q$. 
		We also obtain closed formulae for the generic term of expansions of characteristic numbers and normalization formulae for small and large $q$. Using these formulae one can efficiently generate high-order expansions that can be used for implementation of the algebraic aspects of Mathieu functions in computer algebra systems. These formulae also provide alternative methods for numerical evaluation of Mathieu functions.},
	journal = { J. Phys. A: Math. Gen.},
	volume = {34},
	number = {17},
	pages = {3541 -- 3551},
	numpages = {0},
	year = {2001},
	month = {May},
	publisher = {IOP Publishing},
	doi = {10.1088/0305-4470/34/17/302},
	**url = {https://doi.org/10.1088/0305-4470/34/17/302}**
}

@book{Vila01,
	title = {Hamiltonian Dynamics},
	author = {Vilasi, Gaetano}, 
	editor = {},
	series = {},
	volume = {},
	edition = {},
	year = {2001},
	month = {Mar},
	date = {15},
	publisher = {World Scientific},
	address = {},
	isbn = {978-981-02-3308-2},
	e-isbn = {978-981-4496-73-5},     
	doi = {10.1142/3637},
	**url = {https://doi.org/10.1142/3637}**     
}

@book{Nay04,
	title = {Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods},
	author = {Nayfeh, Ali H. and Balachandran, Balakumar}, 
	editor = {},
	series = {Wiley Series in Nonlinear Science},
	volume = {},
	edition = {2nd},
	publisher = {Wiley-VCH},
	address = {},
	year = {2004},
	month = {},
	isbn = {9780471593485}, 
	e-isbn = {978-3-5276-1754-8},
	doi = {10.1002/9783527617548},
	**url = {https://onlinelibrary.wiley.com/doi/book/10.1002/9783527617548}**	
}

@article{Stra05,
	author = {Sträng, Jan-Eric},
	title = {On the characteristic exponents of Floquet solutions to the Mathieu equation},
	year = {2005},
	journal = {Bull. Acad. R. Belg.},
	volume = {16},
	number = {7},
	doi = {10.3406/barb.2005.28492},
	pages = {269--287},
  **url = {https://www.persee.fr/doc/barb_0001-4141_2005_num_16_7_28492}**
}

@article{Birk07,
	author = {Birkandan, T. and Horta\c{c}su, M.},
	title = {Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces},
	abstract = {We give examples of where the Heun function exists as solutions of wave equations encountered in general relativity. As a new example we find that while the Dirac equation written in the background of Nutku helicoid metric yields Mathieu functions as its solutions in four spacetime dimensions, the trivial generalization to five dimensions results in the double confluent Heun function. We reduce this solution to the Mathieu function with some transformations.},
	journal = {J. Phys. A: Math. Theor.},
	volume = {40},
	number = {5},
	pages = {1105 -- 1116},
	numpages = {0},
	year = {2007},
	month = {Jan},
	publisher = {{IOP P}ublishing},
	doi = {10.1088/1751-8113/40/5/016},
  **url = {https://doi.org/10.1088/1751-8113/40/5/016}**
}

@book{Gla07,
	author = {Glauber, Roy J}, 
	editor = {},
	title = {Quantum Theory of Optical Coherence},
	series = {},
	volume = {},
	edition = {},
	publisher = {Wiley},
	address = {Weinheim},
	year = {2009},
	month = {Oct},
	date = {27},
	isbn = {978-3-527-40687-6},
	e-isbn = {978-3-527-61007-5},
	doi = {10.1002/9783527610075},
	**url = {https://doi.org/10.1002/9783527610075}**
}

@book{Nay08a,
	title = {Nonlinear Oscillations},
	author = {Nayfeh, Ali H. and Mook, Dean T.}, 
	editor = {},
	series = {Wiley Classics Library},
	volume = {},
	edition = {},
	publisher = {Wiley-VCH},
	address = {Weinheim},
	year = {2008},
	month = {Sept},
	isbn = {}, 
	e-isbn = {978-3-527-61759-3},
	doi = {10.1002/9783527617586},
	**url = {https:///onlinelibrary.wiley.com/doi/book/10.1002/9783527617586}**	
}

@book{Nay08b,
	title = {Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods},
	author = {Nayfeh, Ali H. and Balachandran, Balakumar}, 
	editor = {},
	series = {Wiley Series in Nonlinear Science},
	volume = {},
	edition = {},
	publisher = {Wiley-VCH},
	address = {Weinheim},
	year = {2008},
	month = {Nov},
	isbn = {9780471593485}, 
	e-isbn = {9783527617548},
	doi = {10.1002/9783527617548},
	**url = {https://onlinelibrary.wiley.com/doi/book/10.1002/9783527617548}**	
}


@book{Rich09,
	title = {Advanced Mathematical Methods with Maple},
	author = {Richards, Derek}, 
	editor = {},
	series = {},
	volume = {},
	edition = {},
	publisher = {Cambridge Univ. Press},
	address = {Cambridge},
	year = {2009},
	month = {Jun},
	isbn = {9780521135061},
	e-isbn = {}, 
	doi = {}
}

@incollection{Wolf10,
	author = {Wolf, G.}, 
	title = {Mathieu Functions and Hill’s Equation},
	booktitle = {NIST Handbook of Mathematical Functions},
	abstract = {},
	editor = {Olver, Frank W. J. and Lozier, Daniel W. and Boisvert, Ronald F. and Clark, Charles W.},
	volume  = {},
	series = {} ,
	chapter = {28},
	pages  = {651 -- 681},
	edition  = {},
	publisher = {NIST \& Cambridge Univ. Press},
	address = {New York, NY},
	year  = {2010},
	month = {May},
	date = {12},
	isbn  = {978-0-521-19225-5},
	e-isbn = {}, 
	url = {https://dlmf.nist.gov/28}  
}

@article{Quin10,
	author = {Thomas Quinn and Randall P. Perrine and Derek C. Richardson and Rory Barnes},
	title = {A SYMPLECTIC INTEGRATOR FOR HILL'S EQUATIONS},
	abstract = {Hill's equations are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's equations based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV, and tested on some simple orbits. The method demonstrates a lack of secular changes in orbital elements, making it a very useful technique for integrating Hill's equations over many dynamical times. Furthermore, the method allows for efficient collision searching using linear extrapolation of particle positions.},
	journal = {Astron. J.},
	volume = {139},
	number = {2},
	pages = {803},
	numpages = {5},
	year = {2010},
	month = {jan},
	publisher = {The American Astronomical Society - IOP Publishing},
	doi = {10.1088/0004-6256/139/2/803},
**url = {https://dx.doi.org/10.1088/0004-6256/139/2/803}**
}

@book{Kova11,
	author = {}, 
	editor = {Kovacic, Ivana and Brenner, Michael J.},
	title = {The Duffing Equation: Nonlinear Oscillations and their Behaviour},
	series = {Theoretical, Computational, and Statistical Physics},
	volume = {},
	edition = {},
	isbn = {9780470715499},
	e-isbn = {9780470977859}, 
	publisher = {Wiley},
	address = {Chichester, West Sussex},
	year = {2011},
	month = {Mar},
	doi = {10.1002/9780470977859}
}	

@book{Nay11,
	author = {Nayfeh, Ali H.}, 
	editor = {},
	title = {Introduction to Perturbation Techniques},
	series = {Wiley Classics Library},
	volume = {},
	edition = {},
	isbn = {9780470715499},
	e-isbn = {9780470977859}, 
	publisher = {Wiley},
	address = {},
	year = {2011},
	month = {Apr},
	doi = {}
}

@book{Riley11,
	author = {Riley, Kenneth Franklin and Hobson, Michael Paul}, 
	editor = {},
	title = {Essential Mathematical Methods for the Physical Sciences},
	series = {},
	volume = {},
	edition = {},
	publisher = {Cambridge Univ. Press},
	address = {Cambridge},
	year = {2011},
	month = {Feb},
	date = {17},
	isbn = {978-0-521-76114-7},
	e-isbn = {978-0-511-77850-6},
	doi = {},
	**url = {}**
}

@article{Brou11,
	title = {Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping},
	abstract = {A theoretical analysis is presented of the response of a lightly and nonlinearly damped mass–spring system in which the spring constant contains a small randomly fluctuating component. Damping is represented by a combination of linear and nonlinear power-law damping. System response to some initial disturbance at time zero is described by a sinusoidal wave whose amplitude and phase vary slowly and randomly with time. Leading order formulations for the equations of amplitude and phase are obtained through the application of methods of stochastic averaging of Stratonovich. The equations of amplitude and phase are given in two versions: Fokker–Planck equations for transient probability and Langevin equations for response in the time-domain. Solutions in closed-form of these equations are derived by methods of mathematical and theoretical physics involving higher transcendental functions. They are used to study the behavior of system response for ever increasing time applying asymptotic methods of analysis such as the method of steepest descent or saddle-point method. It is found that system behavior depends on the power density of the parametric excitation at twice the natural frequency and on the magnitude and form of the damping. Depending on these parameters different types of system behavior are found to be possible: response which decays exponentially to zero, response which leads to a stationary state of random behavior, and response which can either grow unboundedly or which approaches zero in a finite time.},
	keywords = {Fokker–Planck, Random vibrations, Mathieu instability, Nonlinear damping, Parametric excitation},
	author = {Brouwers, J. J. H.},
	journal = {Physica D},
	volume = {240},
	number = {12},
	pages = {990 -- 1000},
	year = {2011},
	issn = {0167-2789},
	doi = {10.1016/j.physd.2011.02.009},
	**url = {http://www.sciencedirect.com/science/article/pii/S0167278911000418}**
}

@book{Wong13,
	author = {Wong, Chun Wa}, 
	editor = {},
	title = {Introduction to Mathematical Physics: Methods and Concepts},
	series = {},
	volume = {},
	edition = {2nd},
	isbn = {},
	e-isbn = {}, 
	publisher = {Oxford Univ. Press},
	address = {Oxford},
	year = {2013},
	month = {Jan},
	date = {24},
	isbn = {978–0–19–964139–0},
	doi = {10.1093/acprof:oso/9780199641390.002.0003},
  **url = {}**
}

@book{Hassa09,
	author = {Hassani, Sadri}, 
	editor = {},
	title = {Mathematical Methods: For Students of Physics and Related Fields},
	series = {Physics and Astronomy},
	volume = {},
	edition = {2nd},
	publisher = {Springer},
	address = {New York, NY},
	year = {2009},
	month = {Oct},
	date = {27},
	isbn = {978-3-319-01194-3},
	e-isbn = {978-0-387-09504-2},
	doi = {10.1007/978-0-387-09504-2},
	**url = {https://doi.org/10.1007/978-0-387-09504-2}**
}

@book{Hassa13,
	author = {Hassani, Sadri}, 
	editor = {},
	title = {Mathematical Physics: A Modern Introduction to Its Foundations},
	series = {Physics and Astronomy},
	volume = {},
	edition = {2nd},
	publisher = {Springer},
	address = {Cham},
	year = {2013},
	month = {Jul},
	date = {27},
	isbn = {978-3-319-01194-3},
	e-isbn = {978-3-319-01195-0},
	doi = {10.1007/978-3-319-01195-0},
	**url = {https://doi.org/10.1007/978-3-319-01195-0}**
}

@article{Wu14,
	title = {A complicated Duffing oscillator in the surface-electrode ion trap},
	author = {Wu, Hao-Yu and Xie, Yi and Wan, Wei and Chen, Liang and Zhou, Fei and Feng, Mang},
	keywords = {},
	abstract = {The oscillation coupling and different nonlinear effects are observed in a single trapped $^{40}$Ca$^+$ ion confined in our home-built surface-electrode trap (SET). The coupling and the nonlinearity are originated from the high-order multipole potentials, such as hexapole and octopole potentials, due to different layouts and the fabrication asymmetry of the SET. We solve a complicated Duffing equation with coupled oscillation terms by the multiple-scale method, which fits the experimental values very well. Our investigation in the SET helps for exploring multi-dimensional nonlinearity using currently available techniques and for suppressing instability of qubits in quantum information processing with trapped ions.},
	journal = {Appl. Phys. B},
	volume = {114},
	number = {1},
	pages = {81 -- 88},
	year = {2014},
	month = {Jan},
	date = {01},
	publisher = {Springer},
	issn = {1432-0649},
	doi = {10.1007/s00340-013-5541-z},
	**url = {https://doi.org/10.1007/s00340-013-5541-z}**
}

@article{Mous14,
	title = {Generalization of Ince’s Equation},
	author = {Moussa, Ridha A.},
	keywords = {},
	abstract = {We investigate the Hill differential equation $\left(1 + A\left(t \right)y"\left(t \right) + B\left(t \right)y'\left(t \right) +\left(\lambda + D\left( t\right)\right)y\left t\right) = 0$, where A(t), B(t), and D(t) are trigonometric polynomials. We are interested in solutions that are even or odd, and have period $\pi$ or semi-period $\pi$. The above equation with one of the above conditions constitutes a regular Sturm-Liouville eigenvalue problem. We investigate the representation of the four Sturm-Liouville operators by infinite banded matrices.},
	journal = {J. Appl. Math. Phys.},
	volume = {2},
	number = {13},
	pages = {1171 -- 1182},
	year = {2014},
	month = {Dec},
	issn = {},
	doi = {10.4236/jamp.2014.213137},
	**url = {https://doi.org/10.4236/jamp.2014.213137}**
}

@book{Stro15,
	title = {Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering},
	author = {Strogatz, Steven H.}, 
	editor = {},
	series = {Studies in Nonlinearity},
	volume = {},
	edition = {2nd},
	year = {2015},
	month = {Mar},
	publisher = {CRC Press},
	address = {Boca Raton},
	isbn = {9780813349107},
	e-isbn = {9780429492563},     
	doi = {10.1201/9780429492563},
	**url = {https://doi.org/10.1201/9780429492563}**     
}

@article{Gad15,
	title = {Periodic analytic approximate solutions for the Mathieu equation},
	author = {M. Gadella and H. Giacomini and L.P. Lara},
	keywords = {Mathieu equation, A modified harmonic balance method, Leasts squares},
	abstract = {We propose two methods to find analytic periodic approximations intended for differential equations of Hill type. Here, we apply these methods on the simplest case of the Mathieu equation. The former has been inspired in the harmonic balance method and designed to find, making use on a given algebraic function, analytic approximations for the critical values and their corresponding periodic solutions of the Mathieu differential equation. What is new is that these solutions are valid for all values of the equation parameter q, no matter how large. The second one uses truncations of Fourier series and has connections with the least squares method.},
	journal = {Appl. Math. Comput.},
	volume = {271},
	pages = {436 -- 445},
	year = {2015},
	month = {Nov},
	date = {15},
	issn = {0096-3003},
	doi = {10.1016/j.amc.2015.09.018},
	**url = {https://www.sciencedirect.com/science/article/pii/S0096300315012369}**
}

@article{Li16,
	title = {An analytical method for {M}athieu oscillator based on method of variation of parameter},
	abstract = {A simple, but very accurate analytical method for forced Mathieu oscillator is proposed, the idea of which is based on the method of variation of parameter. Assuming that the time-varying parameter in Mathieu oscillator is constant, one could easily obtain its accurately analytical solution. Then the approximately analytical solution for Mathieu oscillator could be established after substituting periodical time-varying parameter for the constant one in the obtained accurate analytical solution. In order to certify the correctness and precision of the proposed analytical method, the first-order and ninth-order approximation solutions by harmonic balance method (HBM) are also presented. The comparisons between the results by the proposed method with those by the numerical simulation and HBM verify that the results by the proposed analytical method agree very well with those by the numerical simulation. Moreover, the precision of the proposed new analytical method is not only higher than the approximation solution by first-order HBM, but also better than the approximation solution by the ninth-order HBM in large ranges of system parameters.},
	keywords = {Mathieu oscillator, Analytical solution, The method of variation of parameter, Harmonic balance method},
	author = {Li, Xianghong and Hou, Jingyu and Chen, Jufeng},
	journal = {Commun. Nonlinear Sci. Numer. Simul.},
	volume = {37},
	pages = {326 -- 353},
	year = {2016},
	issn = {1007-5704},
	doi = {10.1016/j.cnsns.2016.02.003},
	**url = {http://www.sciencedirect.com/science/article/pii/S1007570416300211}**
}

@misc{Rand16,
	author = {Rand, Richard H.},
	title = {CISM Course: Time-Periodic Systems Sept. 5 - 9, 2016},
	address = {Udine, Italy},
	publisher = {CISM},
	howpublished = {\url{http://audiophile.tam.cornell.edu/randpdf/rand_mathieu_CISM.pdf}},
	**month = {Sep},**
	year = {2016},
}

@book{Weyl16,
	author = {Weyl, Hermann}, 
	editor = {},
	title = {Meromorphic Functions and Analytic Curves},
	series = {Annals of Mathematics Studies},
	volume = {12},
	edition = {},
	isbn = {},
	e-isbn = {}, 
	publisher = {Princeton Univ. Press - De Gruyter},
	address = {New Jersey},
	year = {2016},
	month = {Mar},
	date = {2},
	isbn = {},
	e-isbn = {9781400882281},
	doi = {10.1515/9781400882281},
**url = {https://doi.org/10.1515/9781400882281}**
}

@book{Serov17,
	author = {Serov, Valery}, 
	editor = {},
	title = {Fourier Series, Fourier Transform and Their Applications to Mathematical Physics},
	series = {Applied Mathematical Sciences},
	volume = {},
	edition = {},
	publisher = {Springer},
	address = {Cham},
	year = {2017},
	month = {Nov},
	date = {26},
	isbn = {978-3-319-65261-0},
	e-isbn = {978-3-319-65262-7},
	issn = {0066-5452},
	e-issn = {2196-968X},
	doi = {10.1007/978-3-319-65262-7},
  **url = {https://doi.org/10.1007/978-3-319-65262-7}**
}

@article{Buti18,
	author = {Butikov, Eugene},
	title = {Analytical expressions for stability regions in the Ince–Strutt diagram of Mathieu equation},
	abstract = {Simple analytical expressions are suggested for transition curves, which separate in the 	Ince–Strutt diagram different types of solutions to the famous Mathieu equation. Derivation of these expressions in this paper relies on physically meaningful periodic solutions describing various regular motions of a familiar nonlinear mechanical system—the rigid planar pendulum with the vertically oscillating pivot. The paper is accompanied by a relevant simulation program.},
	journal = {Am. J. Phys.},
	volume = {86},
	number = {4},
	pages = {257 -- 267},
	year = {2018},
	month = {04},
	publisher = {Amer. Assoc. Phys. Teachers},
	issn = {},
	doi = {10.1119/1.5021895},
	**url = {https://doi.org/10.1119/1.5021895}**
	}
	
@article{Kova18,
	title = {Mathieu’s Equation and Its Generalizations: Overview of Stability Charts and Their Features},
	author = {Kovacic, Ivana and Rand, Richard and Sah, Si Mohamed},
	abstract = {This work is concerned with Mathieu's equation—a classical differential equation, which has the form of a linear second-order ordinary differential equation (ODE) with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation, or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.},
	journal = {Appl. Mech. Rev.},
	volume = {70},
	number = {2},
	pages = {020802},
	numpages = {22},
	year = {2018},
	month = {Feb},
	date = {14},
	publisher = {Amer. Society Mechanical Engineers},
	address = {},
	issn = {0003-6900},
	e-issn = {},
	doi = {10.1115/1.4039144},
	**url = {https://doi.org/10.1115/1.4039144}**
}

@article{Wilk18,
	title = {Approximate solutions to Mathieu's equation},
	author = {Samuel A. Wilkinson and Nicolas Vogt and Dmitry S. Golubev and Jared H. Cole},
	abstract = {Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.},
	journal = {Physica E: Low Dimens. Syst. Nanostruct.},
	volume = {100},
	pages = {24 -- 30},
	year = {2018},
	issn = {1386-9477},
	publisher = {Elsevier},
	doi = {10.1016/j.physe.2018.02.019},
  **url = {https://www.sciencedirect.com/science/article/pii/S1386947717315990}**
}

@incollection{Krack19,
	author = {Malte Krack and Johann Gross}, 
	editor = {Schr\"{o}der, J\"{o}rg and Weigand, Bernhard},
	booktitle = {Harmonic Balance for Nonlinear Vibration Problems},
	series = {Mathematical Engineering},
	volume = {},
	edition = {},
	year = {2019},
	month = {Apr},
	day = {02},
	publisher = {Springer},
	address = {Cham},
	isbn = {978-3-030-14022-9},
	e-ISBN = {978-3-030-14023-6}, 
	issn = {2192-4732},
	e-issn = {2192-4740},
	doi = {10.1007/978-3-030-14023-6},
	**url = {https://doi.org/10.1007/978-3-030-14023-6}**
}


@book{Tayl19,
	author = {Taylor, Michael E.}, 
	editor = {},
	title = {Introduction to Complex Analysis},
	series = {Graduate Studies in Mathematics},
	volume = {202},
	edition = {},
	publisher = {American Mathematical Society},
	address = {},
	year = {2019},
	month = {Nov},
	date = {18},
	isbn = {978-1-4704-5286-5},
	e-isbn = {978-1-4704-5448-7},
	issn = {},
	e-issn = {},
	doi = {10.1090/gsm/202},
	**url = {}**
}

@article{Dani20,
	author = {Daniel, Derek J.},
	title = {Exact solutions of Mathieu’s equation},
	abstract = {Mathieu’s equation originally emerged while studying vibrations on an elliptical drumhead, so naturally, being a linear second-order ordinary differential equation with a Cosine periodic potential, it has many useful applications in theoretical and experimental physics. Unfortunately, there exists no closed-form analytic solution of Mathieu’s equation, so that future studies and applications of this equation, as evidenced in the literature, are inevitably fraught by numerical approximation schemes and nonlinear analysis of so-called stability charts. The present research work, therefore, avoids such analyses by making exceptional use of Laurent series expansions and four-term recurrence relations. Unexpectedly, this approach has uncovered two linearly independent solutions to Mathie’s equation, each of which is in closed form. An exact and general analytic solution to Mathieu’s equation, then, follows in the usual way of an appropriate linear combination of the two linearly independent solutions.},
	journal = {Prog. Theor. Exp. Phys.},
	volume = {2020},
	number = {4},
	pages = {043A01},
	year = {2020},
	month = {04},
	publisher = {Oxford Academic},
	issn = {2050-3911},
	doi = {10.1093/ptep/ptaa024},
	**url = {https://doi.org/10.1093/ptep/ptaa024},**
	eprint = {https://academic.oup.com/ptep/article-pdf/2020/4/043A01/33114067/ptaa024.pdf}
}

@article{Rodri20,
	author = {Rodriguez, A and Collado, J.},
	title = {Periodic Solutions in Non-Homogeneous Hill Equation},
	abstract = {},
	journal = {Nonlinear Dyn. Syst. Theory},
	volume = {20},
	number = {1},
	pages = {78 -- 91},
	year = {2020},
	month = {},
	publisher = {InforMath Publishing Group},
	issn = {1562-8353},
	e-issn = {1813-7385},
	doi = {},
	url = {https://www.e-ndst.kiev.ua/v20n1/7(71).pdf}
}

@article{Khan20,
	title = {Laplace decomposition for solving nonlinear system of fractional order partial differential equations},
	author = {Khan, Hassan and Shah, Rasool and Kumam, Poom and Baleanu, Dumitru and Arif, Muhammad},
	keywords = {},
	abstract = {In the present article a modified decomposition method is implemented to solve systems of partial differential equations of fractional-order derivatives. The derivatives of fractional-order are expressed in terms of Caputo operator. The validity of the proposed method is analyzed through illustrative examples. The solution graphs have shown a close contact between the exact and LADM solutions. It is observed that the solutions of fractional-order problems converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested technique. Due to better accuracy and straightforward implementation, the extension of the present method can be made to solve other fractional-order problems.},
	journal = {Adv. Differ. Equ.},
	volume = {2020},
	number = {1},
	pages = {375},
	numpages = {18},
	year = {2020},
	month = {Jul},
	date = {22},
	publisher = {Springer},
	issn = {1687-1847},
	doi = {10.1186/s13662-020-02839-y},
	**url = {https://doi.org/10.1186/s13662-020-02839-y}**
}

@book{Pence20,
	title = {Essential Mathematics for Engineers and Scientists},
	author = {Pence, Thomas J. and Wichman, Indrek S.}, 
	editor = {},
	series = {},
	volume = {},	
	edition = {},
	publisher = {Cambridge Univ. Press},
	address = {Cambridge},
	isbn = {978-1-108-42544-5},
	e-isbn = {978-1-108-67135-4},
	year = {2020},
	month = {Apr},
	date = {30},
	doi = {10.1017/9781108671354},
	**url = {https://doi.org/10.1017/9781108671354}**	  
}

@article{Karl20,
	title = {Parametrically amplified Mathieu-Duffing nonlinear energy harvesters},
	author = {Karli\v{c}i\'{c}, Danilo and Chatterjee, Tanmoy and Caji\'{c}, Milan and Adhikari, Sondipon},
	keywords = {Energy harvester, Parametric amplification, Nonlinear response, Mathieu-Duffing oscillator, Incremental harmonic balance method},
	abstract = {The steady-state response of a nonlinear piezoelectric energy harvester subjected to external and parametric excitation is investigated based on the Mathieu-Duffing nonlinear oscillator model. The parametric excitation is introduced to amplify the external harmonic excitation and extend the capabilities of the nonlinear piezoelectric energy harvester device. To obtain the approximated solution of the nonlinear periodic responses for displacement and electrical voltage of the energy harvester, the incremental harmonic balance method in combination with the path-following technique is adopted. It is assumed that the proposed nonlinear model consists of cubic and quadratic nonlinearity, where parametric amplification appears in the form of a trigonometric function. The frequency is tuned as one-to-one and the one-to-two ratio between external and parametric excitation. The effects of quadratic and cubic nonlinearity as well as parametric amplification are studied in detail, and their incredible properties to extend harvester application performance is illustrated. It is explicitly demonstrated that for some particular combination of the system parameters, vibration amplitudes and harvested power can be amplified up to three or five times in comparison to the classical broadband nonlinear energy harvester based on the forced Duffing oscillator. This extraordinary amplification shown to be a key motivation to realize the proposed concept in practice. The presence of combined quadratic and cubic nonlinearities resulted in both hardening and softening spring behavior and leading to the appearance of coexisting periodic solutions in the amplitude-frequency responses. Periodic orbits obtained by the proposed methodology are verified with the results from direct numerical integration and fine agreement is demonstrated. Moreover, a significant influence of the parametric amplification on the instantaneous power is revealed in time response diagrams, thus showing better performance of the proposed energy harvester system.},
	journal = {J. Sound Vib.},
	volume = {488},
	pages = {115677},
	numpages = {18},
	year = {2020},
	month = {Dec},
	publisher = {Elsevier},
	issn = {0022-460X},
	doi = {10.1016/j.jsv.2020.115677},
	**url = {https://www.sciencedirect.com/science/article/pii/S0022460X20305071}**
}

@incollection{Whit21,
	author = {Whittaker, Edmund Taylor and Watson, George Neville}, 
	editor = {Moll, Victor H.},
	booktitle = {A Course of Modern Analysis},
	series = {},
	volume = {} ,
	edition = {5th},
	year = {2021},
	month = {},
	day = {},
	publisher = {Cambridge Univ. Press},
	address = {Cambridge, UK},
	isbn = {978-1-316-51893-9},
	e-ISBN = {}, 
	issn = {},
	e-issn = {},
	doi = {10.1017/9781009004091},
	**url = {https://doi.org/10.1017/9781009004091}**
}

@book{Jazar21,
	title = {Perturbation Methods in Science and Engineering},
	author = {Jazar, Reza N.}, 
	editor = {},
	series = {},
	volume = {},	
	edition = {},
	publisher = {Springer},
	address = {Cham},
	isbn = {978-3-030-73460-2},
	e-isbn = {978-3-030-73462-6},
	year = {2021},
	doi = {10.1007/978-3-030-73462-6},
	**url = {https://doi.org/10.1007/978-3-030-73462-6}**	  
}

@Article{Liu21,
	AUTHOR = {Liu, Chein-Shan and Chen, Yung-Wei},
	TITLE = {A Simplified Lindstedt-Poincar\'{e} Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations},
	ABSTRACT = {In order to improve the Lindstedt-Poincar\'{e} method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in terms of a weight factor to decompose the nonlinear term into two sides. We expand the constant preceding the displacement in powers of the introduced parameter so that the coefficients can be determined to avoid the appearance of secular solutions. The present linearized Lindstedt-Poincar\'{e} method is easily implemented to provide accurate higher order analytic solutions of nonlinear oscillators, such as Duffing and van Der Pol nonlinear oscillators. The accuracy of analytic solutions is evaluated by comparing to the numerical results obtained from the fourth-order Runge-Kutta method. The major novelty is that we can simplify the Lindstedt-Poincar\'{e} method to solve strongly a nonlinear oscillator with a large vibration amplitude.},
	JOURNAL = {Mathematics},
	VOLUME = {9},
	YEAR = {2021},
	NUMBER = {23},
	ARTICLE-NUMBER = {3070},
	URL = {https://www.mdpi.com/2227-7390/9/23/3070},
	ISSN = {2227-7390},
	DOI = {10.3390/math9233070}
}

@article{Brima21,
	author = {Brimacombe, Chris and Corless, Robert M. and Zamir, Mair},
	title = {Computation and Applications of Mathieu Functions: A Historical Perspective},
	abstract = {Mathieu functions of period \$\pi\$ or \$2\pi\$, also called elliptic cylinder functions, were introduced in 1868 by \'{E}mile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today; our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross section. This paper surveys and recapitulates the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly related to double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of antisecularity predates that of Lindstedt. For historical interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: \'{E}mile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.},
	journal = {SIAM Review},
	volume = {63},
	number = {4},
	pages = {653 -- 720},
	year = {2021},
	publisher = {Soc. Industrial Appl. Math.},
	issn = {0036-1445},
	e-issn = {1095-7200},
	doi = {10.1137/20M135786X},
	**URL = {https://doi.org/10.1137/20M135786X}**
}

@article{Kyz21,
	title = {Duffing-type equations: Singular points of amplitude profiles and bifurcations},
	author = {Kyzio{\l}, Jan and Okni{\'n}ski, Andrzej},
	abstract = {We study the Duffing equation and its generalizations with polynomial non-linearities. Recently, we have demonstrated that metamorphoses of the amplitude-response curves, computed by asymptotic methods in implicit form as $F (\Omega, A) = 0$, permit prediction of qualitative changes of dynamics occurring at singular points of the implicit curve $F (\Omega, A) = 0$. In the present work, we determine a global structure of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point. We connect our work with independent research on tangential points on amplitude profiles, associated with jump phenomena, characteristic for the Duffing equation. We also show that our techniques can be applied to solutions of the form of $\Omega_{\pm} = f_{\pm} (A)$, obtained within other asymptotic approaches.},
	journal = {Acta Phys. Pol. B},
	volume = {52},
	number = {10},
	pages = {1239 -- 1262},
	numpages = {0},
	year = {2021},
	month = {},
	date = {},
	publisher = {Jagellonian University; Polish Academy of Sciences},
	address = {Krak\'{o}w},
	issn = {0587-4254},
	e-issn = {1509-5770},
	doi = {10.5506/APhysPolB.52.1239},
	**url = {https://doi.org/10.5506/APhysPolB.52.1239}**
}

@article{Azim22,
	title = {Stability and bifurcation of Mathieu–Duffing equation},
	author = {Mohsen Azimi},
	keywords = {Mathieu-Duffing equation, Pitchfork bifurcation, Subharmonic bifurcation, Supercritical bifurcation, Subcritical bifurcation, Parametric resonance},
	abstract = {Various phenomena in science, physics, and engineering result in the Mathieu equation with cubic nonlinear term, known as the Mathieu–Duffing equation. In previous works, different perturbation methods have been used to investigate the stability and bifurcation of this equation in the vicinity of the first unstable tongue and for relatively small values of natural frequency. The primary goal of this paper is to adapt the Strained Parameters Method to investigate the stability and bifurcation associated with stability change around the second unstable tongue. In addition, this work shows that the Strained Parameters Method is able to obtain the same results previously obtained by other perturbation techniques with minimum computational effort. An inductive approach is used to express the multipliers of the transition curves and the location of the newborn equilibria as a function of the parametric frequency. Lastly, the Floquet theory and Poincar\'{e} map are used to validate the analytical results.},
	journal = {Int. J. Nonlin. Mech.},
	volume = {144},
	pages = {104049},
	numpages = {14},
	year = {2022},
	issn = {0020-7462},
	doi = {10.1016/j.ijnonlinmec.2022.104049},
  **url = {https://www.sciencedirect.com/science/article/pii/S0020746222000890}**
}

@ARTICLE{Ghosh23,
	author = {Ghosh, Ipshit and Saxena, Varun and Krishnamachari, Annangarachari},
	title = {Resonance Curves and Jump Frequencies in a Dual-Frequency Paul Trap on Account of Octopole Field Imperfection}, 
	abstract = {Nonlinear phenomena inside a dual-frequency Paul trap are investigated. The dynamics inside the trap under the influence of an external sinusoidal driving force, collisional damping, and even-order octopole field imperfection can be encapsulated by a damped driven Duffing-type equation.
	Harmonic balance technique is applied to analyze the dynamics around the primary resonance. The amplitude of the resonance curves, the inflection frequency at which the jump phenomena occurs, and transitional frequencies often termed as jump-up and jump-down frequencies are found to depend on the voltage and frequency ratios of the dual-frequency Paul trap.},
	journal = {IEEE Trans. Plasma Sci.}, 
	year = {2023},
	month = {Jul},
	volume = {51},
	number = {7},
	pages = {1924 -- 1931},
	keywords = {Resonant frequency;Voltage;Ions;Electric potential;Mathematical models;Radio frequency;Electrodes;Duffing equation;harmonic balance method;nonlinearity;Paul trap;resonance excitation},
	doi = {10.1109/TPS.2023.3285260}
}

@article{Mora23,
	title = {Reduced-quaternionic {M}athieu functions, time-dependent {M}oisil-{T}eodorescu operators, and the imaginary-time wave equation},
	author = {Morais, J. and Porter, R. Michael},
	keywords = {Quaternionic analysis, Elliptical coordinates, Mathieu functions, Bessel functions, Quaternionic functions, Moisil-Teodorescu operators, Imaginary-time wave equation},
	abstract = {We construct a one-parameter family of generalized Mathieu functions, which are reduced quaternion-valued functions of a pair of real variables lying in an ellipse, and which we call λ-reduced quaternionic Mathieu functions. We prove that the $λ$-RQM functions, which are in the kernel of the Moisil-Teodorescu operator $D + \lambda$ ($D$ is the Dirac operator and $\lambda \in \mathbb R ∖ {0}), form a complete orthogonal system in the Hilbert space of square-integrable $\lambda$-metamonogenic functions with respect to the $L^2$-norm over confocal ellipses. Further, we introduce the zero-boundary $\lambada$-RQM-functions, which are $\lambda$-RQM functions whose scalar part vanishes on the boundary of the ellipse. The limiting values of the $\lambda$-RQM functions as the eccentricity of the ellipse tends to zero are expressed in terms of Bessel functions of the first kind and form a complete orthogonal system for $\lambda$-metamonogenic functions with respect to the $L^2$-norm on the unit disk. A connection between the $\lambda$-RQM functions and the time-dependent solutions of the imaginary-time wave equation in the elliptical coordinate system is shown.},
	journal = {Appl. Math. Comput.},
	volume = {438},
	pages = {127588},
	year = {2023},
	publisher = {Elsevier BV},
	issn = {0096-3003},
	doi = {10.1016/j.amc.2022.127588},
  **url = {https://www.sciencedirect.com/science/article/pii/S0096300322006610}**
}

@article{Moat23,
	author = {Moatimid, Galal M. and Amer, T. S. and Amer, W. S.},
	title = {Dynamical analysis of a damped harmonic forced duffing oscillator with time delay},
	abstract = {This paper is concerned with a time-delayed controller of a damped nonlinear excited Duffing oscillator (DO). Since time-delayed techniques have recently been the focus of numerous studies, the topic of this investigation is quite contemporary. Therefore, time delays of position and velocity are utilized to reduce the nonlinear oscillation of the model under consideration. A much supplementary precise approximate solution is achieved using an advanced Homotopy perturbation method (HPM). The temporal variation of this solution is graphed for different amounts of the employed factors. The organization of the model is verified through a comparison between the plots of the estimated solution and the numerical one which is obtained utilizing the fourth order Runge–Kutta technique (RK4). The outcomes show that the improved HPM is appropriate for a variety of damped nonlinear oscillators since it minimizes the error of the solution while increasing the validation variety. Furthermore, it presents a potential model that deals with a diversity of nonlinear problems. The multiple scales homotopy technique is used to achieve an estimated formula for the suggested time-delayed structure. The controlling nonlinear algebraic equation for the amplitude oscillation at the steady state is gained. The effectiveness of the proposed controller, the time delays impact, controller gains, and feedback gains have been investigated. The realized outcomes show that the controller performance is influenced by the total of the product of the control and feedback gains, in addition to the time delays in the control loop. The analytical and numerical calculations reveal that for certain amounts of the control and feedback signal improvement, the suggested controller could completely reduce the system vibrations. The obtained outcomes are considered novel, in which the used methods are applied on the DO with time-delay. The increase of the time delay parameter leads to a stable case for the DO, which is in harmony with the influence of this parameter. This drawn curves show that the system reaches a stable fixed point which assert the presented discussion.},
	journal = {Sci. Rep.},
	volume = {13},
	number = {1},
	pages = {6507},
	publisher = {Springer Nature},
	issn = {2045-2322},
	year = {2023},
	month = {Apr},
	date = {20},
	keywords = {},
	doi = {10.1038/s41598-023-33461-z},
	**url = {https://doi.org/10.1038/s41598-023-33461-z}**
}

@article{Gad23,
	author = {Gadella, Manuel and Lara, Luis Pedro},
	title = {A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents},
	abstract = {We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one-dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.},
	keywords = {Floquet exponents, harmonic balance, variational methods},
	journal = {Math. Meth. Appl. Sci.},
	volume = {46},
	number = {8},
	pages = {8956 -- 8974},
	year = {2023},
	month = {May},
	doi = {10.1002/mma.9029},
	**url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.9029}**
}

@book{Geze23,
	title = {Numerical Methods in Physics with Python},
	author = {Gezerlis, Alex}, 
	editor = {},
	series = {},
	volume = {},	
	edition = {2nd},
	publisher = {Cambridge Univ. Press},
	address = {Cambridge, UK},
	isbn = {978-1-009-30385-9},
	e-isbn = {978-1-009-30389-7},
	year = {2023},
	doi = {10.1017/9781009303897},
  **url = {https://doi.org/10.1017/9781009303897}**	  
}

@book{Sels24,
	title = {A Computational Introduction to Quantum Physics},
	author = {Selst\o{}, S\o{}lve}, 
	editor = {},
	series = {},
	volume = {},	
	edition = {},
	publisher = {Cambridge Univ. Press},
	address = {Cambridge, UK},
	isbn = {978-1-009-38963-1},
	e-isbn = {978-1-009-38959-4},
	year = {2024},
	month = {Apr},
	doi = {10.1017/9781009389594},
  **url = {https://doi.org/10.1017/9781009389594}**	  
}

@article{Corl24,
	author = {Corless, Robert M.},
	title = {An {H}ermite–{O}breshkov method for 2nd-order linear initial-value problems for {ODE}},
	abstract = {The numerical solution of initial-value problems (IVP) for ordinary differential equations (ODE) is at this time a mature subject, with many high-quality codes freely available. Second-order linear equations without singularities are an especially simple class of problems to solve, even more so if only a single scalar equation such as the Mathieu equation $y'' + (a - 2q\cos 2x)y = 0$ is being considered. Nonetheless, the topic is not yet exhausted, and this paper considers the case of writing an efficient arbitrary-precision code for the solution of such equations. For this purpose, an implicit Hermite–Obreshkov method attains nearly spectral accuracy at a cost only polynomial in the number of bits of accuracy requested. This is interesting for the Mathieu equation in particular because the solutions can be highly oscillatory of variable frequency and be highly ill-conditioned. This paper reports on the details of the prototype Maple implementation of the method and summarizes the approximation theoretic results justifying the choice of a balanced Hermite–Obreshkov method including its backward stability and decent Lebesgue constants. This method may be of especial interest for the solution of so-called D-finite equations, for which Taylor series coefficients up to degree m are available at cost only O(m), instead of the more usual $$O(m^2)$$. This paper celebrates the happy occasion of the 90th birthday of John C. Butcher. },
	journal = {Numer. Algor.},
	volume = {},
	number = {},
	pages = {},
	year = {2024},
	month = {Jan},
	publisher = {Springer},
	e-issn = {1572-9265},
	doi = {10.1007/s11075-023-01738-z},
	**URL = {https://doi.org/10.1007/s11075-023-01738-z}**
}

@article{Viswa01,
	author = {Viswanath, Divakar},
	title = {The Lindstedt--Poincar\'{e} Technique as an Algorithm for Computing Periodic Orbits},
	abstract = {The Lindstedt--Poincar\'{e} technique in perturbation theory is used to calculate periodic orbits of perturbed differential equations. It uses a nearby periodic orbit of the unperturbed differential equation as the first approximation. We derive a numerical algorithm based upon this technique for computing periodic orbits of dynamical systems. The algorithm, unlike the Lindstedt--Poincar\'{e} technique, does not require the dynamical system to be a small perturbation of a solvable differential equation. This makes it more broadly applicable. The algorithm is quadratically convergent. It works with equal facility, as examples show, irrespective of whether the periodic orbit is attracting, or repelling, or a saddle. One of the examples presents what is possibly the most accurate computation of Hill's orbit of lunation since its justly celebrated discovery in 1878.},
	journal = {SIAM Review},
	volume = {43},
	number = {3},
	pages = {478 -- 495},
	year = {2001},
	publisher = {Society for Industrial and Applied Mathematics},
	doi = {10.1137/S0036144500375292},
  **URL = {https://doi.org/10.1137/S0036144500375292},**
	eprint = {https://doi.org/10.1137/S0036144500375292},
}

@article{Wuerk59,
	title = {Electrodynamic Containment of Charged Particles},
	author = {Wuerker, R. F. and Shelton, H. and Langmuir, R. V.},
	abstract = {Electrically charged iron and aluminum particles having diameters of a few microns have been contained in a confined region of space by means of alternating and static electric fields. The theory is essentially that of alternating gradient focusing; here the motion is governed by Mathieu's equation. Under certain circumstances when many particles are confined the three dimensional focusing force and the Coulomb repulsion results in a ``crystaline'' array which can be ``melted'' and reformed.},
	journal = {J. Appl. Phys.},
	volume = {30},
	number = {3},
	pages = {342 -- 349},
	numpages = {0},
	year = {1959},
	publisher = {AIP Publishing},
	doi = {10.1063/1.1735165},
	**url = {https://aip.scitation.org/doi/pdf/10.1063/1.1735165}**
}

@incollection{Deh68,
	title = {Radiofrequency Spectroscopy of Stored Ions I: Storage},
	editor = {D. R. Bates and Immanuel Estermann},
	series = {Advances in Atomic and Molecular Physics},
	publisher = {Academic Press},
	volume = {3},
	pages = {53 -- 72},
	year = {1968},
	issn = {0065-2199},
	author = {H.G. Dehmelt},
	abstract = {This chapter discusses the ways to develop techniques to isolate, contain in a trap, thermalize, and possibly refrigerate the atomic systems under observation. The electrons move along bound orbits that are characterized by three frequencies. In ultrahigh vacuums, a beam of low-energy electrons is reflected upon itself. Temporary trapping occurs by the transformation of longitudinal kinetic energy into transverse because of e–e collisions. By providing simultaneously a radiative damping mechanism, the trapping is made permanent. Because ion–ion collisions cannot transform energy of the motion of the center of mass into the kinetic energy of the relative motion of the ions, no energy absorption from the field can take place. The ions having been formed by fast ions passing on their charge to thermal atoms may, independent of the collision parameter, be assumed to be initially at rest in good approximation. No energy input into the self-regenerating ion cloud occurs because of the collision process.},
	doi = {10.1016/S0065-2199(08)60170-0},
	**url = {https://www.sciencedirect.com/science/article/pii/S0065219908601700}**
}

@article{Lew69,
	author = {Lewis, H. R. and Riesenfeld, W. B.},
	title = {An Exact Quantum Theory of the Time‐Dependent Harmonic Oscillator and of a Charged Particle in a Time‐Dependent Electromagnetic Field},
	journal = {J. Math. Phys.},
	volume = {10},
	number = {8},
	pages = {1458 -- 1473},
	year = {1969},
	month = {},
	abstract = {The theory of explicitly time‐dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schr\"{o}dinger equation. As a specific well‐posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time‐dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time‐dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time‐dependent uniform charge distribution. A class of explicitly time‐dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.},
	doi = {10.1063/1.1664991},
	**URL = {https://doi.org/10.1063/1.1664991}**
}

@article{Baril74,
	title = {Pi{\'e}geage des ions dans un champ quadrupolaire tridimensionnel {\`a} haute fr{\'e}quence},
	author = {Baril, M. and Septier, A.},
	journal = {Rev. Phys. Appl. (Paris)},
	volume = {9},
	number = {3},
	pages = {525 -- 531},
	numpages = {0},
	year = {1974},
	month = {},
	publisher = {{EDP S}ciences},
	doi = {10.1051/rphysap:0197400903052500},
	**url = {}**
}

@book{Kra81,
	author = {},
	editor = {Kramer, Peter and Saraceno, Marcos},
	title = {Geometry of the Time-Dependent Variational Principle in Quantum Mechanics}, 
	series = {Lecture Notes in Physics},
	volume = {140},   
	publisher = {Springer},
	address = {Berlin},  
	year = {1981},
	month = {}, 
	pages = {},
	ISBN = {978-3-540-10579-4},
	e-ISBN = {978-3-540-38576-9},
	doi = {10.1007/3-540-10579-4},
	**url = {https://doi.org/10.1007/3-540-10579-4}** 
}

@inproceedings{Ceau87,
	author = {Ceausescu, V. and Gheorghe, A.},
	title = {Classical limit and quantization of hamiltonian systems},
	booktitle = {Symmetries and Semiclassical Features of Nuclear Dynamics},     
	series = {Lect. Notes Phys.},
	volume = {279},
	editor = {Raduta, A. A.},
	year = {1987},
	month = {Jul}, 
	publisher = {Springer},
	address = {Berlin},
	pages = {69--117},
	ISBN = {978-3-540-17926-9},
	e-ISBN = {978-3-540-47112-7},
	doi = {10.1007/3-540-17926-7_47} 
}

@article{Comb88,
	title = {The quantum stability problem for some class of time-dependent hamiltonians},
	journal = {Ann. Phys. (N. Y.)},
	abstract = {Suitable scaling for the time-dependent Schr\"{o}dinger equation allows reduction of various types of time-dependent Schr\"{o}dinger hamiltonians to the quantum dynamics of the harmonic oscillator. Therefore perturbing any of these time-dependent hamiltonians amounts to perturbing the harmonic oscillator's hamiltonian by suitable time-dependent perturbations. The particular case of time-periodic perturbations of the harmonic oscillator is then considered with regard to the “quantum stability problem”. Quantum stability results are obtained, away from some resonant set of Lebesgue measure smaller than $\varepsilon$ (any $\varepsilon$), provided a suitable norm of the perturbation is dominated by C$\varepsilon$ (some constant C).},
	volume = {185},
	number = {1},
	pages = {86 -- 110},
	year = {1988},
	month = {Jul},
	issn = {0003-4916},
	publisher = {Elsevier},
	doi = {10.1016/0003-4916(88)90259-X},
	author = {M Combescure},
	**url = {https://www.sciencedirect.com/science/article/pii/000349168890259X}**
}

@article{Comb90,
	title = {Crystallization of trapped ions—A quantum approach},
	author = {Combescure, Monique},
	abstract = {We present a quantum approach for a system of two or three ions in a Paul's trap. Under the simplifying assumption that the ionic repulsion is inverse square instead of Coulombic, we explicitly solve the time-dependent Schrödinger equation for this system. The quasi-periodic in time nature of the quasi-energy eigenstates provides a good description of both the “crystalline regime” and the “nonchaotic cloud regime” inside the trap. The characteristic size of the “two-ions crystal” in this quantum approach is compared to its value in the usual classical treatment.},
	journal = {Ann. Phys. (N. Y.)},
	volume = {204},
	number = {1},
	pages = {113 -- 123},
	year = {1990},
	month = {Nov},
	issn = {0003-4916},
	doi = {10.1016/0003-4916(90)90122-5},
	**url = {https://www.sciencedirect.com/science/article/pii/0003491690901225}**
}

@article{Paul90,
	title = {Electromagnetic traps for charged and neutral particles},
	author = {Paul, Wolfgang},
	abstract = {},
	journal = {Rev. Mod. Phys.},
	volume = {62},
	number = {3},
	pages = {531 -- 540},
	numpages = {0},
	year = {1990},
	month = {Jul},
	publisher = {American Physical Society},
	doi = {10.1103/RevModPhys.62.531},
	**url = {https://doi.org/10.1103/RevModPhys.62.531}**
}

@article{Brew90,
	author = {Brewer, R. G. and Hoffnagle, J. and DeVoe, R. G. and Reyna, Luis and Henshaw, William},
	title = {Collision-induced two-ion chaos},
	abstract = {Two ions confined to a radio-frequency trap and cooled by radiation pressure exhibit deterministic chaos. Theoretical analysis of the two-ion dynamics reveals that the route to chaos is due solely to ion-ion collisions. During a collision, the nonlinear Coulomb interaction introduces a transient instability, which gives way to stable single-particle-like motion as the ions move apart. The chaotic dynamics are characterized by a strange attractor that resembles a spiral galaxy.},
	journal = {Nature},
	volume = {344},
	number = {6264},
	pages = {305 -- 309},
	year = {1990},
	month = {Mar},
	date = {01},
	publisher = {Nature},
	issn = {1476-4687},
	doi = {10.1038/344305a0},
	**URL = {https://doi.org/10.1038/344305a0}**
}

@article{Wint91,
	title = {Simple demonstration of storing macroscopic particles in a "Paul trap"},
	author = {Winter, H. and Ortjohann, H. W.},
	abstract = {A simple experimental setup can be used to demonstrate the storage of macroscopic dust particles in a ‘‘Paul trap.’’ The trap is operated at atmospheric pressure, which results in an efficient ‘‘cooling’’ of the stored particles. Under these conditions, a single anthracene dust particle has been confined for over 2 months. The device is well suited to show the formation of ordered structures when a number of ‘‘cooled’’ species are trapped. Due to its overall simplicity, the setup can be used in lectures, student laboratories, etc.},
	journal = {Am. J. Phys.},
	volume = {59},
	number = {9},
	pages = {807 -- 813},
	numpages = {0},
	year = {1991},
	publisher = {Amer. Asoc. Phys. Teachers},
	doi = {10.1119/1.16830},
	**url = {https://aapt.scitation.org/doi/10.1119/1.16830}**
}

@article{Ste92,
	author = {Stenholm, Stig},
	title = {Quantum Motion in a Paul Trap},
	journal = {J. Mod. Optics},
	volume = {39},
	number = {2},
	pages = {279 -- 290},
	year  = {1992},
	publisher = {Taylor & Francis},
	abstract = {This paper discusses the one dimensional quantum motion in a Paul trap maintained by a radio frequency potential. The exact Wigner function is constructed. Similarities and differences with the classical motion and with the motion in a purely harmonic trap are discussed. The stable and unstable cases are both treated.},
	doi = {10.1080/09500349214550281},
	**URL = {https://doi.org/10.1080/09500349214550281}**
}

@incollection{Gla93,
	author = {Glauber, Roy J.}, 
	editor = {Inguva, Ramarao},
	title = {The Quantum Mechanics of Particles in Time-Dependent Quadrupole Fields},
	booktitle = {Recent Developments in Quantum Optics},
	series = {},
	volume = {} ,
	edition = {},
	pages = {1 -- 13},
	year = {1993},
	month = {},
	day = {},
	publisher = {Plenum Press},
	address = {New York, N. Y.},
	ISBN = {978-3-031-55419-3},
	e-ISBN = {}, 
	issn = {},
	e-issn = {},
	doi = {},
	**url = {}**
}

@article{Ita93,
	title = {Quantum projection noise: Population fluctuations in two-level systems},
	author = {Itano, W. M. and Bergquist, J. C. and Bollinger, J. J. and Gilligan, J. M. and Heinzen, D. J. and Moore, F. L. and Raizen, M. G. and Wineland, D. J.},
	abstract = {Measurements of internal energy states of atomic ions confined in traps can be used to illustrate fundamental properties of quantum systems, because long relaxation times and observation times are available. In the experiments described here, a single ion or a few identical ions were prepared in well-defined superpositions of two internal energy eigenstates. The populations of the energy levels were then measured. For an individual ion, the outcome of the measurement is uncertain, unless the amplitude for one of the two eigenstates is zero, and is completely uncertain when the magnitudes of the two amplitudes are equal. In one experiment, a single $^{199}$Hg$^+$ ion, confined in a linear rf trap, was prepared in various superpositions of two hyperfine states. In another experiment, groups of $^9$Be$^+$ ions, ranging in size from about 5 to about 400 ions, were confined in a Penning trap and prepared in various superposition states. The measured population fluctuations were greater when the state amplitudes were equal than when one of the amplitudes was nearly zero, in agreement with the predictions of quantum mechanics. These fluctuations, which we call quantum projection noise, are the fundamental source of noise for population measurements with a fixed number of atoms. These fluctuations are of practical importance, since they contribute to the errors of atomic frequency standards.},
	journal = {Phys. Rev. A},
	volume = {47},
	issue = {5},
	pages = {3554 -- 3570},
	numpages = {0},
	year = {1993},
	month = {May},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.47.3554},
	**url = {https://doi.org/10.1103/PhysRevA.47.3554}**
}

@article{Moore93,
	title = {Quantum manifestations of order and chaos in the Paul trap},
	author = {Moore, M. and Bl{\"u}mel, R.},
	abstract = {In the pseudopotential approximation the type of dynamics of two charged particles confined in a Paul trap depends on the ratio $\lambda = \omega_z/omega_{\rho}$ of the frequencies of the axial and the radial pseudo-oscillators, respectively. For $\lambda =1/2, 1, 2$ the classical dynamics is integrable. The classical dynamics appears to be nonintegrable for $\lambda \neq 1/2, 1, 2$ and shows a mixed phase space with interspersed regular and chaotic regions. The type of classical dynamics manifests itself in the statistics of the quasienergy spectrum of the Paul trap. In the integrable case (\lambda = 1/2, 1, 2$) we find level clustering, whereas level repulsion is obtained in the chaotic regime.},
	journal = {Phys. Rev. A},
	volume = {48},
	number = {4},
	pages = {3082 -- 3091},
	numpages = {0},
	year = {1993},
	month = {Oct},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.48.3082},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.48.3082}**
}

@article{Far94a,
	title = {Double-well dynamics of two ions in the Paul and Penning traps},
	author = {Farrelly, David and Howard, James E.},
	abstract = {A double well is identified in the motion of two ions in a Paul or Penning trap. The associated normal-mode frequencies are found to pass through a series of resonances and an avoided crossing as a control parameter is varied. Chaos induced by the driving field in the vicinity of the separatrix in the Paul trap is also studied.},
	journal = {Phys. Rev. A},
	volume = {49},
	number = {2},
	pages = {1494 -- 1497},
	numpages = {0},
	year = {1994},
	month = {Feb},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.49.1494},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.49.1494}**
}

@article{Wine94,
	title = {Squeezed atomic states and projection noise in spectroscopy},
	author = {Wineland, D. J. and Bollinger, J. J. and Itano, W. M. and Heinzen, D. J.},
	abstract = {We investigate the properties of angular-momentum states which yield high sensitivity to rotation. We discuss the application of these ‘‘squeezed-spin’’ or correlated-particle states to spectroscopy. Transitions in an ensemble of $N$ two-level (or, equivalently, spin-1/2) particles are assumed to be detected by observing changes in the state populations of the particles (population spectroscopy). When the particles’ states are detected with 100% efficiency, the fundamental limiting noise is projection noise, the noise associated with the quantum fluctuations in the measured populations. If the particles are first prepared in particular quantum-mechanically correlated states, we find that the signal-to-noise ratio can be improved over the case of initially uncorrelated particles. We have investigated spectroscopy for a particular case of Ramsey’s separated oscillatory method where the radiation pulse lengths are short compared to the time between pulses. We introduce a squeezing parameter $\xi_R$ which is the ratio of the statistical uncertainty in the determination of the resonance frequency when using correlated states vs that when using uncorrelated states. More generally, this squeezing parameter quantifies the sensitivity of an angular-momentum state to rotation. Other squeezing parameters which are relevant for use in other contexts can be defined. We discuss certain states which exhibit squeezing parameters $\xi_R \backsimeq N^{−1/2}$. We investigate possible experimental schemes for generation of squeezed-spin states which might be applied to the spectroscopy of trapped atomic ions. We find that applying a Jaynes-Cummings–type coupling between the ensemble of two-level systems and a suitably prepared harmonic oscillator results in correlated states with $xi_R < 1$.},
	journal = {Phys. Rev. A},
	volume = {50},
	issue = {1},
	pages = {67--88},
	numpages = {0},
	year = {1994},
	month = {Jul},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.50.67},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.50.67}**
}

@article{Hoff94,
	author = {J. Hoffnagle and R. G. Brewer},
	title = {On the Frequency-Locked Orbits of Two Particles in a Paul Trap},
	abstract = {Calculations are presented that show frequency-locking to be a prominent phenomenon in the dynamics of two ions in a Paul trap, provided that damping is linear and small. The frequency-locked attractors that exist when dissipation is present correspond to stable, periodic orbits of the underlying Hamiltonian system, which appear to be infinite in number. The accuracy of the calculations is illustrated by comparing an orbit observed in a Paul trap for microspheres with the solution of the equations of motion.},
	journal = {Science},
	volume = {265},
	number = {5169},
	pages = {213 -- 215},
	year = {1994},
	doi = {10.1126/science.265.5169.213},
  **URL = {https://www.science.org/doi/abs/10.1126/science.265.5169.213}**
}

@article{Hoff95,
	title = {Stability of two-ion crystals in the Paul trap: A comparison of exact and pseudopotential calculations},
	author = {Hoffnagle, J. A. and Brewer, R. G.},
	abstract = {The stability of two-ion crystals in a Paul trap with a dc component in the quadrupole potential has been studied with the use of the monodromy matrix. The pseudopotential model predicts crystals with the ions at rest either along the trap axis or in the radial plane. The solutions of the full equations of motion disagree with the predictions of the pseudopotential model when the radial and axial secular frequencies are nearly degenerate: the crystal is either unstable (as first noted by Emmert {\em et al.}) or exists in a previously unanticipated configuration in which the ions lie at an angle to the trap axes. A bifurcation diagram near the edge of the crystalline stability range does not support a frequency-doubling route to chaos.},
	journal = {Appl. Phys. B},
	volume = {60},
	number = {2 -- 3},
	pages = {113 -- 117},
	numpages = {0},
	year = {1995},
	month = {Feb},
	publisher = {Springer},
	doi = {10.1007/BF01135851},
	**url = {https://doi.org/10.1007/BF01135851}**
}

@article{Blu95,
	title = {Nonlinear dynamics of trapped ions},
	author = {Bl{\"u}mel, R.},
	abstract = {The first part of this paper discusses the nonlinear dynamics of ions stored in a Paul trap. Four topics are presented in detail: (i) The existence of a new deterministic melting region, (ii) crystallization in a secondary Mathieu stability region, (iii) the existence of hetero-charged ion crystals, and (iv) "cooling induced melting". In the second part, the dynamic Kingdon trap, a close relative of the Paul trap, is suggested as an alternative trap design for the study of nonlinear effects of trapped ions. The dynamic Kingdon trap shows a period doubling route to chaos as well as crystallization.},
	journal = {Phys. Scr.},
	volume = {T59},
	number = {},
	pages = {369 -- 379},
	numpages = {0},
	year = {1995},
	month = {Apr},
	date = {8},
	publisher = {{IOP P}ublishing},
	doi = {10.1088/0031-8949/1995/T59/050},
	**url = {https://doi.org/10.1088/0031-8949/1995/T59/050}**
}

@Article{Hase95,
	author = {Hasegawa, T. and Uehara, K.},
	title = {Dynamics of a single particle in a Paul trap in the presence of the damping force},
	abstract = {The motion of a single charged particle in a Paul trap in the presence of the damping force is investigated theoretically and the modified stability diagrams in the parameter space are calculated. The results show that the stable regions in the $a--q$ parameter plane are not only enlarged but also shifted. Consequently, the damping force causes instability in some cases, contrary to intuition. As a by-product of the calculation, we derive new theoretical approximate expressions for the secular-oscillation frequency. In the limiting case of no damping, these formulas are in good agreement with early measurements done by Wuerker et al.},
	journal = {Appl. Phys. B},
	volume = {61},
	number = {2},
	pages = {159 -- 163},
	publisher = {Springer},
	year = {1995},
	month = {Aug},
	day = {01},
	issn = {1432-0649},
	doi = {10.1007/BF01090937},
	**url = {https://doi.org/10.1007/BF01090937}**
}

@inproceedings{Schli96,
	title = {From a Single Ion to a Mesoscopic System: Crystalization of Ions in Paul Traps},
	author = {Schlipf, Stefan},
	booktitle = {Coherent and Collective Interactions of Particles and Radiation Beams},     
	series = {Proc. Int. School Phys. "Enrico Fermi"},
	volume = {131},
	edition = {},
	editor = {Aspect, Alain and Barletta, W. and Bonifacio, R.},
	pages = {61 -- 99},
	year = {1996},
	publisher = {IOS Press},
	ISBN = {978-90-5199-281-6},
	e-ISBN = {978-1-61499-216-5},
	doi = {10.3254/978-1-61499-216-5-61},
	**url = {http://ebooks.iospress.nl/volume/coherent-and-collective-interactions-of-particles-and-radiation-beams}**
}

@article{Bres97,
	title = {Quantum signatures of chaos in the dynamics of a trapped ion},
	author = {Breslin, J. K. and Holmes, C. A. and Milburn, G. J.},
	journal = {Phys. Rev. A},
	volume = {56},
	number = {4},
	pages = {3022 -- 3027},
	numpages = {0},
	year = {1997},
	month = {Oct},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.56.3022},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.56.3022}**
}

@article{Gard97,
	title = {Quantum Chaos in an Ion Trap: The Delta-Kicked Harmonic Oscillator},
	author = {Gardiner, S. A. and Cirac, J. I. and Zoller, P.},
	abstract = {We propose an experimental configuration, within an ion trap, by which a quantum mechanical delta-kicked harmonic oscillator could be realized, and investigated. We show how to directly measure the sensitivity of the ion motion to small variations in the external parameters.},
	journal = {Phys. Rev. Lett.},
	volume = {79},
	number = {24},
	pages = {4790 -- 4793},
	numpages = {0},
	year = {1997},
	month = {Dec},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevLett.79.4790},
	**url = {https://link.aps.org/doi/10.1103/PhysRevLett.79.4790}**
}

@article{Alhe96b,
	title = {Higher order non-linear resonances in a Paul trap},
	author = {Alheit, R. and Kleineidam, S. and Vedel, F. and Vedel, M. and Werth, G.},
	keywords = {Ion trap, Non-linear resonances, Paul trap},
	abstract = {We report some detailed investigations on non-linear resonances of high order as they appear in a radio-frequency ion trap, whose potential deviates from the ideal quadrupolar shape. The strength and shape of these resonances have been measured throughout the first stable region of the trap. Moreover, space charge shifts and also effects of a modulation of the trapping voltage on the shape of the resonances have been observed. Some of the resonances cannot be explained by the general rule that the sums of multiples of the ion macrofrequencies $n_r\omega_r + n_z\omega_z$ ($n_r, n_z$ are integers) are multiples of the trapping field frequency $\Omega$.},
	journal = {Int. J. Mass Spectrom. Ion Proc.},
	volume = {154},
	number = {3},
	pages = {155 -- 169},
	publisher = {Elsevier},
	year = {1996},
	month = {Jul},
	date = {31},
	issn = {0168-1176},
	doi = {10.1016/0168-1176(96)04380-7},
	**url = {http://www.sciencedirect.com/science/article/pii/0168117696043807}**
}

@conference{Ghe97,
	title        = {Ion stability in laser fields and anharmonic RF potentials},
	author       = {Gheorghe, V. N. and Mihalcea, B. M. and Gheorghe, A.},
	year         = 1997,
	month        = {June},
	booktitle    = {29th EGAS Conference Abstracts},
	publisher    = {European Physical Society},
	address      = {Berlin},
	pages        = {427},
	note         = {},
	editor       = {Kronfeldt, H. D.},
	organization = {European Physical Society}
}

@article{Wine98a,
	author = {Wineland, D. J. and Monroe, C. and Meekhof, D. M. and King, B. E. and Leibfried, D. and Itano, W. M. and Bergquist, J. C. and Berkeland, D. and Bollinger, J. J. and Miller, J.},
	title = {Quantum state manipulation of trapped atomic ions},
	journal = {Proc. R. Soc. Lond. A},
	volume = {454},
	number = {1969},
	pages = {411 -- 429},
	numpages = {0},
	publisher = {The Royal Society Publishing},
	year = {1998},
	month = {Jan},
	doi = {10.1098/rspa.1998.0168}
}

@article{Blu98,
	title = {Chaos and bifurcations in ion traps of cylindrical and spherical design},
	author = {Bl\"umel, R. and Bonneville, E. and Carmichael, A.},
	abstract = {With the help of analytical and numerical methods we analyze the nonlinear dynamics of a single ion stored in periodically driven dynamical traps of cylindrical and spherical design. Sinusoidal and impulsive drives are investigated. Both traps exhibit a mixed phase space for both drives. Additionally there is a route to chaos via period-doubling bifurcations of the fundamental stable trapping island. We demonstrate that the bifurcation scenarios of the kicked and cw-driven traps are quantitatively close and qualitatively identical.},
	journal = {Phys. Rev. E},
	volume = {57},
	number = {2},
	pages = {1511 -- 1518},
	numpages = {0},
	year = {1998},
	month = {Feb},
	date = {1},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevE.57.1511},
	**url = {https://link.aps.org/doi/10.1103/PhysRevE.57.1511}**
}

@article{Ghe98,
	title = {Ordered Structures in a Variable Length AC Trap},
	author = {Gheorghe, V. N. and Giurgiu, L. and Stoican, O. and Cacicovschi, D. and Molnar, R. and Mihalcea, B.},
	abstract = {A variable length linear trap operating between 20-1000 Hz under standard temperature and pressure conditions is reported. We simulated and observed various kinds of stable patterns: Iinear strings, planar zig-zag and volume structures. The interparticle distances depend on the trapped particle number and trap length. Our setup allows trapped microparticle diagnosis by various methods.},
	journal = {Acta Phys. Pol. A},
	volume = {93},
	number = {4},
	pages = {625 -- 629},
	numpages = {5},
	year = {1998},
	month = {Apr},
	publisher = {Polish Academy of Sciences},
	doi = {10.12693/aphyspola.93.625},
	**url = {http://przyrbwn.icm.edu.pl/APP/PDF/93/a093z4p05.pdf}**
}

@article{Dodon98,
	title = {Quantum singular oscillator as a model of a two-ion trap: An amplification of transition probabilities due to small-time variations of the binding potential},
	author = {Dodonov, V. V. and Man'ko, V. I. and Rosa, L.},
	abstract = {Following the paper by Combescure [Ann. Phys. (N.Y.) 204, 113 (1990)], we apply the quantum singular time-dependent oscillator model to describe the relative one-dimensional motion of two ions in a trap. We argue that the model can be justified for low-energy excited states with the quantum numbers $n \ll n_{max} \sim 100$, provided the dimensionless constant characterizing the strength of the repulsive potential is large enough $g* \sim 105$. Time-dependent Gaussian-like wave packets generalizing odd coherent states of the harmonic oscillator and excitation number eigenstates are constructed. We show that the relative motion of the ions, in contradistinction to its center-of-mass counterpart, is extremely sensitive to the time dependence of the binding harmonic potential since the large value of $g*$ results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.},
	journal = {Phys. Rev. A},
	volume = {57},
	number = {4},
	pages = {2851 -- 2858},
	numpages = {0},
	year = {1998},
	month = {Apr},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.57.2851},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.57.2851}**
}

@article{VanEik99,
	title = {A driven, trapped, laser cooled ion cloud: a forced damped oscillator},
	author = {M. A. {van Eijkelenborg} and K. Dholakia and M. E. M Storkey and D. M. Segal and R. C. Thompson},
	abstract = {We have studied the efficiency of laser cooling on clouds of magnesium and beryllium ions held in a Penning trap. We applied a driving voltage to the trap electrodes to drive the ions near one of their motional resonances, and studied the difference between the phase of the applied driving voltage and that of the resultant ion motion. Just as expected for a forced damped oscillator, we find that the phase difference changes by π when the driving frequency is scanned through resonance. From the width of the measured phase change curves we determine the relative strength of the laser cooling on each of the three characteristic ion motions.},
	journal = {Opt. Commun.},
	volume = {159},
	number = {1 -- 3},
	pages = {169 -- 176},
	year = {1999},
	month = {Jan},
	date = {1},	
	issn = {0030-4018},
	doi = {https://doi.org/10.1016/S0030-4018(98)00587-2},
**url = {https://www.sciencedirect.com/science/article/pii/S0030401898005872}**
}

@article{Sevu00,
	title = {Frequency perturbation in nonlinear Paul traps: A simulation study of the effect of geometric aberration, space charge, dipolar excitation, and damping on ion axial secular frequency},
	abstract = {This article develops an expression that relates perturbation in ion axial secular frequency to geometric aberration, space charge, dipolar excitation, and collisional damping in nonlinear Paul trap mass spectrometers. A multipole superposition model incorporating hexapole and octopole superposition has been adopted to represent field inhomogeneities. A uniform charge density distribution has been assumed for characterizing space charge. Dipolar excitation has been represented as a forcing term weighted by dipole superposition, and damping is represented in terms of reduced collision frequency in the equation of ion motion. The perturbed secular frequency of the ion has been obtained by using a modified Lindstedt–Poincar{\'e} perturbation technique. The expression for perturbed frequency adequately reflects the reported experimental and simulation results. Perturbation is sign sensitive for octopole superposition and sign insensitive for hexapole superposition. Larger shifts occur with octopole aberrations. Perturbation of secular frequency based on the number of ions is mass dependent. Lower masses show larger negative frequency shifts with an increase in the number of ions within the trap. Dipolar excitation potential shifts the secular frequency in the positive direction and is larger for lower masses than for higher masses. Damping plays a minor role in shifting the secular frequencies. The shift increases as we increase the pressure of the bath gas. The shift in ion secular frequency with the axial distance from the center of the trap shows quadratic variation.},
	keywords = {Nonlinear traps, Perturbed secular frequency, Frequency shift, Space charge, Duffing equation},
	author = {Sevugarajan, S. and Menon, A. G.},
	journal = {Int. J. Mass Spectrom.},
	volume = {197},
	number = {1 -- 3},
	pages = {263 -- 278},
	publisher = {Elsevier},
	year = {2000},
	month = {Feb},
	day = {29},
	issn = {1387-3806},
	doi = {10.1016/S1387-3806(99)00265-1},
	**url = {http://www.sciencedirect.com/science/article/pii/S1387380699002651}**
}

@article{Sevu02,
	title = {Transition curves and $iso-\beta_u$ lines in nonlinear Paul traps},
	abstract = {This paper has the motivation to understand the role of field inhomogeneties in altering stability boundaries in nonlinear Paul traps mass spectrometers. With the inclusion of higher order terms in the equation of motion, the governing equation takes the form of a nonlinear Mathieu equation. The harmonic balance technique has been used to obtain periodic solutions which represents the transition curves, $\beta_u = 0$ and $\beta_u = 1$. A continuous fraction expression, similar in form to the linear case, has also been derived to plot iso-$\beta_u$ lines within the stability region. The expression qualitatively reflects experimental observations in literature related to ion stabilities in nonlinear traps. The role of hexapole and octopole superposition in shifting the stable region as well as ion secular frequencies in nonlinear Paul traps has been discussed using the analytical expression derived in this paper.},
	keywords = {Nonlinear Paul traps, Nonlinear Mathieu equation, Harmonic balance method, Transition curves and lines},
	author = {Sevugarajan, S. and Menon, A. G.},
	journal = {Int. J. Mass Spectrom.},
	volume = {218},
	number = {2},
	pages = {181 -- 196},
	publisher = {Elsevier},
	year = {2002},
	month = {Jul},
	date = {1},
	issn = {1387-3806},
	doi = {10.1016/S1387-3806(02)00692-9},
	**url = {http://www.sciencedirect.com/science/article/pii/S1387380602006929}**
}

@article{Dodon02,
	author = {Dodonov, V. V.},
	title = {`Nonclassical states' in quantum optics: a `squeezed' review of the first 75 years},
	abstract = {Seventy five years ago, three remarkable papers by Schr\"{o}dinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called ‘nonclassical states’ (squeezed states are their simplest representatives) over the past seventy five years, both in
	quantum optics and in other branches of quantum physics. 
	My starting point is to elucidate who introduced different concepts, notions and terms,
	when, and what were the initial motivations of the authors. Many new references have been found which enlarge the ‘standard citation package’ used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and ‘nonclassical’ states commenced. The 1990s are described in a more ‘squeezed’ manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.},
	journal = {J. Opt. B: Quantum Semiclass. Opt.},
	volume = {4},
	number = {1},
	pages = {R1 -- R33},
	day = {8},
	month = {Jan},
	year = {2002},
	doi = {10.1088/1464-4266/4/1/201},
	**url = {https://doi.org/10.1088/1464-4266/4/1/201}**
}


@article{Leibf03,
	title = {Quantum dynamics of single trapped ions},
	author = {Leibfried, D. and Blatt, R. and Monroe, C. and Wineland, D.},
	abstract = {Single trapped ions represent elementary quantum systems that are well isolated from the environment. They can be brought nearly to rest by laser cooling, and both their internal electronic states and external motion can be coupled to and manipulated by light fields. This makes them ideally suited for quantum-optical and quantum-dynamical studies under well-controlled conditions. Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions trapped in radio-frequency (Paul) traps.},
	journal = {Rev. Mod. Phys.},
	volume = {75},
	number = {1},
	pages = {281 -- 324},
	numpages = {0},
	year = {2003},
	month = {Mar},
	publisher = {American {P}hysical {S}ociety},
	doi = {10.1103/RevModPhys.75.281},
	**url = {https://link.aps.org/doi/10.1103/RevModPhys.75.281}**
}

@article{Lond03,
	title = {Mass selective axial ion ejection from a linear quadrupole ion trap},
	author = {Londry, F. A. and Hager, James W.},
	abstract = {The electric fields responsible for mass-selective axial ejection (MSAE) of ions trapped in a linear quadrupole ion trap have been studied using a combination of analytic theory and computer modeling. Axial ejection occurs as a consequence of the trapped ions' radial motion, which is characterized by extrema that are phase-synchronous with the local RF potential. As a result, the net axial electric field experienced by ions in the fringe region, over one RF cycle, is positive. This axial field depends strongly on both the axial and radial ion coordinates. The superposition of a repulsive potential applied to an exit lens with the diminishing quadrupole potential in the fringing region near the end of a quadrupole rod array can give rise to an approximately conical surface on which the net axial force experienced by an ion, averaged over one RF cycle, is zero. This conical surface has been named the cone of reflection because it divides the regions of ion reflection and ion ejection. Once an ion penetrates this surface, it feels a strong net positive axial force and is accelerated toward the exit lens. As a consequence of the strong dependence of the axial field on radial displacement, trapped thermalized ions can be ejected axially from a linear ion trap in a mass-selective way when their radial amplitude is increased through a resonant response to an auxiliary signal.},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {14},
	number = {10},
	pages = {1130 -- 1147},
	year = {2003},
	month = {Oct},
	issn = {1044-0305},
	doi = {10.1016/S1044-0305(03)00446-X},
	**url = {http://www.sciencedirect.com/science/article/pii/S104403050300446X}**,
}

@book{Major05,
	title = {Charged Particle Traps: Physics and Techniques of Charged Particle Field Confinement},
	author = {Major, Fouad G. and Gheorghe, Viorica N. and Werth, G{\"u}nther}, 
	editor = {},
	series = {Springer Series on Atomic, Optical and Plasma Physics},
	volume = {37},
	edition = {},
	publisher = {Springer},    
	address = {Berlin, Heidelberg},
	isbn = {978-3-540-22043-5},
	e-isbn = {978-3-540-26576-4},   
	year = {2005},
	doi = {10.1007/b137836}
}

@article{Micha05,
	title = {Ion Excitation in a Linear Quadrupole Ion Trap with an Added Octopole Field},
	author = {Michaud, A. L. and Frank, A. J. and Ding, C. and Zhao, XianZhen and Douglas, D. J.},
	abstract = {Modeling of ion motion and experimental investigations of ion excitation in a linear quadrupole trap with a 4 added octopole field are described. The results are compared with those obtained with a conventional round rod set. Motion in the effective potential of the rod set can explain many of the observed phenomena. The frequencies of ion oscillation in the x and y directions shift with amplitude in opposite directions as the amplitudes of oscillation increase. Excitation profiles for ion fragmentation become asymmetric and in some cases show bistable behavior where the amplitude of oscillation suddenly jumps between high and low values with very small changes in excitation frequency. Experiments show these effects. Ions are injected into a linear trap, stored, isolated, excited for MS/MS, and then mass analyzed in a time-of-flight mass analyzer. Frequency shifts between the x and y motions are observed, and in some cases asymmetric excitation profiles and bistable behavior are observed. Higher MS/MS efficiencies are expected when an octopole field is added. MS/MS efficiencies (N$_2$ collision gas) have been measured for a conventional quadrupole rod set and a linear ion trap with a 4% added octopole field. Efficiencies are chemical compound dependent, but when an octopole field is added, efficiencies can be substantially higher than with a conventional rod set, particularly at pressures of $1.4 \time 10^{−4}$ torr or less.},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {16},
	number = {6},
	pages = {835 -- 849},
	year = {2005},
	month = {Jun},
	publisher = {Elsevier},
	issn = {1044-0305},
	doi = {10.1016/j.jasms.2005.02.006},
	**url = {http://www.sciencedirect.com/science/article/pii/S1044030505001212}**
}

@incollection{March05,
	booktitle = {Quadrupole Ion Trap Mass Spectrometry},
	author = {March, Raymond E. and Todd, John F. J.}, 
	editor = {Winefordner, J. D.},
	series = {Chemical Analysis},
	volume = {165},	
	edition = {2nd},
	publisher = {Wiley},
	address = {Hoboken, NJ},
	year = {2005},
	month = {Aug},
	ISBN = {978-0-471-48888-0},
	e-ISBN = {978-0-471-71798-0},
	doi = {10.1002/0471717983},
	**url = {https://onlinelibrary.wiley.com/doi/book/10.1002/0471717983}**
}

@article{Pedro05,
	author = {Pedrosa, I. A. and Rosas, Alexandre and Guedes, I.},
	title = {Exact quantum motion of a particle trapped by oscillating fields},
	journal = {J. Phys. A: Math. Gen.},
	volume = {38},
	number = {35},
	pages = {7757 -- 7763},
	year = {2005},
	doi = {10.1088/0305-4470/38/35/009}
}


@book{Haro06,
	title = {Exploring the Quantum: Atoms, Cavities and Photons},
	author = {Haroche, Serge and Raimond, Jean-Michel}, 
	editor = {},
	series = {Oxford Graduate Texts},    
	volume = {},
	edition = {},
	year = {2006},
	month = {Aug},
	date = {10},
	publisher = {Oxford Univ. Press},   
	address = {Clarendon, Oxford},
	isbn = {978-0-19968031-3},
	e-isbn = {9780191708626},
	doi = {10.1093/acprof:oso/9780198509141.001.0001},
	**url = {https://doi.org/10.1093/acprof:oso/9780198509141.001.0001}**	
}

@Article{Raja07,
	author = {Rajanbabu, N. and Marathe, Amol and Chatterjee, Anindya and Menon, A. G.},
	title = {Multiple scales analysis of early and delayed boundary ejection in Paul traps},
	abstract = {We use the method of multiple scales to elucidate dynamics associated with early and delayed ejection of ions in mass selective ejection experiments in Paul traps. We develop a slow flow equation to approximate the solution of a weakly nonlinear Mathieu equation to describe ion dynamics in the neighborhood of the stability boundary of ideal traps (where the Mathieu parameter $q_z = q_z^* = 0.908046$). The method of multiple scales enables us to incorporate higher order multipoles, extend computations to higher orders, and generate phase portraits through which weview early and delayed ejection.Our use of the method of multiple scales is atypical in two ways. First, because we look at boundary ejection, the solution to the unperturbed equation involves linearly growing terms, requiring some care in identification and elimination of secular terms. Second, due to analytical difficulties, we make additional harmonic balance approximations within the formal implementation of the method. For positive even multipoles in the ion trapping field, in the stable region of trap operation, the phase portrait obtained from the slow flow consists of three fixed points, two of which are saddles and the third is a center. As the $q_z$ value of an ion approaches $q_z^*$, the saddles approach each other, and a point is reached where all nonzero solutions are unbounded, leading to an observation of early ejection.The phase portraits for negative even multipoles and odd multipoles of either sign are qualitatively similar to each other and display bounded solutions even for $q_z > q_z^*$, resulting in the observation of delayed ejection associated with a more gentle increase in ion motion amplitudes, a mechanism different from the case of the positive even multipoles.},
	journal = {Int. J. Mass Spectrom.},
	publisher = {Elsevier},
	year = {2007},
	month = {Mar},
	day = {15},
	volume = {261},
	number = {2--3},
	pages = {170 -- 182},
	issn = {},
	doi = {10.1016/j.ijms.2006.09.009},
	**url = {https://doi.org/10.1016/j.ijms.2006.09.009}**
}

@article{Har07,
	title = {Detection of Chemical Warfare-Related Species on Complex Aerosol Particles Deposited on Surfaces Using an Ion Trap-Based Aerosol Mass Spectrometer},
	author = {Harris, William A. and Reilly, Peter T. A. and Whitten, William B.},
	abstract = {A new type of aerosol mass spectrometer was developed by minimal modification of an existing commercial ion trap to analyze the semivolatile components of aerosols in real time. An aerodynamic lens-based inlet system created a well-collimated particle beam that impacted into the heated ionization volume of the commercial ion trap mass spectrometer. The semivolatile components of the aerosols were thermally vaporized and ionized by electron impact or chemical ionization in the source. The nascent ions were extracted and injected into the ion trap for mass analysis. The utility of this instrument was demonstrated by identifying semivolatile analytes in complex aerosols. This study is part of an ongoing effort to develop methods for identifying chemical species related to CW agent exposure. Our efforts focused on detection of CW-related species doped on omnipresent aerosols such as house dust particles vacuumed from various surfaces found in any office building. The doped aerosols were sampled directly into the inlet of our mass spectrometer from the vacuumed particle stream. The semivolatile analytes were deposited on house dust and identified by positive ion chemical ionization mass spectrometry up to 2.5 h after deposition. Our results suggest that the observed semivolatile species may have been chemisorbed on some of the particle surfaces in submonolayer concentrations and may remain hours after deposition. This research suggests that identification of trace CW agent-related species should be feasible by this technique.},
	journal = {Anal. Chem.},
	volume = {79},
	number = {6},
	pages = {2354 -- 2358},
	numpages = {0},
	year = {2007},
	month = {Mar},
	date = {1},
	publisher = {American Chemical Society},
	issn = {0003-2700},
	doi = {10.1021/ac0620664},
	**url = {https://doi.org/10.1021/ac0620664}**
}

@article{Meni07,
	title = {Single trapped ion as a time-dependent harmonic oscillator},
	author = {Menicucci, Nicolas C. and Milburn, G. J.},
	abstract = {We show how a single trapped ion may be used to test a variety of important physical models realized as time-dependent harmonic oscillators. The ion itself functions as its own motional detector through laser-induced electronic transitions. Alsing et al., [Phys. Rev. Lett. 94, 220401 (2005)] proposed that an exponentially decaying trap frequency could be used to simulate (thermal) Gibbons-Hawking radiation in an expanding universe, but the Hamiltonian used was incorrect. We apply our general solution to this experimental proposal, correcting the result for a single ion and showing that while the actual spectrum is different from the Gibbons-Hawking case, it nevertheless shares an important experimental signature with this result.},
	journal = {Phys. Rev. A},
	volume = {76},
	number = {5},
	pages = {052105},
	numpages = {5},
	year = {2007},
	month = {Nov},
	date = {},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.76.052105},
	**url = {https://journals.aps.org/pra/pdf/10.1103/PhysRevA.76.052105}**
}

@article{Tak07,
	author = {Takai, Ryota and Nakayama, Kenji and Saiki, Wataru and Ito, Kiyokazu and Okamoto, Hiromi},
	title = {Nonlinear Resonance Effects in a Linear Paul Trap},
	abstract = {The effects of nonlinear resonances in a linear Paul trap have been investigated through systematic experiments and numerical simulations. The main causes of the nonlinearity that affects the stability of the ion motion are the use of circular (rather than hyperbolic) electrodes, their misalignments, and Coulomb interactions among the ions. A particle tracking code based on a two-dimensional model is employed to study the efficiency of plasma storage and to confirm the existence of nonlinear resonance stop bands. Experiments are performed with Ar$^+$ plasmas that are eventually detected by a Faraday cup after a short storage. The obtained data are compared with numerical simulations in which the exact three-dimensional structure of the trap system has been incorporated. Several nonlinear stop bands have been experimentally identified inside the Mathieu stability region. It is demonstrated that the location of a resonance stop band moves due to the space-charge potential depending on the number of confined ions.},
	journal = {J. Phys. Soc. Japan},
	volume = {76},
	number = {1},
	pages = {014802},
	numpages = {8},
	year = {2007},
	doi = {10.1143/JPSJ.76.014802}
}

@article{Zhao08b,
	author = {Zhao, XianZhen and Granot, Ori and Douglas, D. J.},
	title = {Quadrupole Excitation of Ions in Linear Quadrupole Ion Traps with Added Octopole Fields},
	abstract = {Modeling and experimental studies of quadrupole excitation of ions in linear quadrupole traps with added octopole fields are described. An approximate solution to the equations of motion of ions trapped in a quadrupole with added octopole and dodecapole fields, with quadrupole excitation and damping is given. The solutions give the steady-state or stationary amplitudes of oscillation with different excitation frequencies. Trajectory calculations of the oscillation amplitudes are also presented. The calculations show that there can be large changes in the amplitude of ion oscillation with small changes in excitation frequency, on both the low and high-frequency sides of a resonance. Results of experiments with quadrupole excitation of reserpine ions in linear quadrupole traps with 2.0 %, 2.6 %, and 4.0 % added octopole fields are given. It is found that as the excitation frequency is changed, two resonances are generally observed, which are attributed to the motion in the x and y directions. The two resonances can have quite different intensities. Sudden jumps or sharp sided resonances are not observed, although in some cases asymmetric resonances are seen. The calculated frequency differences between the two resonances are in approximate agreement with the experiments.},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {19},
	number = {4},
	pages = {510 -- 519},
	month = {April},
	year = {2008},
	publisher = {Am. Soc. Mass Spectrom.},
	issn = {1879-1123},
	doi = {10.1016/j.jasms.2007.12.007},
	**url = {https://link.springer.com/article/10.1016/j.jasms.2007.12.007}*
}


@article{Mih08,
	author = {Mihalcea, B. M. and Visan, G. T. and Giurgiu, L. C. and Radan, S.},
	title = {Optimization of ion trap geometries and of the signal to noise ratio for high resolution spectroscopy},
	abstract = {Physicists have always focused on designing and implementing methods which would eventually allow them to confine and levitate a particle in a well defined region in space, under conditions of minimal perturbations, in an almost interaction free environment. This led to the development of radically new techniques for trapping atomic particles. One major advantage of such a system lies in the fact that the apparatus used to prepare and manipulate atomic quantum states is already in place. It has been refined and developed for high precision spectroscopy, quantum logic and high accuracy frequency standards. This paper deals with optimization of ion trap geometries which are widely used in modern physics.},
	journal = {J. Optoelectron. Adv. Mat.},
	volume = {10},
	number = {8},
	pages = {1994 -- 1998},
	publisher = {INOE Publishing House},
	address = {Magurele, Romania},
	year = {2008},
	month = {Jul},
	url = {https://old.joam.inoe.ro/download.php?idu=1538}
}    

@article{Haff08,
	title = {Quantum computing with trapped ions},
	author = {H{\"a}ffner, H. and Roos, C. F. and Blatt, R.},
	abstract = {Quantum computers hold the promise of solving certain computational tasks much more efficiently than classical computers. We review recent experimental advances towards a quantum computer with trapped ions. In particular, various implementations of qubits, quantum gates and some key experiments are discussed. Furthermore, we review some implementations of quantum algorithms such as a deterministic teleportation of quantum information and an error correction scheme.},
	keywords = {Quantum computing and information, Entanglement, Ion traps},
	journal = {Phys. Rep.},
	volume = {469},
	number = {4},
	pages = {155 -- 203},
	publisher = {Elsevier},
	year = {2008},
	month = {Dec},
	issn = {0370-1573}, 
	doi = {10.1016/j.physrep.2008.09.003},
	**url = {http://www.sciencedirect.com/science/article/pii/S0370157308003463}**
}

@article{Mih09,
	title = {A quantum parametric oscillator in a radiofrequency trap},
	author = {Mihalcea, B. M.},
	abstract = {An ion confined within a Paul trap can be assimilated, in good approximation, with a harmonic oscillator. A method which enables the construction of an invariant operator based on the Lewis and Riesenfeld approach is reported. The method is then applied to the case of an ion confined in a Paul trap, treated as a quantum harmonic oscillator. An invariant operator is associated with the system. The spectrum of the quasienergy operator finally results. The eigenvectors of the system are found using the Fock state basis.},
	journal = {Phys. Scr.},
	volume = {T135},
	number = {},
	pages = {014006},
	numpages = {4}, 	
	year = {2009},
	month = {Jul},
	date = {31},
	publisher = {{IOP P}ublishing}, 
	doi = {10.1088/0031-8949/2010/T135/0140006},
	**url = {https://iopscience.iop.org/article/10.1088/0031-8949/2009/T135/014006/pdf}**
}

@article{Oka10,
	title = {Characterization of ion Coulomb crystals in a linear Paul trap},
	author = {Okada, K. and Wada, M. and Takayanagi, T. and Ohtani, S. and Schuessler, H. A.},
	abstract = {We describe a simple and fast method for simulating observed images of ion Coulomb crystals. In doing so, cold elastic collisions between Coulomb crystals and virtual very light atoms are implemented in a molecular dynamics (MD) simulation code. Such an approach reproduces the observed images of Coulomb crystals by obtaining density plots of the statistics of existence of each ion. The simple method has the advantage of short computing time in comparison with previous calculation methods. As a demonstration of the simulation, the formation of a planar Coulomb crystal with a small number of ions has been investigated in detail in a linear ion trap both experimentally and by simulation. However, also large Coulomb crystals including up to 1400 ions have been photographed and simulated to extract the secular temperature and the number of ions. For medium-sized crystals, a comparison between experiments and calculations has been performed. Moreover, an MD simulation of the sympathetic cooling of small molecular ions was performed in order to test the possibility of extracting the temperature and the number of refrigerated molecular ions from crystal images of laser-cooled ions. Such information is basic to studying ultracold ion-molecule reactions using ion Coulomb crystals including sympathetically cooled molecular ions.},
	journal = {Phys. Rev. A},
	volume = {81},
	number = {1},
	pages = {013420},
	numpages = {10},
	year = {2010},
	month = {Jan},
	date = {28},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.81.013420},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.81.013420}**
}

@article{Pedre10,
	title = {Anharmonic contributions in real RF linear quadrupole traps},
	author = {Pedregosa, J. and Champenois, C. and Houssin, M. and Knoop, M.},
	abstract = {The radiofrequency quadrupole linear ion trap is a widely used device in physics and chemistry. When used for trapping of large ion clouds, the presence of anharmonic terms in the radiofrequency potential limits the total number of stored ions. In this paper, we have studied the anharmonic content of the trapping potential for different implementations of a quadrupole trap, searching for the geometry best suited for the trapping of large ion clouds. This is done by calculating the potential of a real trap using SIMION8.0, followed by a fit, which allows us to obtain the evolution of anharmonic terms for a large part of the inner volume of the trap.},
	journal = {Int. J. Mass Spectrom.},
	volume = {290},
	number = {2 -- 3},
	pages = {100 -- 105},
	numpages = {0},
	year = {2010},
	month = {Feb},
	date = {15},
	publisher = {Elsevier},
	doi = {10.1016/j.ijms.2009.12.009},
	**url = {https://www.sciencedirect.com/science/article/pii/S1387380609003832}**
}

@article{Bla10,
	author = {Blaum, K. and  Novikov, Yu. N. and  Werth, G.},
	title = {Penning traps as a versatile tool for precise experiments in fundamental physics},
	abstract = { This review article describes the trapping of charged particles. The main principles of electromagnetic confinement of various species from elementary particles to heavy atoms are briefly described. The preparation and manipulation with trapped single particles, as well as methods of frequency measurements, providing unprecedented precision, are discussed. Unique applications of Penning traps in fundamental physics are presented. Ultra-precise trap-measurements of masses and magnetic moments of elementary particles (electrons, positrons, protons and antiprotons) confirm CPT-conservation, and allow accurate determination of the fine-structure constant α and other fundamental constants. This together with the information on the unitarity of the quark-mixing matrix, derived from the trap-measurements of atomic masses, serves for assessment of the Standard Model of the physics world. Direct mass measurements of nuclides targeted to some advanced problems of astrophysics and nuclear physics are also presented. },
	journal = {Contemp. Phys.},
	volume = {51},
	number = {2},
	pages = {149 -- 175},
	year  = {2010},
	month = {Feb},
	publisher = {Taylor \& Francis},
	doi = {10.1080/00107510903387652},
	**URL = {https://doi.org/10.1080/00107510903387652}**
}

@article{Aust10,
	title = {Multipole expansion in quadrupolar devices comprised of planar electrode arrays},
	keywords = {Quadrupole theory, Quadrupole ion trap, Higher-order multipole},
	note = {Harsh Environment Mass Spectrometry: New Developments and Applications},
	abstract = {Ion trap mass analyzers made using arrays of independent electrodes allow unprecedented control and variability of electric field shapes. We present a method to select and implement specific values for higher-order multipoles, which are known to affect ion trapping and mass analysis in quadrupole ion traps. Electrode arrays are amenable to microfabrication techniques, hence this method can be used to improve performance in miniaturized ion trap mass spectrometers. With ion traps made using two opposing electrode array plates, both even- and odd-order multipoles can be independently adjusted.},
	author = {Austin, Daniel E. and Hansen, Brett J. and Peng, Ying and Zhang, Zhiping},
	journal = {Int. J. Mass Spectrom.},
	volume = {295},
	number = {3},
	pages = {153 -- 158},
	publisher = {Elsevier},
	year = {2010},
	month = {Aug},
	date = {1},
	issn = {1387-3806},
	doi = {10.1016/j.ijms.2010.05.009},
	**url = {http://www.sciencedirect.com/science/article/pii/S1387380610001351}**
}

@article{Zhou10,
	title = {Characteristics of stability boundary and frequency in nonlinear ion trap mass spectrometer},
	author = {Zhou, Xiaoyu and Zhu, Zhiqiang and Xiong, Caiqiao and Chen, Rui and Xu, Wenjun and Qiao, Haoxue and Peng, Wen-Ping and Nie, Zongxiu and Chen, Yi},
	abstract = {In this article, the Poincare-Lighthill-Kuo (PLK) method is used to derive an analytical expression on the stability boundary and the ion trajectory. A multipole superposition model mainly including octopole component is adopted to represent the inhomogeneities of the field. In this method, both the motional displacement and secular frequency of ions have been expanded to asymptotic series by the scale of nonlinear term $\varepsilon$, which represents a weak octopole field. By solving the zero and first-order approximate equations, it is found that a frequency shift exists between the ideal and nonlinear conditions. The motional frequency of ions in nonlinear ion trap depends on not only Mathieu parameters, $a$ and $q$, but also the percentage of the nonlinear field and the initial amplitude of ions. In the same trap, ions have the same mass-to-charge ratio $(m/z)$ but they have different initial amplitudes or velocities. Consequently, they will be ejected at different time through after a mass-selective instability scan. The influences on the mass resolution in quadrupole ion trap, which is coupled with positive or negative octopole fields, have been discussed respectively.},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {21},
	number = {9},
	pages = {1588 -- 1595},
	numpages = {0},
	year = {2010},
	month = {Sep},
	date = {01},
	publisher = {American Chemical Society},
	issn = {1044-0305},
	doi = {10.1016/j.jasms.2010.04.013},
	**url = {https://doi.org/10.1016/j.jasms.2010.04.013}**,
}

@book{March10,
	title = {Practical Aspects of Ion Trap Mass Spectrometry: Applications of Ion Trapping Devices},
	author = {}, 
	editor = {March, Raymond E. and Todd, John F. J.},
	series = {Chemical Analysis},
	volume = {5},	
	edition = {},
	year = {2010},
	publisher = {CRC Press},
	address = {Boca Raton},
	ISBN = {978-0-471-48888-0},
	doi = {}
}

@article{Mih10a,
	title = {Nonlinear harmonic boson oscillator},
	author = {Mihalcea, Bogdan M.},
	abstract = {This paper deals with the study of the quantum Hamiltonian for a boson confined in a nonlinear ion trap. The expected value of the quantum Hamiltonian on coherent states is obtained. Then, the equation of motion for the boson results using the time-dependent variational principle (TDVP) and the Hamilton equations of motion. The resulting equation of motion is equivalent to the one describing a perturbed classical oscillator.},
	journal = {Phys. Scr.},
	volume = {T140},
	number = {},
	pages = {014056},
	numpages = {4}, 	
	year = {2010},
	month = {Sep},
	date = {30},
	publisher = {IOP Publishing}, 
	doi = {10.1088/0031-8949/2010/T140/014056},
	**url = {https://doi.org/10.1088/0031-8949/2010/T140/014056}**
}

@article{Mih10b,
	title = {Nonlinear ion trap stability analysis},
	author = {Mihalcea, Bogdan M. and Vi{\c s}an, Gina G.},
	abstract = {This paper investigates the dynamics of an ion confined in a nonlinear Paul trap. The equation of motion for the ion is shown to be consistent with the equation describing a damped, forced Duffing oscillator. All perturbing factors are taken into consideration in the approach. Moreover, the ion is considered to undergo interaction with an external electromagnetic field. The method is based on numerical integration of the equation of motion, as the system under investigation is highly nonlinear. Phase portraits and Poincar\'{e} sections show that chaos is present in the associated dynamics. The system of interest exhibits fractal properties and strange attractors. The bifurcation diagrams emphasize qualitative changes of the dynamics and the onset of chaos.},
	journal = {Phys. Scr.},
	volume = {T140},
	number = {},
	pages = {014057},
	numpages = {6}, 	
	year = {2010},
	month = {Sep},
	date = {30},
	publisher = {IOP Publishing}, 
	doi = {10.1088/0031-8949/2010/T140/014057},
	**url = {https://doi.org/10.1088/0031-8949/2010/T140/014057}**
}

@article{Ake10,
	author = {Akerman, N. and Kotler, S. and Glickman, Y. and Dallal, Y. and Keselman, A. and Ozeri, R.},
	title = {Single-ion nonlinear mechanical oscillator},
	abstract = {We study the steady-state motion of a single trapped ion oscillator driven to the nonlinear regime. Damping is achieved via Doppler laser cooling. The ion motion is found to be well described by the Duffing oscillator model with an additional nonlinear damping term. We demonstrate here the unique ability of tuning both the linear as well as the nonlinear damping coefficients by controlling the laser-cooling parameters. Our observations pave the way for the investigation of nonlinear dynamics on the quantum-to-classical interface as well as mechanical noise squeezing in laser-cooling dynamics.},
	journal = {Phys. Rev. A},
	volume = {82},
	number = {6},
	pages = {061402(R)},
	numpages = {4},
	year = {2010},
	month = {Dec},
	publisher = {Amer. Phys. Soc.},
	doi = {10.1103/PhysRevA.82.061402},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.82.061402}**
}

@article{Zhang11,
	title = {Effects of higher-order multipoles on the performance of a two-plate quadrupole ion trap mass analyzer},
	keywords = {Microfabricated ion trap mass spectrometer, Two-plate quadrupole ion trap, Planar ion trap, Octopole, Dodecapole, Higher-order multipole},
	abstract = {We report on the effects of varying higher-order multipole components on the performance of a novel radiofrequency quadrupole ion-trap mass analyzer, named the planar Paul trap. The device consists of two parallel ceramic plates, the opposing surfaces of which are lithographically imprinted with 24 concentric metal rings. Using this device, the magnitude and sign of different multipole components, including octopole and dodecapole, can be independently adjusted through altering the voltages applied to each ring. This study presents a systematic investigation of the effects of the octopole and dodecapole field components on the mass resolution and signal intensity of the planar Paul trap. Also, the effect of dipole amplitude and scan speed under both forward and reverse scan modes have been investigated for various combinations of octopole and dodecapole. A trapping field in which the magnitudes of the octopole and dodecapole are, respectively, set to 0% and +8% of the magnitude of the quadrupole yields the highest mass resolution under the conditions studied. A small threshold voltage for dipole resonance ejection is observed for positive octopole, and to a lesser extent for positive dodecapole, but not for negative poles. When both octopole and dodecapole are negative, a reverse scan produces higher resolution, but this effect is not observed when only one of the components is negative.},
	author = {Zhang, Zhiping and Quist, Hannah and Peng, Ying and Hansen, Brett J. and Wang, Junting and Hawkins, Aaron R. and Austin, Daniel E.},
	journal = {Int. J. Mass Spectrom.},
	volume = {299},
	number = {2 -- 3},
	pages = {151 -- 157},
	publisher = {Elsevier},
	issn = {1387-3806},
	year = {2011},
	month = {Jan},
	date = {15},
	doi = {10.1016/j.ijms.2010.10.022},
	**url = {http://www.sciencedirect.com/science/article/pii/S1387380610004033}**
}

@article{Mih11,
	title = {Semiclassical dynamics for an ion confined within a nonlinear electromagnetic trap},
	author = {Mihalcea, Bogdan M.},
	abstract = {We investigate the spectral properties of the Hamiltonian function that describes an ion confined within a nonlinear trap. The Hamiltonian is then particularized for the case of dynamic traps and we introduce algebraic models with the aim to characterize the associated dynamics in such traps. The coherent states formalism for dynamic groups and the time-dependent variational principle are applied in order to study the semiclassical behaviour of the confined ion. We deal with the bosonic realization of the Lie algebra of the SU(1,1) group, which we particularize for the case of an ion confined in a combined (Paul and Penning) trap. We infer the equations of motion for the ion in a semiclassical approach. We suggest an algorithm by means of which we can associate a classical Hamiltonian with the quantum Hamiltonian that describes the ion. The classical Hamiltonian implicitly contains spectral information on the quantum system, which means we can dequantify the system.},
	journal = {Phys. Scr.},
	volume = {T143},
	pages = {014018},
	numpages = {6},
	year = {2011},
	month = {Feb},
	date = {21},
	publisher = {IOP Publishing},
	doi = {10.1088/0031-8949/2011/T143/014018},
	**url = {https://doi.org/10.1088/0031-8949/2011/T143/014018}**
}

@article{Wine11,
	author = {Wineland, D. J. and Leibfried, D.},
	title = {Quantum information processing and metrology with trapped ions},
	journal = {Laser Phys. Lett.},
	volume = {8},
	number = {3},
	pages = {175 -- 188},
	month = {Mar},
	year = {2011},
	publisher = {Wiley-VCH},
	doi = {10.1002/lapl.201010125} 
}

@article{Ishi11,
	author = {Ishizaki, Ryuji and Sata, Hiroki and Shoji, Tatsuo},
	title = {Chaos-Induced Diffusion in a Nonlinear Dissipative Mathieu Equation for a Charged Fine Particle in an AC Trap},
	abstract = {Charged fine particles confined in an AC trap exhibit either periodic motion or irregular motion, depending on the frequency and amplitude of the AC electric field. This motion was analyzed using an idealized electric field model with a nonlinear term in the radial direction ( $r$ ) and an angular (\theta-dependent) term. The potential U($r, \theta, z, t$) generates a rotational diffusion of chaotic orbits, and a transition from ballistic motion to diffusive motion was observed in the mean square displacement (MSD) of $\theta$. The distribution function f($\tau$) for the lifetime of angular unidirectional motion is exponential. This exponential distribution is produced by the chaotic switching between clockwise and anticlockwise rotations of orbits on the $x y$ -plane. The time-correlation function C($\tau$) of v$_θ$ also has an exponential decay form as a result of the lifetime distribution function f($\tau$). The scaling function of the MSD of $\theta(\tau)$ is derived using the correlation time $\tau_c$ of C ($\tau$).},
	journal = {J. Phys. Soc. Jpn.},
	volume = {80},
	number = {4},
	pages = {044001},
	numpages = {6},
	year = {2011},
	month = {Mar},
	date = {25},
	publisher = {The Physical Society of Japan},
	doi = {10.1143/JPSJ.80.044001}
	**url = {https://journals.jps.jp/doi/abs/10.1143/JPSJ.80.044001}**
}

@article{Lee11,
	title = {Pattern Formation with Trapped Ions},
	author = {Lee, Tony E. and Cross, M. C.},
	abstract = {Ion traps are a versatile tool to study nonequilibrium statistical physics, due to the tunability of dissipation and nonlinearity. We propose an experiment with a chain of ions, where dissipation is provided by laser heating and cooling, while nonlinearity is provided by trap anharmonicity and beam shaping. The dynamics are governed by an equation similar to the complex Ginzburg-Landau equation, except that the reactive nature of the coupling leads to qualitatively different behavior. The system has the unusual feature of being both oscillatory and excitable at the same time. The patterns are observable for realistic experimental parameters despite noise from spontaneous emission. Our scheme also allows controllable experiments with noise and quenched disorder.},
	journal = {Phys. Rev. Lett.},
	volume = {106},
	number = {14},
	pages = {143001},
	numpages = {4},
	year = {2011},
	month = {Apr},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevLett.106.143001},
	**url = {https://link.aps.org/doi/10.1103/PhysRevLett.106.143001}**
}

@inproceedings{Sinc11,
	author = {Sinclair, Alastair},
	title = {An {I}ntroduction to {T}rapped {I}ons, {S}calability and {Q}uantum {M}etrology},
	booktitle = {Quantum Information and Coherence},     
	series = {Scottish Graduate Series},
	volume = {67},
	editor = {Andersson, Erika and {\"O}hberg, Patrik},
	year = {2011}, 
	publisher = {Springer},
	address = {},
	pages = {211 -- 246},
	ISBN = {978-3-319-04062-2},
	e-ISBN = {978-3-319-04063-9},  
	doi = {10.1007/978-3-319-04063-9_9}
	**url = {}**
}

@article{Home11,
	title = {Normal modes of trapped ions in the presence of anharmonic trap potentials},
	author = {Home, J. P. and Hanneke, D. and Jost, J. D. and Leibfried, D. and Wineland, D. J.},
	abstract = {We theoretically and experimentally examine the effects of anharmonic terms in the trapping potential for linear chains of trapped ions. We concentrate on two different effects that become significant at different levels of anharmonicity. The first is a modification of the oscillation frequencies and amplitudes of the ions' normal modes of vibration for multi-ion crystals, resulting from each ion experiencing a different curvature in the potential. In the second effect, which occurs with increased anharmonicity or higher excitation amplitude, amplitude-dependent shifts of the normal-mode frequencies become important. We evaluate normal-mode frequency and amplitude shifts, and comment on the implications for quantum information processing and quantum state engineering. Since the ratio of the anharmonic to harmonic terms typically increases as the ion–electrode distance decreases, anharmonic effects will become more significant as ion trap sizes are reduced. To avoid unwanted problems, anharmonicities should therefore be taken into account at the design stage of trap development.},
	journal = {New J. Phys.},
	volume = {13},
	number = {7},
	pages = {073026},
	numpages = {25},
	year = {2011},
	month = {Jul},
	publisher = {IOP Publishing},
	doi = {10.1088/1367-2630/13/7/073026},
	**url = {https://doi.org/10.1088/1367-2630/13/7/073026}**
}

@book{Kul11,
	title = {Aerosol Measurement: Principles, Techniques and Applications},
	author = {}, 
	editor = {Kulkarni, Pramod and Baron, Paul A. and Willeke, Klaus},
	series = {},
	volume = {},
	edition = {3rd},
	publisher = {Wiley},
	address = {Hoboken, New Jersey},
	ISBN = {9780470387412}, 
	e-ISBN = {9781118001684},
	year = {2011},
	month = {Jun},
	date = {27},
	doi  = {10.1002/9781118001684},
	**url = {https://onlinelibrary.wiley.com/doi/book/10.1002/9781118001684}**
}

@article{Li12,
	title = {Space-Time Crystals of Trapped Ions},
	author = {Li, Tongcang and Gong, Zhe-Xuan and Yin, Zhang-Qi and Quan, H. T. and Yin, Xiaobo and Zhang, Peng and Duan, L.-M. and Zhang, Xiang},
	abstract = {Spontaneous symmetry breaking can lead to the formation of time crystals, as well as spatial crystals. Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions 	produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. We also study the effects of finite temperatures on the persistent rotation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.},
	journal = {Phys. Rev. Lett.},
	volume = {109},
	number = {16},
	pages = {163001},
	numpages = {5},
	year = {2012},
	month = {Oct},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevLett.109.163001},
	**url = {https://link.aps.org/doi/10.1103/PhysRevLett.109.163001}**
}

@article{Doro12c,
	title = {Application of a Modified Homotopy Perturbation Method for Calculation of Secular Axial Frequencies in a Nonlinear Ion Trap with Hexapole, Octopole and Decapole Superpositions},
	author = {Alireza Doroudi},
	abstract = {In this paper we have used a modified homotopy perturbation method used previously by A. Belendez and his coworkers, for calculation of axial secular frequencies of a nonlinear ion trap with hexapole, octopole and decapole superpositions. We transform the motion of the ion in a rapidly oscillating field to the motion in an effective potential and obtain a nonlinear differential equation in the form of a Duffing-like equation. With only octopole superposition the resulted nonlinear equations are symmetric; however, in the presence of hexapole and decapole superpositions, they are asymmetric. For asymmetric oscillators, it has been pointed out that the angular frequency for positive
	amplitudes is different from the angular frequency for negative amplitudes. Considering this problem, the modified homotopy perturbation method is used for solving the resulted nonlinear equations. As a result, the ion axial secular frequencies as a function of nonlinear field parameters are obtained. The calculated secular frequencies are compared with the results of modified Lindstedt-Poincare approximation and the exact results. There is an excellent agreement between the results of this paper and the exact results.},
	keywords = {Modified homotopy perturbation; Nonlinear ion trap; Secular frequency; Hexapole; Octopole; Decapole; Symmetric oscillator; Asymmetric oscillator},
	journal = {J. Bioanal. Biomed.},
	volume = {4},
	number = {5},
	pages = {85 -- 91},
	numpages = {0},
	year = {2012},
	month = {Oct},
	publisher = {Hilaris},
	issn = {1948-593X},
	doi = {10.4172/1948-593x.1000068},
	**url = {https://doi.org/10.4172/1948-593x.1000068}**
}

@article{Land12a,
	author = {Landa, H. and Drewsen, M. and Reznik, B. and Retzker, A.},
	title = {Modes of oscillation in radiofrequency Paul traps},
	abstract = {We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general three-dimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We analytically derive the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a recently proposed method to find the modes that diagonalize the linearized time-dependent dynamical problem. This allows one to obtain explicitly the (‘Floquet–Lyapunov’) transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small ‘peculiar’ crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.},
	journal = {New J. Phys.},
	volume = {14},
	number = {9},
	pages = {093023},
	numpages = {21},
	year = {2012},
	month = {Sep},
	publisher = {IOP Publishing},
	doi = {10.1088/1367-2630/14/9/093023},
	**url = {https://iopscience.iop.org/article/10.1088/1367-2630/14/9/093023/pdf}**
}

@article{Land12b,
	author = {H. Landa and M. Drewsen and B. Reznik and A. Retzker},
	title = {Classical and quantum modes of coupled Mathieu equations},
	abstract = {We expand the solutions of linearly coupled Mathieu equations in terms of infinite-continued matrix inversions, and use it to find the modes which diagonalize the dynamical problem. This allows obtaining explicitly the (Floquet–Lyapunov) transformation to coordinates in which the motion is that of decoupled linear oscillators. We use this transformation to solve the Heisenberg equations of the corresponding quantum-mechanical problem, and find the quantum wavefunctions for stable oscillations, expressed in configuration space. The obtained transformation and quantum solutions can be applied to more general linear systems with periodic coefficients (coupled Hill equations, periodically driven parametric oscillators), and to nonlinear systems as a starting point for convenient perturbative treatment of the nonlinearity.},
	journal = {J. Phys. A: Math. Theor.},
	volume = {45},
	number = {45},
	pages = {455305},
	month = {October},
	year = {2012},
	publisher = {IOP Publishing},
	doi = {10.1088/1751-8113/45/45/455305},
	**url = {https://doi.org/10.1088/1751-8113/45/45/455305}**
}

@article{Sha12,
	author = {Shaikh, Fayaz A. and Ozakin, Arkadas},
	title = {Stability analysis of ion motion in asymmetric planar ion traps},
	abstract = {Motivated by recent developments in ion trap design and fabrication, we investigate the stability of the motion of an ion in asymmetrical, planar versions of the linear Paul trap. The equations of motion of an ion in such a trap are generally coupled due to a nonzero relative angle $\theta$ between the principal axes of RF and DC fields, invalidating the assumptions behind the standard stability analysis for symmetric Paul traps. Using numerical methods, we obtain stability diagrams for the coupled system for various values of $\theta$, generalizing the standard $q-a$ stability diagrams. We then use multi-scale perturbation theory to obtain approximate formulas for the boundaries of the primary stability region and check our formulas against results from numerical analysis. Our results show that while the primary stability region is quite robust to changes in $\theta$, a secondary stability region is highly variable, joining the primary stability region at the special case of $\theta = 45^{\circ}$, which results in a significantly enlarged stability region for this particular angle. We conclude that while the standard stability diagrams for classical, symmetric Paul traps are not entirely accurate for asymmetric surface traps (or for other types of traps with a relative angle between the RF and DC axes), they are “safe” in the sense that operating conditions deemed stable according to standard stability plots are in fact stable for asymmetric traps, as well. By ignoring the coupling in the equations, one only underestimates the size of the primary stability region.},
	journal = {J. Appl. Phys.},
	volume = {112},
	number = {7},
	pages = {074904},
	year = {2012},
	month = {Oct},
	publisher = {{AIP P}ublishing},
	doi = {10.1063/1.4752404},
	**URL = {https://doi.org/10.1063/1.4752404}**
}

@article{Wine13,
	title = {Nobel Lecture: Superposition, entanglement, and raising Schr{\"o}dinger’s cat},
	author = {Wineland, David J.},
	journal = {Rev. Mod. Phys.},
	volume = {85},
	number = {3},
	pages = {1103 -- 1114},
	publisher = {American Physical Society},
	year = {2013},
	month = {Jul},
	issn = {0034-6861},
	doi = {10.1103/RevModPhys.85.1103},
	**url = {https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.85.1103}** 
}

@article{Wang13,
	author = {Wang, Yuzhuo and Huang, Zejian and Jiang, You and Xiong, Xingchuang and Deng, Yulin and Fang, Xiang and Xu, Wei},
	title = {The coupling effects of hexapole and octopole fields in quadrupole ion traps: a theoretical study},
	abstract = {A theoretical method, the harmonic balance method, was introduced to study the coupling effects of hexapole and octopole fields on ion motion in a quadrupole ion trap. Ion motion characteristics, such as ion motion center displacement, ion secular frequency shift, nonlinear resonance curve and buffer gas damping effects, have been studied with the presence of both hexapole and octopole fields. It is found that hexapole fields have bigger impacts on ion motion center displacement, while octopole fields dominate ion secular frequency shift. Furthermore, the nonlinear features originated from hexapole and octopole fields could enhance or cancel each other, which provide us more space in a practical ion trap design process. As an example, an ion trap with improved performance was designed using a specific combination of hexapole and octopole fields. In this ion trap, a hexapole field was used to achieve efficient ion directional ejection, while an octopole field was added to correct the chemical mass shift and resolution degradation introduced by the hexapole field.},
	keywords = {quadrupole ion trap, hexapole, octopole, coupling effects, harmonic balance method},
	journal = {J. Mass Spectrom.},
	volume = {48},
	number = {8},
	pages = {937 -- 944},
	publisher = {Wiley},
	year = {2013},
	month = {Aug},
	date = {},
	doi = {10.1002/jms.3239},
	**url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/jms.3239}**
}

@inbook{Arfk13,
    booktitle = {Mathematical Methods for Physicists},
	author = {Arfken, G. B. and Weber, H. J. and Harris, F. E.}, 
	editor = {},
	series = {},
	volume = {},	
	chapter = {32},
	pages = {1 -- 29},
	edition = {7th},
	publisher = {Academic Press},
	address = {Waltham},
	isbn = {},
	year = {2013},
	month = {},
	doi = {},
    url = {https://booksite.elsevier.com/9780123846549/Chap\_Mathieu.pdf}	  
}

@article{Poli13,
	title = {Optical atomic clocks},
	author = {Poli, N. and Oates, C. W. and Gill, P. and Tino, G. M.},
	journal = {Nuovo Cimento},
	volume = {36},
	number = {12},
	pages = {555 -- 624},
	numpages = {70},
	year = {2013},
	month = {Dec},
	date = {},
	publisher = {Soc. It. Fisica}, 
	doi = {10.1393/ncr/i2013-10095-x}
}

@incollection{Marg15,
	booktitle = {Ion Traps for Tomorrow’s Applications},
	title = {Frequency standards with trapped ions},
	author = {Margolis, Helen S.}, 
	editor = {},
	series = {Proc. Intl. School Phys. “Enrico Fermi”},
	volume = {189 “Ion Traps for Tomorrow’s Applications”},	
	edition = {},
	pages = {139 -- 158},
	publisher = {IOS \& Societa Italiana di Fisica},
	address = {Amsterdam \& Bologna},
	isbn = {},
	e-isbn = {},
	year = {2015},
	month = {},
	doi = {10.3254/978-1-61499-526-5-139},
	**url = {https://doi.org/10.3254/978-1-61499-526-5-139}**	  
}

@incollection{Bibby14,
	booktitle = {Accurate Computation of Mathieu Functions},
	author = {Bibby, Malcom M. and Peterson, Andrew F.}, 
	editor = {Balanis, Constantine A.},
	series = {Synthesis Lectures on Computational Electromagnetics},
	volume = {Lecture 32},	
	edition = {},
	publisher = {Morgan\&Claypool},
	address = {},
	isbn = {9781627050852},
	e-isbn = {9781627050869},
	year = {2014},
	month = {},
	doi = {10.2200/S00526ED1V01Y201307CEM032},
	**url = {https://doi.org/10.2200/S00526ED1V01Y201307CEM032}**	  
}

@book{Kno14,
	title = {Physics with Trapped Charged Particles: Lectures from the Les Houches Winter School},
	author = {}, 
	editor = {Knoop, Martina and Madsen, Niels and Thompson, Richard C.},
	series = {},
	volume = {},
	edition = {},
	publisher = {Imperial College Press \& World Scientific},   
	address = {London},
	isbn = {978-1-78326-404-9},
	e-isbn = {978-1-78326-407-0},  
	year =  {2014},
	month = {Mar},
	doi = {10.1142/p928},
	**url = {https://www.worldscientific.com/worldscibooks/10.1142/p928}**
}

@article{Pan14,
	author = {Pan, Yong-Le and Wang,Chuji and Hill,Steven C. and Coleman,Mark and Beresnev,Leonid A. and Santarpia,Joshua L.},
	title = {Trapping of individual airborne absorbing particles using a counterflow nozzle and photophoretic trap for continuous sampling and analysis},
	abstract = {We describe an integrated opto-aerodynamic system and demonstrate that it enables us to trap absorbing airborne micron-size particles from air, hold them and then release them, and to repeat this sequence many times as would be appropriate for continuous sampling of particles from air. The key parts of the system are a conical photophoretic optical trap and a counter-flow coaxial-double-nozzle that concentrates and then slows particles for trapping. This technology should be useful for on-line applications that require monitoring (by single particle analyses) of a series of successively arriving particles (e.g., from the atmosphere or pharmaceutical or other production facilities) where the total sampling time may last from minutes to days, but where each particle must be held for a short time for measurements (e.g., Raman scattering).},
	journal = {Appl. Phys. Lett.},
	volume = {104},
	number = {11},
	pages = {113507},
	numpages = {4},
	year = {2014},
	month = {Mar},
	date = {19},
	doi = {10.1063/1.4869105},
	**URL = {https://doi.org/10.1063/1.4869105}**
}

@article{Gras14,
	author = {Gra\ss{}, Tobias and Lewenstein, Maciej},
	title = {Trapped-ion quantum simulation of tunable-range Heisenberg chains},
	abstract = {Quantum-optical techniques allow for generating controllable spin-spin interactions between ions, making trapped ions an ideal quantum simulator of Heisenberg chains. A single parameter, the detuning of the Raman coupling, allows to switch between ferromagnetic and antiferromagnetic chains, and to modify the range of the interactions. On the antiferromagnetic side, the system can be tuned from an extreme long-range limit, in which any pair of ions interacts with almost equal strength, to interactions with a $1/r^3$ decay. By exact diagonalization, we study how a system of up to 20 ions behaves upon tuning the interactions. We find that it undergoes a transition from a dimerized state with extremely short-ranged correlations towards a state with quasi long-range order, that is, algebraically decaying correlations. The dynamical evolution of the system after a local quench is shown to strongly vary in the two regimes: While in the dimerized limit, the excitation remains localized for long times, propagating spinons characterize the dynamics of the quasi-long-range ordered system. Taking a look onto the ferromagnetic side of the system, we demonstrate the feasibility of witnessing non-locality of quantum correlations by measuring two-particle correlators.},
	journal = {EPJ Quant. Technol.},
	volume = {1},
	number = {1},
	pages = {8},
	numpages = {20},
	year = {2014},
	month = {Jun},
	date = {03},
	publisher = {Springer},
	issn = {2196-0763},
	doi = {10.1140/epjqt8},
	**URL = {https://doi.org/10.1140/epjqt8}**
}

@article{Lama14,
	title = {A high-precision segmented Paul trap with minimized micromotion for an optical multiple-ion clock},
	author = {Lamata, Lucas and Mezzacapo, Antonio and Casanova, Jorge and Solano, Enrique},
	abstract = {We analyze the efficiency of quantum simulations of fermionic and bosonic models in trapped ions. In particular, we study the optimal time of entangling gates and the required number of total elementary gates. Furthermore, we exemplify these estimations in the light of quantum simulations of quantum field theories, condensed-matter physics, and quantum chemistry. Finally, we show that trapped-ion technologies are a suitable platform for implementing quantum simulations involving interacting fermionic and bosonic modes, paving the way for overcoming classical computers in the near future.},
	journal = {EPJ Quantum Technol.},
	volume = {1},
	number = {1},
	pages = {9},
	numpages = {13},
	year = {2014},
	month = {Jun},
	date = {18},
	publisher = {Springer},
	issn = {2196-0763},
	doi = {10.1140/epjqt9},
	**url = {https://doi.org/10.1140/epjqt9}**
}

@article{Wang14,
	title = {Characterization of geometry deviation effects on ion trap mass analysis: A comparison study},
	abstract = {A high precision electric field simulation method is employed to investigate micron-size geometry deviation effects on the electric field distributions as well as on mass resolutions in linear ion traps. Translational and angular displacements within three linear ion traps with different electrode geometries, hyperbolic, circular and rectangular electrodes, were studied. It is found that the linear ion trap with rectangular electrodes is more sensitive to translational displacements, while the linear ion trap with circular electrodes is more sensitive to angular displacements in general. As a guideline, the results could be used in the design and fabrication process of a linear ion trap.},
	keywords = {Linear ion trap, Geometry deviation, High-order fields, Mass resolution},
	author = {Wang, Yuzhuo and Zhang, Xiaohua and Feng, Yan and Shao, Ruiting and Xiong, Xingchuang and Fang, Xiang and Deng, Yulin and Xu, Wei},
	journal = {Int. J. Mass Spectrom.},
	volume = {370},
	pages = {125 -- 131},
	publisher = {Elsevier},
	year = {2014},
	month = {Sept},
	date = {15},
	issn = {1387-3806},
	doi = {10.1016/j.ijms.2014.07.014},
	**url ={http://www.sciencedirect.com/science/article/pii/S1387380614002747}**
}

@article{McAne14,
	title = {Intrinsic anharmonic effects on the phonon frequencies and effective spin-spin interactions in a quantum simulator made from trapped ions in a linear Paul trap},
	abstract = {The Coulomb repulsion between ions in a linear Paul trap gives rise to anharmonic terms in the potential energy when expanded about the equilibrium positions. We examine the effect of these anharmonic terms on the accuracy of a quantum simulator made from trapped ions. To be concrete, we consider a linear chain of Yb$^{171+}$ ions stabilized close to the zigzag transition. We find that for typical experimental temperatures, frequencies change by no more than a factor of 0.01\% due to the anharmonic couplings. Furthermore, shifts in the effective spin-spin interactions (driven by a spin-dependent optical dipole force) are also, in general, less than 0.01% for detunings to the blue of the transverse center-of-mass frequency. However, detuning the spin interactions near other frequencies can lead to non-negligible anharmonic contributions to the effective spin-spin interactions. We also examine an odd behavior exhibited by the harmonic spin-spin interactions for a range of intermediate detunings, where nearest-neighbor spins with a larger spatial separation on the ion chain interact more strongly than nearest neighbors with a smaller spatial separation.},
	author = {McAneny, M. and Freericks, J. K.},
	journal = {Phys. Rev. A},
	volume = {90},
	issue = {5},
	pages = {053405},
	numpages = {9},
	year = {2014},
	month = {Nov},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.90.053405},
	url = {https://link.aps.org/doi/10.1103/PhysRevA.90.053405}
}

@book{Kno15,
	title = {Ion Traps for Tomorrow Applications},
	author = {}, 
	editor = {Knoop, Martina and Marzoli, Irene and Morigi, Giovanna},
	series = {Proc. Int. School of Physics Enrico Fermi},
	volume = {189},
	edition = {},
	publisher = {Societ\`a Italiana di Fisica and IOS Press},
	address = {Bologna},
	isbn = {978-1-61499-525-8},
	e-isbn = {978-1-61499-526-5},  
	year = {2015},
	month = {Jul},
	doi = {}
}

@book{Van15,
	author = {Vanier, Jacques and Tomescu, Cipriana}, 
	editor = {},
	title = {The Quantum Physics of Atomic Frequency Standards: Recent Developments},
	volume = {},
	series = {},
	edition = {},
	publisher = {CRC Press},
	address = {Boca Raton},
	year = {2015},
	month = {Aug},
	date = {5},
	isbn = {},
	e-isbn = {9780429101403},
	doi = {10.1201/b18738},
	**url = {https://doi.org/10.1201/b18738}**  
}

@inproceedings{Drew15,
	author = {Drewsen, Michael },
	title = {Ion Coulomb crystals and their applications},
	abstract = {The following text will give a brief introduction to the physics of the spatially ordered structures, so-called Coulomb crystals, that appear when confined ions are cooled to sufficiently low temperatures. It will as well briefly comment on the very diverse scientific applications of such crystals, which have emerged in the past two decades. While this document lacks figures, it includes a substantial number of references in which more detailed information can be found. It is our hope that the text will stimulate the readers to dig deeper into one or more of the discussed subjects, and inspire them to think about new potential applications.},
	keywords = {},
	volume = {189},
	booktitle = {Ion Traps for Tomorrow’s Applications},
	editor = {Knoop, Martina and Marzoli, Irene and Morigi, Giovanna},
	organization = { International School of Physics “Enrico Fermi”},
	publisher = {Inst. of Physics, Amsterdam, and Societa Italiana di Fisica, Bologna},
	pages = {81 -- 101},
	year = {2015},
	month = {},
	doi = {10.3254/10.3254/978-1-61499-526-5-81},
	**URL = {https://doi.org/10.3254/10.3254/978-1-61499-526-5-81}**
}

@inproceedings{Fish15,
	author = {Drewsen, Michael },
	title = {The linear-zigzag structural transition in cold ion chains},
	abstract = {A chain of singly charged particles confined by a harmonic potential exhibits a structural transition to a zigzag configuration when the radial potential frequency is at a critical value, which depends on the particle number. This structural change is a phase transition of second order, whose order parameter is the ions’ transverse displacement from the chain axis. The transition is driven by transverse, short-wavelength vibrational modes. At ultra-low temperatures the linear-zigzag instability is a quantum phase transition of the same universality class of Ising models.},
	keywords = {},
	volume = {189},
	booktitle = {Ion Traps for Tomorrow’s Applications},
	editor = {Knoop, Martina and Marzoli, Irene and Morigi, Giovanna},
	organization = { International School of Physics “Enrico Fermi”},
	publisher = {Inst. of Physics, Amsterdam, and Societa Italiana di Fisica, Bologna},
	pages = {103 -- 114},
	year = {2015},
	month = {},
	doi = {10.3254/10.3254/978-1-61499-526-5-103},
	**URL = {https://doi.org/10.3254/10.3254/10.3254/978-1-61499-526-5-103}**
}

@inproceedings{Burg15,
	author = {Burgermeister, T. and Mehlst\"{a}ubler, Tanja E.},
	title = {Creation and dynamics of topological defects in ion Coulomb crystals},
	abstract = {We discuss our recent studies of topological defects (kinks) in ion Coulomb crystals. Experimentally two different types of kinks are created by non-adiabatically driving the second-order phase transition from a linear to a zigzag phase. The kink creation rates are investigated in relation to the inhomogeneous Kibble-Zurek mechanism. Stability and dynamic properties of both types of kinks are explained by the Peierls-Nabarro potentials. In addition, we report on the
	influence of mass defects on kinks. We show how the application of electric fields can change the influence of mass defects in a controlled way and present a first evidence for a deterministic creation and manipulation of kinks.},
	keywords = {},
	volume = {189},
	booktitle = {Ion Traps for Tomorrow’s Applications},
	editor = {Knoop, Martina and Marzoli, Irene and Morigi, Giovanna},
	organization = { International School of Physics “Enrico Fermi”},
	publisher = {Inst. of Physics, Amsterdam, and Societa Italiana di Fisica, Bologna},
	pages = {115 -- 125},
	year = {2015},
	month = {},
	doi = {10.3254/978-1-61499-526-5-115},
	**URL = {https://doi.org/10.3254/978-1-61499-526-5-115}**
}

@inproceedings{Lem15,
	author = {Lemmer, A. and Plenio, M. B. and Bermudez, A.},
	title = {Noise studies of driven geometric phase gates with trapped ions},
	abstract = {We present a study of the performance of the trapped-ion driven geometric phase gates (New J. Phys., 15 (2013) 083001) when realized using a stimulated Raman transition. We show that the gate can achieve errors below the fault-tolerance threshold in the presence of laser intensity fluctuations. We also find that, in order to reduce the errors due to photon scattering below the fault-tolerance threshold, very intense laser beams are required to allow for large detunings in the
	Raman configuration without compromising the gate speed.},
	keywords = {},
	volume = {189},
	booktitle = {Ion Traps for Tomorrow’s Applications},
	editor = {Knoop, Martina and Marzoli, Irene and Morigi, Giovanna},
	organization = { International School of Physics “Enrico Fermi”},
	publisher = {Inst. of Physics, Amsterdam, and Societa Italiana di Fisica, Bologna},
	pages = {229 -- 244},
	year = {2015},
	month = {},
	doi = {10.3254/978-1-61499-526-5-229},
	**URL = {https://doi.org/10.3254/978-1-61499-526-5-229}**
}

@article{Joshi15,
	title = {Influence of a geometrical perturbation on the ion dynamics in a 3D Paul trap},
	author = {Joshi, Manoj Kumar and Satyajit, K. T. and Rao, Pushpa M.},
	keywords = {Quadrupole ion trap, Motional frequencies, Multipolar expansion, Trapped ion trajectories, Non-ideal trap, Geometrical perturbations},
	abstract = {In ion traps, the purity of the Quadrupole potential is essential. Any perturbations in potential caused by geometrical imperfections or the presence of other ions alter the dynamics of trapped ions. In this paper the effect of a particular perturbation on dynamics of trapped ions in a 3-D quadrupole trap is analyzed and we have compared the experimental findings with simulations. To see the effect of geometrical perturbation, the position of a filament reaching into the trap (which acts as the perturbing element) is altered. The equi-frequency line on the a−q Mathieu plot was scanned at different filament insertion heights. We studied the effect of filament current and the duration of loading. The distorted potential within this trap has been theoretically calculated and simulated using SIMION. An analytical expression for potential inside the trap is fitted to the potential values obtained from the simulation and the numerical solutions of the equation of motion of trapped ions is obtained. The motional frequencies are then calculated from the Fourier transformation of the simulated ion trajectories at any given trapping potential and compared with our experimental findings. The shift in the secular frequency with respect to the level of insertion of the filament within the trap is evaluated.},
	journal = {Nucl. Instrum. Methods Phys. Res. A},
	volume = {800},
	pages = {111 -- 118},
	year = {2015},
	month = {Nov},
	date = {11},
	issn = {0168-9002},
	doi = {10.1016/j.nima.2015.07.046},
	**url = {https://www.sciencedirect.com/science/article/pii/S016890021500875X}**
}

@article{Xiong14,
	author = {Xiong, Caiqiao and Zhou, Xiaoyu and Zhang, Ning and Zhan, Lingpeng and Chen, Yongtai and Chen, Suming and Nie, Zongxiu},
	title = {Nonlinear Effects in Paul Traps Operated in the Second Stability Region: Analytical Analysis and Numerical Verification},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {25},
	number = {11},
	pages = {1882 -- 1889},
	keywords = {Paul trap, Second stability region, Nonlinear effects, Poincare-Lighthill-Kuo, Runge–Kutta},
	abstract = {Paul trap working in the second stability region has long been recognized as a possible approach for achieving high-resolution mass spectrometry (MS), which however is still far away from the experimental implementations because of the narrow working area and inefficient ion trapping. Full understanding of the ion motional behavior is helpful for solving the problem. In this article, the ion motion in a superimposed octopole field, which was characterized by the nonlinear Mathieu equation, was solved analytically using Poincare-Lighthill-Kuo (PLK) method. This method equivalently described the nonlinear disturbance by an effective quadrupole field with perturbed Mathieu parameters, $a^u$ and $q^u$, which would bring huge convenience in the studies of nonlinear ion dynamics and was, therefore, used for rapid 	evaluation of the nonlinear effects of ion motion. Fourth-order Runge-Kutta method (4th R-K) indicated the error of PLK for characterizing the frequency shift of ion motion was within 15%.},
	year = {2014},
	month = {Nov},
	date = {},
	publisher = {Springer},
	e-issn = {1879-1123},
	doi = {10.1007/s13361-014-0979-8},
	**url = {https://doi.org/10.1007/s13361-014-0979-8}**
}

@article{Xiong15,
	author = {Xiong, Caiqiao and Zhou, Xiaoyu and Zhang, Ning and Zhan, Lingpeng and Chen, Yongtai and Chen, Suming and Nie, Zongxiu},
	title = {A Theoretical Method for Characterizing Nonlinear Effects in Paul Traps with Added Octopole Field},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {26},
	number = {8},
	pages = {1338 -- 1348},
	keywords = {},
	abstract = {In comparison with numerical methods, theoretical characterizations of ion motion in the nonlinear Paul traps always suffer from low accuracy and little applicability. To overcome the difficulties, the theoretical harmonic balance (HB) method was developed, and was validated by the numerical fourth-order Runge-Kutta (4th RK) method. Using the HB method, analytical ion trajectory and ion motion frequency in the superimposed octopole field, $\varepsilon$, were obtained by solving the nonlinear Mathieu equation (NME). The obtained accuracy of the HB method was comparable with that of the 4th RK method at the Mathieu parameter, $q = 0.6$, and the applicable $q$ values could be extended to the entire first stability region with satisfactory accuracy. Two sorts of nonlinear effects of ion motion were studied, including ion frequency shift, $\Delta \beta$, and ion amplitude variation, $\Delta\left(C_{2n}/C_0\right) \left(n \neq 0\right). New phenomena regarding $\Delta \beta$ were observed, although extensive studies have been performed based on the pseudo-potential well (PW) model. For instance, the $|Δ\Delta \beta|$ at $\varepsilon = 0.1$ and $\varepsilon = – 0.1$ were found to be different, but they were the same in the PW model. This is the first time the nonlinear effects regarding $\Delta(C_{2n}/C_0) (n \neq\ 0)$ are studied, and the associated study has been a challenge for both theoretical and numerical methods. The nonlinear effects of $\Delta (C_{2n}/C_0) (n \neq 0)$ and $\Delta \beta$ were found to share some similarities at $q < 0.6$: both of them were proportional to $\varepsilon$, and the square of the initial ion displacement, $z(0)^2$.},
	year = {2015},
	month = {Apr},
	date = {30},
	publisher = {Springer},
	e-issn = {1879-1123},
	doi = {10.1007/s13361-015-1145-7},
	**url = {https://doi.org/10.1007/s13361-015-1145-7}**
}

@article{Lud15,
	title = {Optical atomic clocks},
	author = {Ludlow, Andrew D. and Boyd, Martin M. and Ye, Jun and Peik, E. and Schmidt, P. O.},
	journal = {Rev. Mod. Phys.},
	volume = {87},
	number = {2},
	pages = {637 -- 701},
	numpages = {65},
	year = {2015},
	month = {Jun},
	publisher = {American Physical Society},
	doi = {10.1103/RevModPhys.87.637},
	**url = {https://link.aps.org/doi/10.1103/RevModPhys.87.637}**
}

@article{Schu16,
	title = {Quantum Algorithmic Readout in Multi-Ion Clocks},
	author = {Schulte, M. and L\"orch, N. and Leroux, I. D. and Schmidt, P. O. and Hammerer, K.},
	abstract = {Optical clocks based on ensembles of trapped ions promise record frequency accuracy with good short-term stability. Most suitable ion species lack closed transitions, so the clock signal must be read out indirectly by transferring the quantum state of the clock ions to cotrapped logic ions of a different species. Existing methods of quantum logic readout require a linear overhead in either time or the number of logic ions. Here we describe a quantum algorithmic readout whose overhead scales logarithmically with the number of clock ions in both of these respects. The scheme allows a quantum nondemolition readout of the number of excited clock ions using a single multispecies gate operation which can also be used in other areas of ion trap technology such as quantum information processing, quantum simulations, metrology, and precision spectroscopy.},
	journal = {Phys. Rev. Lett.},
	volume = {116},
	issue = {1},
	pages = {013002},
	numpages = {5},
	year = {2016},
	month = {Jan},
	date = {8},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevLett.116.013002},
  **url = {https://link.aps.org/doi/10.1103/PhysRevLett.116.013002}**
}

@article{Kiri16,
	title = {Rotating saddle trap as Foucault’s pendulum},
	author = {Kirillov, Oleg N. and Levi, Mark},
	abstract = {One of the many surprising results found in the mechanics of rotating systems is the stabilization of a particle in a rapidly rotating planar saddle potential. Besides the counterintuitive stabilization, an unexpected precessional motion is observed. In this note, we show that this precession is due to a Coriolis-like force caused by the rotation of the potential. To our knowledge, this is the first example where such a force arises in an inertial reference frame. We also propose a simple mechanical demonstration of this effect.},
	journal = {Am. J. Phys.},
	volume = {84},
	number = {1},
	pages = {26 -- 31},
	numpages = {0},
	year = {2016},
	month = {Jan},
	date = {01},
	publisher = {American Association Phys. Teachers},
	doi = {10.1119/1.4933206},
	**url = {https://aapt.scitation.org/doi/10.1119/1.4933206}**
}

@article{Schu16,
	title = {Quantum Algorithmic Readout in Multi-Ion Clocks},
	author = {Schulte, M. and L\"orch, N. and Leroux, I. D. and Schmidt, P. O. and Hammerer, K.},
	abstract = {Optical clocks based on ensembles of trapped ions promise record frequency accuracy with good short-term stability. Most suitable ion species lack closed transitions, so the clock signal must be read out indirectly by transferring the quantum state of the clock ions to cotrapped logic ions of a different species. Existing methods of quantum logic readout require a linear overhead in either time or the number of logic ions. Here we describe a quantum algorithmic readout whose overhead scales logarithmically with the number of clock ions in both of these respects. The scheme allows a quantum nondemolition readout of the number of excited clock ions using a single multispecies gate operation which can also be used in other areas of ion trap technology such as quantum information processing, quantum simulations, metrology, and precision spectroscopy.},
	journal = {Phys. Rev. Lett.},
	volume = {116},
	number = {1},
	pages = {013002},
	numpages = {5},
	year = {2016},
	month = {Jan},
	date = {8},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevLett.116.013002},
	**url = {https://link.aps.org/doi/10.1103/PhysRevLett.116.013002}**
}

@book{Kno16,
	title = {Trapped Charged Particles: A Graduate Textbook with Problems and Solutions},
	author = {}, 
	editor = {Knoop, Martina and Madsen, Niels and Thompson, Richard C.},
	series = {Advanced Textbooks in Physics},
	volume = {},
	edition = {},
	publisher = {World Scientific Europe},   
	address = {London},
	isbn = {978-1-78634-011-5},
	e-isbn = {978-1-78634-014-6},  
	year =  {2016},
	month =  {Jun},
	doi = {10.1142/q0004},
	**url = {https://doi.org/10.1142/q0004}**
}

@book{Orsz16,
	title = {Quantum Optics: Including Noise Reduction, Trapped Ions, Quantum Trajectories, and Decoherence},
	author = {Orszag, Miguel}, 
	editor = {},
	publisher = {Springer Intl. Publishing},
	edition = {3rd},
	address = {Cham}, 
	year = {2016},
	month = {Apr},
	date = {18},
	volume = {},
	series = {},
	ISBN = {978-3-319-29035-5},
	e-ISBN = {978-3-319-29037-9},
	doi = {10.1007/978-3-319-29037-9},
	**url = {https://doi.org/10.1007/978-3-319-29037-9}**
}

@article{Trypo16,
	title = {Cotrapping different species in ion traps using multiple radio frequencies},
	author = {Trypogeorgos, Dimitris and Foot, Christopher J.},
	abstract = {We consider the stability of systems subjected to periodic parametric driving in the context of ions confined by oscillating electric fields. The behavior of these systems can be understood in terms of a pseudopotential approximation and resonances arising from parametric excitation. We investigate the key properties of a way of operating a linear Paul trap with two radio frequencies that simultaneously confines two species with {\emph extremely different} charge-to-mass ratios. The theoretical calculations have been verified by molecular dynamics simulations and normal modes analysis.},
	journal = {Phys. Rev. A},
	volume = {94},
	number = {2},
	pages = {023609},
	numpages = {9},
	year = {2016},
	month = {Aug},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.94.023609},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.94.023609}**
}

@article{Xio16,
	title = {Nonlinear Ion Harmonics in the Paul Trap with Added Octopole Field: Theoretical Characterization and New Insight into Nonlinear Resonance Effect},
	author = {Xiong, Caiqiao and Zhou, Xiaoyu and Zhang, Ning and Zhan, Lingpeng and Chen, Yongtai and Nie, Zongxiu},
	abstract = {The nonlinear harmonics within the ion motion are the fingerprint of the nonlinear fields. They are exclusively introduced by these nonlinear fields and are responsible to some specific nonlinear effects such as nonlinear resonance effect. In this article, the ion motion in the quadrupole field with a weak superimposed octopole component, described by the nonlinear Mathieu equation (NME), was studied by using the analytical harmonic balance (HB) method. Good accuracy of the HB method, which was comparable with that of the numerical fourth-order Runge-Kutta (4th RK), was achieved in the entire first stability region, except for the points at the stability boundary (i.e., $\beta = 1$) and at the nonlinear resonance condition (i.e., $\beta = 0.5$). Using the HB method, the nonlinear $3\beta$ harmonic series introduced by the octopole component and the resultant nonlinear resonance effect were characterized. At nonlinear resonance, obvious resonant peaks were observed in the nonlinear $3\beta$ series of ion motion, but were not found in the natural harmonics. In addition, both resonant excitation and absorption peaks could be observed, simultaneously. These are two unique features of the nonlinear resonance, distinguishing it from the normal resonance. Finally, an approximation equation was given to describe the corresponding working parameter, $q_{nr}$, at nonlinear resonance. This equation can help avoid the sensitivity degradation due to the operation of ion traps at the nonlinear resonance condition.},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {27},
	number = {2},
	pages = {344 -- 351},
	numpages = {0},
	year = {2016},
	month = {Oct},
	date = {23},
	publisher = {American Society for Mass Spectrometry},
	doi = {10.1007/s13361-015-1291-y},
	**url = {https://doi.org/10.1007/s13361-015-1291-y}**
}

@article{Zha16b,
	title = {Lie transformation method on quantum state evolution of a general time-dependent driven and damped parametric oscillator},
	author = {Zhang, Lin and Zhang, Weiping},
	keywords = {Time-dependent harmonic oscillator, Lie transformation, Quantum control, Algebraic method, Invariant operator},
	abstract = {A variety of dynamics in nature and society can be approximately treated as a driven and damped parametric oscillator. An intensive investigation of this time-dependent model from an algebraic point of view provides a consistent method to resolve the classical dynamics and the quantum evolution in order to understand the time-dependent phenomena that occur not only in the macroscopic classical scale for the synchronized behaviors but also in the microscopic quantum scale for a coherent state evolution. By using a Floquet U-transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamic behaviors of a driven and damped parametric oscillator to obtain the optimal solutions by means of invariant parameters of Ks to combine with Lewis–Riesenfeld invariant method. This approach can discriminate the external dynamics from the internal evolution of a wave packet by producing independent parametric equations that dramatically facilitate the parametric control on the quantum state evolution in a dissipative system. In order to show the advantages of this method, several time-dependent models proposed in the quantum control field are analyzed in detail.},
	journal = {Ann. Phys.},
	volume = {373},
	pages = {424 -- 455},
	year = {2016},
	month = {Oct},
	issn = {0003-4916},
	doi = {10.1016/j.aop.2016.07.032},
	**url = {https://www.sciencedirect.com/science/article/pii/S0003491616301336}**
}


@article{Berm17,
	title = {Long-range Heisenberg models in quasiperiodically driven crystals of trapped ions},
	author = {Bermudez, A. and Tagliacozzo, L. and Sierra, G. and Richerme, P.},
	abstract = {We introduce a theoretical scheme for the analog quantum simulation of long-range XYZ models using current trapped-ion technology. In order to achieve fully tunable Heisenberg-type interactions, our proposal requires a state-dependent dipole force along a single vibrational axis, together with a combination of standard resonant and detuned carrier drivings. We discuss how this quantum simulator could explore the effect of long-range interactions on the phase diagram by combining an adiabatic protocol with the quasiperiodic drivings, and test the validity of our scheme numerically. At the isotropic Heisenberg point, we show that the long-range Hamiltonian can be mapped onto a nonlinear sigma model with a topological term that is responsible for its low-energy properties, and we benchmark our predictions with matrix-product-state numerical simulations.},
	journal = {Phys. Rev. B},
	volume = {95},
	number = {2},
	pages = {024431},
	numpages = {14},
	year = {2017},
	month = {Jan},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevB.95.024431},
  **url = {https://link.aps.org/doi/10.1103/PhysRevB.95.024431}**
}

@article{Kiri17,
	author = {Oleg Kirillov and Mark Levi},
	title = {A Coriolis force in an inertial frame},
	abstract = {Particles in rotating saddle potentials exhibit precessional motion which, up to now, has been explained by explicit computation. We show that this precession is due to a hidden Coriolis-like force which, unlike the standard Coriolis force, is present in the inertial frame. We do so by finding a hodograph-like ‘guiding center’ transformation using the method of normal form. We also point out that the transformation cannot be of contact type in principle, thus showing that the standard (in applied literature) heuristic averaging obscures the fact that the transformation of the position must involve the velocity.},
	journal = {Nonlinearity},
	volume = {30},
	number = {3},
	pages = {1109},
	numpages = {},
	year = {2017},
	month = {Feb},
	publisher = {IOP Publishing \& London Math. Soc.},
	doi = {10.1088/1361-6544/aa59a0},
	**url = {https://dx.doi.org/10.1088/1361-6544/aa59a0}**
}

@article{Sny17d,
	title = {Unique capabilities of AC frequency scanning and its implementation on a Mars Organic Molecule Analyzer linear ion trap },
	author = {Snyder, Dalton T. and Kaplan, Desmond A. and Danell, Ryan M. and van Amerom, Friso H. W. and Pinnick, Veronica T. and Brinkerhoff, William B. and Mahaffy, Paul R. and Cooks, R. Graham},
	journal = {Analyst},
	volume = {142},
	number = {12},
	pages = {2109 -- 2117},
	numpages = {0},
	year = {2017},
	month = {May},
	date = {22},
	publisher = {Royal Society Chemistry},
	doi = {10.1039/C7AN00664K},
	**url = {https://doi.org/10.1039/C7AN00664K}**
}

@article{Kot17,
	title = {Computation of Mathieu stability plot for an arbitrary toroidal ion trap mass analyser},
	author = {Kotana, Appala Naidu and Mohanty, Atanu K.},
	 abstract = {This paper presents a method to construct Mathieu stability plot for any arbitrary toroidal ion trap mass analyser. The proposed method is demonstrated on two toroidal ion trap mass analyser geometries. Two additional studies have also been carried out. These include prediction of ion secular frequency and ion trajectories at resonances in these traps. 
	In this numerical study, toroidal multipole coefficients of the respective traps are first evaluated and these are used for computing Mathieu parameters, a and q. These Mathieu parameters are used for constructing the stability plot, predicting secular frequency of ion motion and evaluating nonlinear resonances.
	
	The stability regions of both the traps considered in this paper are qualitatively similar to the corresponding plot of the linear ion trap mass analyser. There are, however two significant differences. First, the stability plots for toroidal ion trap mass analysers have prominent resonances. Second the apices of the stability plots of the toroidal ion trap mass analysers are different from those of the linear ion trap mass analyser.
	
	The secular frequency obtained using the Mathieu parameters in this paper agrees well with those obtained numerically, except close to the stability boundaries.
	
	Finally, the prominent nonlinear resonances have been identified as $\beta_r = \frac 23$ and $\beta_z = \frac 23$ in both the traps. In one of the two traps there is an additional prominent nonlinear resonance at $\beta_r + \beta_z = 1$.},
	journal = {Int. J. Mass Spectrom.},
	volume = {414},
	number = {},
	pages = {13 -- 22},
	numpages = {0},
	year = {2017},
	month = {Mar},
	publisher = {Elsevier BV},
	doi = {10.1016/j.ijms.2016.11.021},
	**url = {https://www.sciencedirect.com/science/article/pii/S1387380616302202}**
}

@article{Fach17,
	title = {The ion trap aerosol mass spectrometer: field intercomparison with the ToF-AMS and the capability of differentiating organic compound classes via MS-MS},
	author = {Fachinger, Johannes R. W. and Gallavardin, St{\'e}phane J. and Helleis, Frank and Fachinger, Friederike and Drewnick, Frank and Borrmann, Stephan},
	journal = {Atmos. Meas. Tech.},
	volume = {10},
	number = {4},
	pages = {1623 -- 1637},
	year = {2017},
	month = {Apr},
	date = {27},
	issn = {},
	publisher = {Copernicus Publications},
	doi = {10.5194/amt-10-1623-2017},
	**url = {https://www.atmos-meas-tech.net/10/1623/2017/amt-10-1623-2017.pdf}**
}

@article{Mih17,
	title = {Study of quasiclassical dynamics of trapped ions using the coherent state formalism and associated algebraic groups},
	author = {Mihalcea, Bogdan M.},
	journal = {Rom. J. Phys.},
	volume = {62},
	number = {5 -- 6},
	pages = {113},
	numpages = {21},
	year = {2017},
	month = {Jun},
	publisher = {Romanian Academy Printing House},
	doi = {}, 
	url = {https://www.nipne.ro/rjp/2017_62_5-6/RomJPhys.62.113.pdf}	
}

@article{Fan17,
	author = {Fan, Wenkai and Du, Li and Wang, Sihui and Zhou, Huijun},
	title = {Confining rigid balls by mimicking quadrupole ion trapping},
	abstract = {The rotating saddle not only is an interesting system that is able to trap a ball near its saddle point, but can also intuitively illustrate the operating principles of quadrupole ion traps in modern physics. Unlike the conventional models based on the mass-point approximation, we study the stability of a ball in a rotating-saddle trap using rigid-body dynamics. The stabilization condition of the system is theoretically derived and subsequently verified by experiments. The results are compared with the previous mass-point model, giving large discrepancy as the curvature of the ball is comparable to that of the saddle. We also point out that the spin angular velocity of the ball is analogous to the cyclotron frequency of ions in an external magnetic field utilized in many prevailing ion-trapping schemes.},
	journal = {Am. J. Phys.},
	volume = {85},
	number = {11},
	pages = {821 -- 829},
	year = {2017},
	month = {Nov},
	issn = {0002-9505},
	doi = {10.1119/1.5005927},
	**url = {https://doi.org/10.1119/1.5005927},**
	eprint = {https://pubs.aip.org/aapt/ajp/article-pdf/85/11/821/19485870/821\_1\_1.5005927.pdf}
}

@article{Rei17,
	title = {Dissipative quantum error correction and application to quantum sensing with trapped ions},
	author = {Reiter, F. and S{\o}rensen, A. S. and Zoller, P. and Muschik, C. A.},
	abstract = {Quantum-enhanced measurements hold the promise to improve high-precision sensing ranging from the definition of time standards to the determination of fundamental constants of nature. However, quantum sensors lose their sensitivity in the presence of noise. To protect them, the use of quantum error-correcting codes has been proposed. Trapped ions are an excellent technological platform for both quantum sensing and quantum error correction. Here we present a quantum error correction scheme that harnesses dissipation to stabilize a trapped-ion qubit. In our approach, always-on couplings to an engineered environment protect the qubit against spin-flips or phase-flips. Our dissipative error correction scheme operates in a continuous manner without the need to perform measurements or feedback operations. We show that the resulting enhanced coherence time translates into a significantly enhanced precision for quantum measurements. Our work constitutes a stepping stone towards the paradigm of self-correcting quantum information processing.},
	journal = {Nature Comm.},
	volume = {8},
	number = {},
	pages = {1822},
	numpages = {11},
	year = {2017},
	month = {Nov},
	date = {28},
	publisher = {Nature},
	doi = {10.1038/s41467-017-01895-5},
	**url = {https://www.nature.com/articles/s41467-017-01895-5.pdf}**
}

@article{Mih18,
	title = {Squeezed coherent states of motion for ions confined in quadrupole and octupole ion traps},
	author = {Mihalcea, Bogdan M.},
	abstract = {Quasiclassical dynamics of trapped ions is characterized by applying the time dependent variational principle (TDVP) on coherent state orbits, in case of quadrupole and octupole combined (Paul and Penning) or radiofrequency (RF) traps. A dequantization algorithm is proposed, by which the classical Hamilton (energy) function associated to the system results as the expectation value of the quantum Hamiltonian on squeezed coherent states. We develop such method and particularize the quantum Hamiltonian for both combined and RF nonlinear traps, that exhibit axial symmetry. We also build the classical Hamiltonian functions for the particular traps we considered and find the classical equations of motion.},
	journal = {Ann. Phys. (N. Y.)},
	volume = {388},
	number = {},
	pages = {100 -- 113},
	numpages = {0},
	year = {2018},
	month = {Jan},
	date = {},
	publisher = {Elsevier},
	issn = {0003-4916},
	doi = {10.1016/j.aop.2017.11.004},
	**url = {https://www.sciencedirect.com/science/article/pii/S0003491617303196}**
}

@article{Raja18,
	title = {A Linear Ion Trap with an Expanded Inscribed Diameter to Improve Optical Access for Fluorescence Spectroscopy},
	author = {Rajagopal, Vaishnavi and Stokes, Chris and Ferzoco, Alessandra},
	abstract = {We report a custom-geometry linear ion trap designed for fluorescence spectroscopy of gas-phase ions at ambient to cryogenic temperatures. Laser-induced fluorescence from trapped ions is collected from between the trapping rods, orthogonal to the excitation laser that runs along the axis of the linear ion trap. To increase optical access to the ion cloud, the diameter of the round trapping rods is 80% of the inscribed diameter, rather than the roughly 110% used to approximate purely quadrupolar electric fields. To encompass as much of the ion cloud as possible, the first collection optic has a 25.4 mm diameter and a numerical aperture of 0.6. The choice of geometry and collection optics yields $10^7$ detected photons/s from trapped rhodamine 6G ions. The trap is coupled to a closed-cycle helium refrigerator, which in combination with two 50 Ohm heaters enables temperature control to below 25 K on the rod electrodes. The purpose of the instrument is to broaden the applicability of fluorescence spectroscopy of gas-phase ions to cases where photon emission is a minority relaxation pathway. Such studies are important to understand how the microenvironment of a chromophore influences excited state charge transfer processes.},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {29},
	number = {2},
	pages = {260 -- 269},
	numpages = {0},
	year = {2018},
	month = {Feb},
	publisher = {Am. Soc. Mass Spectrom. - Springer US},
	doi = {10.1007/s13361-017-1763-3},
	**url = {https://doi.org/10.1007/s13361-017-1763-3}**
}

@article{Pedre18a,
	title = {Symmetry breaking in linear multipole traps},
	author = {Pedregosa-Gutierrez, J. and Champenois, C. and Kamsap, M. R. and Hagel, G. and Houssin, M. and Knoop, M.},
	abstract = {Radiofrequency multipole traps have been used for some decades in cold collision experiments and are gaining interest for precision spectroscopy due to their low micromotion contribution and the predicted unusual cold-ion structures. However, the experimental realisation is not yet fully controlled, and open questions in the operation of these devices remain. We present experimental observations of symmetry breaking of the trapping potential in a macroscopic octupole trap with laser-cooled ions. Numerical simulations have been performed in order to explain the appearance of additional local potential minima and be able to control them in a next step. We characterise these additional potential minima, in particular with respect to their position, their potential depth and their probability of population as a function of the radial and angular displacement of the trapping rods.},
	journal = {J. Mod. Optics},
	volume = {65},
	number = {5 -- 6},
	pages = {529 -- 537},
	numpages = {0},
	year = {2018},
	month = {Mar},
	date = {1},
	publisher = {Taylor /& Francis},
	doi = {10.1080/09500340.2017.1408866},
	**eprint = {atom-ph/1705.08133}**,
	**url = {https://www.tandfonline.com/doi/abs/10.1080/09500340.2017.1408866}**    
}

@article{Tian18,
	title = {Experimental Observation of the Effects of Translational and Rotational Electrode Misalignment on a Planar Linear Ion Trap Mass Spectrometer},
	author = {Tian, Yuan and Decker, Trevor K. and McClellan, Joshua S. and Wu, Qinghao and De la Cruz, Abraham and Hawkins, Aaron R. and Austin, Daniel E.},
	abstract = {The performance of miniaturized ion trap mass analyzers is limited, in part, by the accuracy with which electrodes can be fabricated and positioned relative to each other. Alignment of plates in a two-plate planar LIT is ideal to characterize misalignment effects, as it represents the simplest possible case, having only six degrees of freedom (DOF) (three translational and three rotational). High-precision motorized actuators were used to vary the alignment between the two ion trap plates in five DOFs—x, y, z, pitch, and yaw. A comparison between the experiment and previous simulations shows reasonable agreement. Pitch, or the degree to which the plates are parallel along the axial direction, has the largest and sharpest impact to resolving power, with resolving power dropping noticeably with pitch misalignment of a fraction of a degree. Lateral displacement (x) and yaw (rotation of one plate, but plates remain parallel) both have a strong impact on ion ejection efficiency, but little effect on resolving power. The effects of plate spacing (y-displacement) on both resolving power and ion ejection efficiency are attributable to higher-order terms in the trapping field. Varying the DC (axial) trapping potential can elucidate the effects where more misalignments in more than one DOF affect performance. Implications of these results for miniaturized ion traps are discussed.},
	journal = {J. Am. Soc. Mass. Spectrom.},
	volume = {29},
	number = {7},
	pages = {1376 -- 1385},
	numpages = {0},
	year = {2018},
	month = {Apr},
	date = {5},
	publisher = {American Society for Mass Spectrometry},
	doi = {10.1007/s13361-018-1942-x},
	**url = {https://doi.org/10.1007/s13361-018-1942-x}**
}

@article{Safro18,
	title = {Search for new physics with atoms and molecules},
	author = {Safronova, M. S. and Budker, D. and DeMille, D. and Kimball, Derek F. Jackson and Derevianko, A. and Clark, Charles W.},
	abstract = {This article reviews recent developments in tests of fundamental physics using atoms and molecules, including the subjects of parity violation, searches for permanent electric dipole moments, tests of the CPT theorem and Lorentz symmetry, searches for spatiotemporal variation of fundamental constants, tests of quantum electrodynamics, tests of general relativity and the equivalence principle, searches for dark matter, dark energy, and extra forces, and tests of the spin-statistics theorem. Key results are presented in the context of potential new physics and in the broader context of similar investigations in other fields. Ongoing and future experiments of the next decade are discussed.},
	journal = {Rev. Mod. Phys.},
	volume = {90},
	number = {2},
	pages = {025008},
	numpages = {106},
	year = {2018},
	month = {Jun},
	publisher = {American Physical Society},
	doi = {10.1103/RevModPhys.90.025008},
	**url = {https://link.aps.org/doi/10.1103/RevModPhys.90.025008}**
}

@article{Foot18,
	title = {Two-frequency operation of a Paul trap to optimise confinement of two species of ions},
	author = {Foot, C. J. and Trypogeorgos, D. and Bentine, E. and Gardner, A. and Keller, M.},
	abstract = {We describe the operation of an electrodynamic ion trap in which the electric quadrupole field oscillates at two frequencies. This mode of operation allows simultaneous tight confinement of ions with extremely different charge-to-mass ratios, e.g., singly ionised atomic ions together with multiply charged nanoparticles. We derive the stability conditions for two-frequency operation from asymptotic properties of the solutions of the Mathieu equation and give a general treatment of the effect of damping on parametric resonances. Two-frequency operation is effective when the two species’ mass ratios and charge ratios are sufficiently large, and further when the frequencies required to optimally trap each species are widely separated. This system resembles two coincident Paul traps, each operating close to a frequency optimised for one of the species, such that both species are tightly confined. This method of operation provides an advantage over single-frequency Paul traps, in which the more weakly confined species forms a sheath around a central core of tightly confined ions. We verify these ideas using numerical simulations and by measuring the parametric heating induced in experiments by the additional driving frequency.},
	journal = {Int. J. Mass. Spectrom.},
	volume = {430},
	number = {},
	pages = {117 -- 125},
	numpages = {0},
	year = {2018},
	month = {Jul},
	date = {},
	publisher = {Elsevier},
	doi = {10.1016/j.ijms.2018.05.007},
	**url = {https://www.sciencedirect.com/science/article/abs/pii/S1387380618300010}**
}

@article{Benk18,
	title = {Painlev\'{e} analysis and integrability of the trapped ionic system},
	author = {Benkhali, Mohamed and Kharbach, Jaouad and {El Fakkousy}, Idriss and Chatar, Walid and Rezzouk, Abdellah and Ouazzani-Jamil, Mohammed},
	keywords = {Hamiltonian system, Integrability, Painlev\'{e} analysis, Poincar\'{e} section, Chaos-order-chaos},
	abstract = {Integrability in the Painlev\'{e} sense of the trapped ionic system in the quadrupole field with superpositions of rotationally symmetric hexapole and octopole fields is studied. Five integrable cases of the system are reported. First Integrals of the planar motion are founded. Confirming three-dimensional integrability of the equations of motion, the third explicit integrals of motion are constructed directly for each case. We carried out a numerical study to observe the regularity and chaotic regions via the Poincar\'{e} surface of sections, and corroborate the analytical results.},
	journal = {Phys. Lett. A},
	volume = {382},
	number = {36},
	pages = {2515 -- 2525},
	year = {2018},
	month = {Sep},
	date = {14},
	issn = {0375-9601},
	doi = {10.1016/j.physleta.2018.06.034},
	**url = {https://www.sciencedirect.com/science/article/pii/S0375960118307060}**
}

@article{Paga18,
	author = {Pagano, G. and Hess, P. W. and Kaplan, H. B. and Tan, W. L. and Richerme, P. and Becker, P. and Kyprianidis, A. and Zhang, J. and Birckelbaw, E and Hernandez, M. R. and Wu, Y. and Monroe, C.},
	title = {Cryogenic trapped-ion system for large scale quantum simulation},
	journal = {Quantum Sci. Technol.},
	abstract = {Trapped-ion systems are among the most promising hardware candidates for large scale quantum computing and quantum simulation. In order to scale up such devices, it is necessary to engineer extreme-high vacuum (XHV) environments to prevent background gas from disrupting the ion crystal. Here we present a new cryogenic ion trapping system designed for long time storage of large ion chains. Our apparatus is based on a segmented-blade ion trap enclosed in a 4 K cryostat, which enables us to routinely trap and hold over 100 $^{171}$Yb$^+$ ions for hours in a linear configuration, due to low background gas pressure from differential cryo-pumping. We characterize the XHV cryogenic environment measuring pressures below $10^{-11}$ Torr by recording both inelastic and elastic collisions between the ion chain and the molecular background gas. We also demonstrate coherent one and two-qubit operations and nearly equidistant ion spacing for chains of up to 44 ions using anharmonic axial potentials, in order to enable better detection and single ion addressing in large ion arrays. We anticipate that this reliable production and lifetime enhancement of large linear ion chains will enable quantum simulation of models that are intractable with classical computer modeling.},
	volume = {4},
	number = {1},
	pages = {014004},
	year = 2018,
	month = {oct},
	publisher = {{IOP} Publishing},
	doi = {10.1088/2058-9565/aae0fe},
	**url = {https://doi.org/10.1088/2058-9565/aae0fe}**
}

@article{Tome18,
	title = {Atomic Clocks and Time Keeping in Romania},
	author = {Tomescu, Cipriana and Giurgiu, Liviu},
	journal = {Rom. Rep. Phys.},
	volume = {70},
	number = {4},
	pages = {205},
	numpages = {29},
	year = {2018},
	month = {Nov},
	publisher = {Romanian Academy Publishing House},
	doi = {},
	url = {https://rrp.nipne.ro/2018/AN70205.pdf} 
}

@ARTICLE{Fou19,
	title = {Classical and quantum dynamics of a trapped ion coupled to a charged nanowire},
	author = {Fountas, P. N. and Poggio, M. and Willitsch, Stephan},
	abstract = {We study theoretically the mechanical drive of a trapped ultracold ion by a charged nanowire through their mutual Coulomb interaction. We characterize the perturbation of the trapping potential for the ion by the nanowire and discuss the parameters determining the dynamics of the ion under the action of the nanooscillator. We explore the classical dynamics as well as motional quantum states of the ion which can be generated and manipulated with the resonant drive of the nanowire and the effects of anharmonicities of the ion-trap potential on the system. Our modelling indicates that unusual quantum states of the ion motion can be generated with this approach and that sympathetic cooling and quantum entanglement can be realised when both subsystems operate in the quantum regime. The present ion-mechanical hybrid system might prove interesting as a new quantum device, for quantum sensing experiments, for spectroscopy and for mass spectrometry.},
	journal = {New J. Phys.},
	volume = {21},
	number = {1},
	pages = {013030},
	numpages = {12},
	year = {2019},
	month = {Jan},
	date = {30},
	publisher = {I{OP P}ublishing},
	doi = {10.1088/1367-2630/aaf8f5},
	**url = {https://doi.org/10.1088/1367-2630/aaf8f5}**
}

@article{Bize19,
	title = {The unit of time: Present and future directions},
	author = {Bize, S\'{e}bastien},
	keywords = {Atomic fountain, Timescale, Optical frequency standard, Fundamental physics test, Chronometric geodesy, Redefinition of the SI second, Fontaine atomique, Échelle de temps, Étalon de fréquence optique, Test de physique fondamentale, Géodésie chronométrique, Redéfinition de la seconde du SI},
	abstract = {Some 50 years ago, physicists, and after them the entire world, started to found their time reference on atomic properties instead of motions of the Earth that have been in use since the origin. Far from being an arrival point, this decision marked the beginning of an adventure characterized by an improvement by 6 orders of magnitude in the uncertainty of realization of atomic frequency and time references. Ever-progressing atomic frequency standards and time references derived from them are key resources for science and for society. We will describe how the unit of time is realized with a fractional accuracy approaching $10^{−16}$ and how it is delivered to users via the elaboration of the international atomic time. We will describe the tremendous progress of optical frequency metrology over the last 20 years that led to a novel generation of optical frequency standards with fractional uncertainties of $10^{−18}$. We will describe work toward a possible redefinition of the SI second based on such standards. We will describe existing and emerging applications of atomic frequency standards in science.},
	journal = {C. R. Phys.},
	volume = {20},
	number = {1},
	pages = {153 -- 168},
	year = {2019},
	month = {jan},
	note = {The new International System of Units / Le nouveau Système international d’unités},
	issn = {1631-0705},
	doi = {https://doi.org/10.1016/j.crhy.2019.02.002},
	**url = {https://www.sciencedirect.com/science/article/pii/S1631070519300167}**
}

@article{Kel19,
	title = {Controlling systematic frequency uncertainties at the ${10}^{\ensuremath{-}19}$ level in linear {C}oulomb crystals},
	author = {Keller, J. and Burgermeister, T. and Kalincev, D. and Didier, A. and Kulosa, A. P. and Nordmann, T. and Kiethe, J. and Mehlst\"aubler, T. E.},
	abstract = {Trapped ions are ideally suited for precision spectroscopy, as is evident from the remarkably low systematic uncertainties of single-ion clocks. The major weakness of these clocks is the long averaging time, necessitated by the low signal of a single atom. An increased number of ions can overcome this limitation and allow for the implementation of novel clock schemes. However, this presents the challenge to maintain the excellent control over systematic shifts of a single particle in spatially extended and strongly coupled many-body systems. We measure and deduce systematic frequency uncertainties related to spectroscopy with ion chains in an rf trap array designed for precision spectroscopy on simultaneously trapped ion ensembles. For the example of an In$^+$ clock, sympathetically cooled with Yb$^+$ ions, we show in our system that the expected systematic frequency uncertainties related to multi-ion operation can be below $1 \times 10^{−19}$. Our results pave the way to advanced spectroscopy schemes such as entangled clock spectroscopy and cascaded clock operation.},
	journal = {Phys. Rev. A},
	volume = {99},
	number = {1},
	pages = {013405},
	numpages = {12},
	year = {2019},
	month = {Jan},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.99.013405},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.99.013405}**
}

@article{McGrew19,
	author = {W. F. McGrew and X. Zhang and H. Leopardi and R. J. Fasano and D. Nicolodi and K. Beloy and J. Yao and J. A. Sherman and S. A. Sch\"{a}ffer and J. Savory and R. C. Brown and S. R\"{o}misch and C. W. Oates and T. E. Parker and T. M. Fortier and A. D. Ludlow},
	title = {Towards the optical second: verifying optical clocks at the SI limit},
	abstract = {The pursuit of ever more precise measures of time and frequency motivates redefinition of the second in terms of an optical atomic transition. To ensure continuity with the current definition, based on the microwave hyperfine transition in $^{133}$Cs, it is necessary to measure the absolute frequency of candidate optical standards relative to primary cesium references. Armed with independent measurements, a stringent test of optical clocks can be made by comparing ratios of absolute frequency measurements against optical frequency ratios measured via direct optical comparison. Here we measure the $^1$S$_0${\textrightarrow}$^3$P$_0$ transition of $^{171}$Yb using satellite time and frequency transfer to compare the clock frequency to an international collection of national primary and secondary frequency standards. Our measurements consist of 79 runs spanning eight months, yielding the absolute frequency to be 518 295 836 590 863.71(11) Hz and corresponding to a fractional uncertainty of $2.1 \times 10^{-16}$. This absolute frequency measurement, the most accurate reported for any transition, allows us to close the Cs-Yb-Sr-Cs frequency measurement loop at an uncertainty $\lt 3 \times 10^{-16}$, limited for the first time by the current realization of the second in the International System of Units (SI). Doing so represents a key step towards an optical definition of the SI second, as well as future optical time scales and applications. Furthermore, these high accuracy measurements distributed over eight months are analyzed to tighten the constraints on variation of the electron-to-proton mass ratio, $\mu = m_e/m_p$. Taken together with past Yb and Sr absolute frequency measurements, we infer new bounds on the coupling coefficient to gravitational potential of $k_{\mu} = \left(- 1.9 \pm 9.4\right) \times 10^{-7}$ and a drift with respect to time of $\frac{\dot \mu}{\mu} = \left(5.3 \pm 6.5\right)\times 10^{-17}$/yr.},
	journal = {Optica},
	keywords = {Frequency measurement; Optical clocks; Optical measurement; Optical standards; Optical testing; Precision metrology},
	volume = {6},
	number = {4},
	pages = {448 -- 454},
	publisher = {Optical Society of America},
	month = {Apr},
	year = {2019},
	doi = {10.1364/OPTICA.6.000448},
	**url = {http://opg.optica.org/optica/abstract.cfm?URI=optica-6-4-448}**
}

@article{Nolt19,
	author = {Nolting, Dirk and Malek, Robert and Makarov, Alexander},
	title = {Ion traps in modern mass spectrometry},
	journal = {Mass Spectrom. Rev.},
	volume = {38},
	number = {2},
	pages = {150 -- 168},
	keywords = {Fourier transform, high resolution, ion trap, mass accuracy, mass spectrometry, Orbitrap analyzer},
	abstract = {This review is devoted to trapping mass spectrometry wherein ions are confined by electromagnetic fields for prolonged periods of time within limited volume, with mass measurement taking place within the same volume. Three major types of trapping mass spectrometers are discussed, specifically radiofrequency ion trap, Fourier transform ion cyclotron resonance and Orbitrap. While these three branches are intricately interwoven with each other over their recent history, they also differ greatly in their fundamentals, roots and historical origin. This diversity is reflected also in the difference of viewpoints from which each of these directions is addressed in this review. Following the theme of the issue, we focus on developments mainly associated with the country of Germany but, at the same time, we use this review as an illustration of the rapidly increasing globalization of science and expanding multi-national collaborations.},
	year = {2019},
	month = {Mar},
	doi = {10.1002/mas.21549},
	**  url = {https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/mas.21549}**
}

@article{John19,
	author = {Johnston, Murray V. and Kerecman, Devan E.},
	title = {Molecular Characterization of Atmospheric Organic Aerosol by Mass Spectrometry},
	abstract = {Atmospheric aerosol, particulate matter suspended in the air we breathe, exerts a strong impact on our health and the environment. Controlling the amount of particulate matter in air is difficult, as there are many ways particles can form by both natural and anthropogenic processes. We gain insight into the sources of particulate matter through chemical composition measurements. A substantial portion of atmospheric aerosol is organic, and this organic matter is exceedingly complex on a molecular scale, encompassing hundreds to thousands of individual compounds that distribute between the gas and particle phases. Because of this complexity, no single analytical technique is sufficient. However, mass spectrometry plays a crucial role owing to its combination of high sensitivity and molecular specificity. This review surveys the various ways mass spectrometry is used to characterize atmospheric organic aerosol at a molecular level, tracing these methods from inception to current practice, with emphasis on current and emerging areas of research. Both offline and online approaches are covered, and molecular measurements with them are discussed in the context of identifying sources and elucidating the underlying chemical mechanisms of particle formation. There is an ongoing need to improve existing techniques and develop new ones if we are to further advance our knowledge of how to mitigate the unwanted health and environmental impacts of particles.},
	journal = {Annu. Rev. Anal. Chem.},
	volume = {12},
	number = {1},
	pages = {247 -- 274},
	year = {2019},
	month = {Jun},
	doi = {10.1146/annurev-anchem-061516-045135},
	**URL = {https://doi.org/10.1146/annurev-anchem-061516-045135}**
}

@article{Oral19,
	author = {Oral, Martin and \v{C}\'{\i}p, Ond\v{r}ej and Slodi\v{c}ka, Luk\'{a}\v{s}},
	title = {Simulation of motion of many ions in a linear Paul trap},
	abstract = {The quadrupole linear Paul trap is one of the key instruments in building highly stable atomic clocks. However, a frequency reference based on a single trapped ion is limited in stability due to the time needed for the interrogation cycle which cannot be further shortened. A promising strategy is the utilization of multiple trapped ions. The ions of the same kind then repulse each other with the Coulomb force, which is countered by the ponderomotive force of the time depended field in the trap. A few ions form a chain along the axis of a linear Paul trap. Adding more ions (a few tens or hundreds) gives rise to Coulomb crystals. We created an efficient simulation code which calculates the motion of such collections of ions in quasi-static radiofrequency fields of real linear quadrupole traps (including the micromotion). We include a model for laser cooling of the ions. The simulation tool can be used to study the formation and the dynamics of Coulomb crystals under conditions corresponding to various experimental setups.},
	journal = {Int. J. Mod. Phys. A},
	volume = {34},
	number = {36},
	pages = {1942003},
	numpages = {},
	year = {2019},
	publisher = {World Scientific},
	doi = {10.1142/S0217751X1942003X},
	**URL = {https://doi.org/10.1142/S0217751X1942003X}**
}

@article{Bruz19,
	author = {Bruzewicz, Colin D. and Chiaverini, John and McConnell, Robert and Sage, Jeremy M.},
	title = {Trapped-ion quantum computing: Progress and challenges},
	abstract = {Trapped ions are among the most promising systems for practical quantum computing (QC). The basic requirements for universal QC have all been demonstrated with ions, and quantum algorithms using few-ion-qubit systems have been implemented. We review the state of the field, covering the basics of how trapped ions are used for QC and their strengths and limitations as qubits. In addition, we discuss what is being done, and what may be required, to increase the scale of trapped ion quantum computers while mitigating decoherence and control errors. Finally, we explore the outlook for trapped-ion QC. In particular, we discuss near-term applications, considerations impacting the design of future systems of trapped ions, and experiments and demonstrations that may further inform these considerations.},
	journal = {Appl. Phys. Rev.},
	volume = {6},
	number = {2},
	pages = {021314},
	numpages = {46},
	publisher = {{AIP P}ublishing},
	year = {2019},
	month = {Jun},
	doi = {10.1063/1.5088164},
	**URL = {https://doi.org/10.1063/1.5088164}**
}

@article{Lee19,
	title = {Time fractals and discrete scale invariance with trapped ions},
	author = {Lee, Dean and Watkins, Jacob and Frame, Dillon and Given, Gabriel and He, Rongzheng and Li, Ning and Lu, Bing-Nan and Sarkar, Avik},
	abstract = {We show that a one-dimensional chain of trapped ions can be engineered to produce a quantum mechanical system with discrete scale invariance and fractal-like time dependence. By discrete scale invariance we mean a system that replicates itself under a rescaling of distance for some scale factor, and a time fractal is a signal that is invariant under the rescaling of time. These features are reminiscent of the Efimov effect, which has been predicted and observed in bound states of three-body systems. We demonstrate that discrete scale invariance in the trapped ion system can be controlled with two independently tunable parameters. We also discuss the extension to $n$-body states where the discrete scaling symmetry has an exotic heterogeneous structure. The results we present can be realized using currently available technologies developed for trapped ion quantum systems.},
	journal = {Phys. Rev. A},
	volume = {100},
	number = {1},
	pages = {011403},
	numpages = {4},
	year = {2019},
	month = {Jul},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.100.011403},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.100.011403}**
}

@article{Vin19,
	title = {Testing collapse models with levitated nanoparticles: Detection challenge},
	author = {Vinante, A. and Pontin, A. and Rashid, M. and Toro{\v s}, M. and Barker, P. F. and Ulbricht, H.},
	abstract = {We consider a nanoparticle levitated in a Paul trap in ultrahigh cryogenic vacuum, and look for the conditions which allow for a stringent noninterferometric test of spontaneous collapse models. In particular we compare different possible techniques to detect the particle motion. Key conditions which need to be achieved are extremely low residual pressure and the ability to detect the particle at ultralow power. We compare three different detection approaches based, respectively, on an optical cavity, an optical tweezer, and an electrical readout, and for each one we assess advantages, drawbacks, and technical challenges.},
	journal = {Phys. Rev. A},
	volume = {100},
	number = {1},
	pages = {012119},
	numpages = {13},
	year = {2019},
	month = {Jul},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.100.012119},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.100.012119}**
}

@article{Wolf19,
	author = {Wolf, Fabian and Shi, Chunyan and Heip, Jan C. and Gessner, Manuel and Pezz\`{e}, Luca and Smerzi, Augusto and Schulte, Marius and Hammerer, Klemens and Schmidt, Piet O.},
	title = {Motional {F}ock states for quantum-enhanced amplitude and phase measurements with trapped ions},
	abstract = {The quantum noise of the vacuum limits the achievable sensitivity of quantum sensors. In non-classical measurement schemes the noise can be reduced to overcome this limitation. However, schemes based on squeezed or Schr\"{o}dinger cat states require alignment of the relative phase between the measured interaction and the non-classical quantum state. Here we present two measurement schemes on a trapped ion prepared in a motional Fock state for displacement and frequency metrology that are insensitive to this phase. The achieved statistical uncertainty is below the standard quantum limit set by quantum vacuum fluctuations, enabling applications in spectroscopy and mass measurements.},
	journal = {Nat. Commun.},
	volume = {10},
	number = {1},
	pages = {2929},
	numpages = {8},
	year = {2019},
	month = {Jul},
	issn = {2041-1723},
	doi = {10.1038/s41467-019-10576-4},
	**URL = {https://doi.org/10.1038/s41467-019-10576-4}**
}

@article{Vasi19,
	title = {Effective Rotational Potential of a Molecular Ions in a Plane Radio-Frequency Trap},
	author = {Vasil’ev, I. A. and Kushchenko, O. M. and Rudyi, S. S. and Rozhdestvenskii, Yu. V.},
	abstract = {A model of rotational potential that represents an analog of the pseudopotential for localization of a model of diatomic particle with rigid bond in a plane quadrupole radio-frequency trap is considered. It is shown that the effective rotational potential can be used to describe dynamics of various diatomic particles with different centers of mass and charges. Comparison of the model of pseudopotential for localization of a single ion and the proposed model of the effective rotational potential for diatomic structure is used to determine additional positions of quasi-equilibrium for the center of mass of the diatomic particle and orientation angle of the molecule. Additional positions of quasi-equilibrium substantially affect dynamics of charged structure localized in the field of a plane quadrupole ion trap.},
	journal = {Tech. Phys.},
	volume = {64},
	number = {9},
	pages = {1379 -- 1385},
	numpages = {0},
	year = {2019},
	month = {Sep},
	date = {16},
	publisher = {Pleiades Publishing-Springer},
	doi = {10.1134/S1063784219090202},
	**url = {https://doi.org/10.1134/S1063784219090202}**
}

@Incollection{Ortiz19,
	author = {Rosas-Ortiz, Oscar},
	editor = {Kuru, {\c{S}}eng{\"u}l and Negro, Javier and Nieto, Luis M.},
	title = {Coherent and {S}queezed {S}tates: Introductory {R}eview of {B}asic {N}otions, {P}roperties, and {G}eneralizations},
	bookTitle = {Integrability, Supersymmetry and Coherent States: A Volume in Honour of Professor V{\'e}ronique Hussin},
	year = {2019},
	publisher = {Springer International Publishing},
	address = {Cham},
	chapter = {7},
	pages = {187--230},
	abstract = {A short review of the main properties of coherent and squeezed states is given in the introductory form. The efforts are addressed to clarify concepts and notions, including some passages of the history of science, with the aim of facilitating the subject for nonspecialists. In this sense, the present work is intended to be complementary to other papers of the same nature and subject in current circulation.},
	isbn = {978-3-030-20086-2},
	e-isbn = {978-3-030-20087-9},
	doi = {10.1007/978-3-030-20087-9_7},
	**url = {https://doi.org/10.1007/978-3-030-20087-9_7}**
}

@Article{Ozd19,
	author = {{\"O}zdemir, Abdil and Lin, Jung-Lee and G{\"u}lfen, Mustafa and Hsiao, Chun-Jen and Chen, Chung-Hsuan},
	title  = {A quadrupole ion trap mass spectrometer for dry microparticle analysis},
	abstract  = {In this work we report a new design of a charge detection quadrupole ion trap mass spectrometer (QIT-MS) for the analysis of micro-sized dry inorganic and bioparticles including red blood cells (RBCs) and different sizes of MCF-7 breast cancer cells. The developed method is one of the fastest methods to measure the mass of micro-sized particles. This system allows the online analysis of various micro-sized particles up to $1 \times 10^{17}$ Da. The calibration of the mass spectrometer has been done by using different sizes of polystyrene (PS) particles (2 – 15 $mu$m). The measured masses of RBCs were around $1.8 \times 10^{13}$ Da and MCF-7 cancer cells were between $1 \times 10^{14}$ and $4 \times 10^{14}$ Da. The calculated mass distribution profiles of the particles and cells were given as histogram profiles. The statistical data were summarized after Gaussian type fitting to the experimental histogram profiles. The new method gives very promising results for the analysis of particles and has very broad application.},
	journal  = {Analyst},
	volume  = {144},
	number  = {18},
	pages  = {5608 -- 5616},
	year  = {2019},
	month = {Aug},
	publisher  = {The Royal Society of Chemistry},
	doi  = {10.1039/C9AN01431D},
	**url  = {https://doi.org/10.1039/C9AN01431D}**,
}

@article{Bahra19,
	author = {Bahrami, Abasalt and M{\"u}ller, Matthias and Drechsler, Martin and Joger, Jannis and Gerritsma, Rene and Schmidt-Kaler, Ferdinand},
	title = {Operation of a Microfabricated Planar Ion-Trap for Studies of a Yb$^+$–Rb Hybrid Quantum System},
	abstract = {In order to study interactions of atomic ions with ultracold neutral atoms, it is important to have sub-$\mu$m control over positioning ion crystals. Serving for this purpose, a microfabricated planar ion trap featuring 21 DC electrodes is introduced. The ion trap is controlled by a home-made FPGA voltage source providing independently variable voltages to each of the DC electrodes. To assure stable positioning of ion crystals with respect to trapped neutral atoms, the authors integrate into the overall design a compact mirror magneto optical chip trap (mMOT) for cooling and confining neutral 87Rb atoms. The trapped atoms will be transferred into an also integrated chip-based Ioffe-Pritchard trap potential formed by a Z-shaped wire and an external bias magnetic field. The authors introduce the hybrid atom–ion chip, the microfabricated planar ion trap, and use trapped ion crystals to determine ion lifetimes, trap frequencies, positioning ions, and the accuracy of the compensation of micromotion.},
	keywords = {atom–ion interaction, atom traps, ion traps},
	journal = {Phys. Status Solidi B},
	volume = {256},
	number = {9},
	pages = {1800647},
	numpages = {9},
	publisher = {Wiley},
	year = {2019},
	month = {Sep},
	doi = {10.1002/pssb.201800647},
	**url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/pssb.201800647}**,
}

@article{Sny19a,
	title = {Two-Dimensional Tandem Mass Spectrometry in a Single Scan on a Linear Quadrupole Ion Trap},
	author = {Snyder, Dalton T. and Szalwinski, Lucas J. and St. John, Zachary and Cooks, R. Graham},
	abstract = {A two-dimensional tandem mass spectrometry (2D MS/MS) scan has been developed for the linear quadrupole ion trap. Precursor ions are mass-selectively excited using a nonlinear ac frequency sweep at constant rf voltage, while simultaneously, all product ions of the excited precursor ions are ejected from the ion trap using a broad-band waveform. The fragmentation time of the precursor ions correlates with the precursor $m/z$ value (the first mass dimension) and also with the ejection time of the product ions, allowing the correlation between precursor and product ions. Additionally, the second mass dimension (product ions’ $m/z$ values) is recovered through fast Fourier transform of each mass spectral peak, revealing either intentionally introduced “frequency tags” or the product ion micropacket frequencies, both of which can be converted to product ion $m/z$ through the classical Mathieu parameters, thereby revealing a product ion mass spectrum for every precursor ion without prior isolation. We demonstrate the utility of this method for analyzing a broad range of structurally related precursor ions, including chemical warfare agent simulants, fentanyls and other opioids, amphetamines, cathinones, antihistamines, and tetracyclic antidepressants.},
	journal = {Anal. Chem.},
	volume = {91},
	number = {21},
	pages = {13752 -- 13762},
	numpages = {0},
	year = {2019},
	month = {Oct},
	date = {8},
	publisher = {American Chemical Society},
	issn = {0003-2700},
	doi = {10.1021/acs.analchem.9b03123},
	**url = {https://doi.org/10.1021/acs.analchem.9b03123}**
}

@article{Reece19,
	title = {Digital Mass Analysis in a Linear Ion Trap without Auxiliary Waveforms},
	author = {Reece, Margaret P. and Huntley, Adam P. and Moon, Ashley M. and Reilly, Peter T. A.},
	abstract = {Mass analysis in a linear ion trap is traditionally performed using resonant ejection induced by auxiliary waveforms. For sinusoidally driven ion traps without resonant ejection, resolution and sensitivity are poor because mass-selected instability yields excitation along both the x and y axes simultaneously. Digital ion traps, on the other hand, have the advantage of duty cycle manipulation that can be used to change the ion excitation along the x and y axes. Consequently, the duty cycle can be used to enhance the resolution and sensitivity for mass-selected instability in a linear ion trap without the application of an auxiliary waveform. This work introduces and explores mass-selected instability in a linear trap without the use of auxiliary waveforms.},
	journal = {J. Am. Soc. Mass Spectrom.},
	volume = {31},
	number = {1},
	pages = {103 -- 108},
	year = {2019},
	month = {Nov},
	date = {21},
	issn = {},
	publisher = {Am. Chem. Soc.},
	doi = {10.1021/jasms.9b00012},
	**url = {https://pubs.acs.org/doi/10.1021/jasms.9b00012}**
}

@incollection{Mokh20,
	title = {Chapter Four - Trapped Rydberg ions: A new platform for quantum information processing},
	author = {Mokhberi, Arezoo and Hennrich, Markus and Schmidt-Kaler, Ferdinand},
	keywords = {Quantum information processing, Quantum simulation, Trapped ions, Rydberg states},
	abstract = {In this chapter, we present an overview of experiments with trapped Rydberg ions and outline the advantages and challenges of developing applications of this new platform for quantum computing, sensing, and simulation. Trapped Rydberg ions feature several important properties, unique in their combination: they are tightly bound in a harmonic potential of a Paul trap, in which their internal and external degrees of freedom can be controlled in a precise fashion. High fidelity state preparation of both internal and motional states of the ions has been demonstrated, and the internal states have been employed to store and manipulate qubit information. Furthermore, strong dipolar interactions can be realized between ions in Rydberg states and be explored for investigating correlated many body systems. By laser coupling to Rydberg states, the polarizability of the ions can be both enhanced and tuned. This can be used to control the interactions with the trapping fields in a Paul trap as well as dipolar interactions between the ions. Thus, trapped Rydberg ions present an attractive alternative for fast entangling operations as compared to those mediated by normal modes of trapped ions, which are advantageous for a future quantum computer or a quantum simulator.},
	booktitle = {Adv. In Atomic, Molecular, and Optical Phys.},
	volume = {69},
	pages = {233 -- 306},
	editor = {Dimauro, Louis F. and Perrin, H\'{e}l\`{e}ne and Yelin, Susanne F.},
	series = {},
	publisher = {Academic Press},
	year = {2020},
	issn = {1049-250X},
	doi = {10.1016/bs.aamop.2020.04.004},
	**url = {https://www.sciencedirect.com/science/article/pii/S1049250X20300045}**
}

@article{Pere20,
	title = {Electro-Optical Ion Trap for Experiments with Atom-Ion Quantum Hybrid Systems},
	author = {Perego, Elia and Duca, Lucia and Sias, Carlo},
	abstract = {In the development of atomic, molecular, and optical (AMO) physics, atom-ion hybrid systems are characterized by the presence of a new tool in the experimental AMO toolbox: atom-ion interactions. One of the main limitations in state-of-the-art atom-ion experiments is represented by the micromotion component of the ions’ dynamics in a Paul trap, as the presence of micromotion in atom-ion collisions results in a heating mechanism that prevents atom-ion mixtures from undergoing a coherent evolution. Here, we report the design and the simulation of a novel ion trapping setup especially conceived of for integration with an ultracold atoms experiment. The ion confinement is realized by using an electro-optical trap based on the combination of an optical and an electrostatic field, so that no micromotion component will be present in the ions’ dynamics. The confining optical field is generated by a deep optical lattice created at the crossing of a bow-tie cavity, while a static electric quadrupole ensures the ions’ confinement in the plane orthogonal to the optical lattice. The setup is also equipped with a Paul trap for cooling the ions produced by photoionization of a hot atomic beam, and the design of the two ion traps facilitates the swapping of the ions from the Paul trap to the electro-optical trap.},
	journal = {Appl. Sci.},
	volume = {10},
	number = {7},
	pages = {2222},
	numpages = {17},
	year = {2020},
	month = {Mar},
	date = {25},
	issn = {2076-3417},
	publisher = {MDPI},
	address = {Basel}, 
	doi = {10.3390/app10072222}
}

@Incollection{Wils20,
	author = {Wilson, E. and Holzer, B. J.},
	editor = {Myers, Stephen and Schopper, Herwig},
	title = {Beam Dynamics},
	bookTitle = {Particle Physics Reference Library : Volume 3: Accelerators and Colliders},
	year = {2020},
	month = {May},
	date = {15},
	publisher = {Springer International Publishing},
	address = {Cham},
	pages = {15--50},
	abstract = {Now let us look in detail at the motion of particles in the transverse coordinates of the coordinate system defined in Fig. 2.1.},
	isbn = {978-3-030-34245-6},
	e-isbn = {978-3-030-34245-6},
	doi = {10.1007/978-3-030-34245-6_2},
	**url = {https://doi.org/10.1007/978-3-030-34245-6_2}**
}

@article{News20, 
	title={Real-time Detection and Tandem Mass Spectrometry of Secondary Organic Aerosols with a Quadrupole Ion Trap},
	author={Newsome, G. Asher and Rosen, Elias P. and Kamens, Richard M. and Glish, Gary L.},
	abstract = {An aerosol quadrupole ion trap mass spectrometer is reported that is sensitive, has unique capabilities to perform chemical ionization, is operated in real-time, and is able to perform tandem mass spectrometry. The instrument samples particles with an aerodynamic lens and volatilizes them within the heated ion trap electrode assembly. Analyte molecules are ionized within the ion trap by proton transfer from reagent ions, and resultant fragmentation is reduced compared to vacuum UV photoionization. Particle concentrations can be detected linearly over two orders of magnitude and as low as 5 μg/m3. To demonstrate the real-time analysis capability of the instrument, secondary organic aerosol particles were produced by reaction of 100 ppb $\alpha$-pinene and 200 ppb ozone in an aerosol bag and observed in real-time to monitor the progress of the reaction. Pinic acid and pinonic acid are two of the many components of the secondary aerosol mixture that form and gradually decrease in concentration. Individual concentrations are calculated using pinic acid as an internal standard and vary from 4-36 ppb. The identities of analyte ions from both compounds are confirmed by tandem mass spectrometry in real-time.},
	journal={ChemRxiv},
	year={2020},
	month = {Jul},
	publisher={Cambridge Open Engage},
	place={Cambridge}, 
	DOI={10.26434/chemrxiv.12633836.v1}, 
} 

@article{Witt20,
	author = {Wittemer, Matthias and Schr\"{o}der, Jan-Philipp and Hakelberg, Frederick and Kiefer, Philip and Fey, Christian and Schuetzhold, Ralf and Warring, Ulrich and Schaetz, Tobias},
	title = {Trapped-ion toolkit for studies of quantum harmonic oscillators under extreme conditions},
	abstract = {Many phenomena described in relativistic quantum field theory are inaccessible to direct observations, but analogue processes studied under well-defined laboratory conditions can present an alternative perspective. Recently, we demonstrated an analogy of particle creation using an intrinsically robust motional mode of two trapped atomic ions. Here, we substantially extend our classical control techniques by implementing machine-learning strategies in our platform and, consequently, increase the accessible parameter regime. As a proof of methodology, we present experimental results of multiple quenches and parametric modulation of an unprotected motional mode of a single ion, demonstrating the increased level of real-time control. In combination with previous results, we enable future experiments that may yield entanglement generation using a process in analogy to Hawking radiation. This article is part of a discussion meeting issue ‘The next generation of analogue gravity experiments’.},
	journal = {Phil. Trans. R. Soc. A},
	volume = {378},
	number = {2177},
	pages = {20190230},
	numpages = {9},
	publisher = {Royal Society},
	address = {},
	year = {2020},
	month = {Jul},
	date = {},
	doi = {10.1098/rsta.2019.0230},
	**URL = {https://doi.org/10.1098/rsta.2019.0230}**
}

@Article{Notz20,
	AUTHOR = {N\"{o}tzold, Markus and Hassan, Saba Zia and Tauch, Jonas and Endres, Eric and Wester, Roland and Weidem\"{u}ller, Matthias},
	TITLE = {Thermometry in a Multipole Ion Trap},
	ABSTRACT = {We present a characterization of the ions&rsquo; translational energy distribution in a multipole ion trap. A linear mapping between the energy distribution of the trapped ions onto the ions&rsquo; time-of-flight (TOF) to a detector is demonstrated. For low ion temperatures, a deviation from linearity is observed and can be attributed to the emergence of multiple potential minima. The potential landscape of the trapped ions is modeled via the finite element method, also accounting for subtleties such as surface-charge accumulation. We demonstrate the validity of our thermometry method by simulating the energy distribution of the ion ensemble thermalized with buffer gas using a Molecular Dynamics (MD) simulation. A comparison between the energy distribution of trapped ions in different multipole trap configurations&mdash;i.e., with hyperbolic rods, cylindrical rods, and cylindrical wires&mdash;is provided. With these findings, one can map the temperature of the trapped ions down to the Kelvin regime using their TOF distributions. This enables future studies on sympathetic cooling and chemical reactions involving ions in multipole traps.},
	JOURNAL = {Appl. Sci.},
	VOLUME = {10},
	NUMBER = {15},
	ARTICLE-NUMBER = {5264},
	numpages = {17},
	YEAR = {2020},
	MONTH = {Jul},
	DATE = {30},
	ISSN = {2076-3417},
	PUBLISHER = {MDPI BASEL},
	DOI = {10.3390/app10155264},
	**URL = {https://www.mdpi.com/2076-3417/10/15/5264}**
}

@article{Rud20b,
	title = {Single-phase multipole radiofrequency trap},
	author = {Rudyi, S. S. and Vovk, T. A. and Kosternoi, I. A. and Rybin, V. V. and Rozhdestvensky, Yu. V.},
	abstract = {We present a theoretical basis and simple experimental realization of a multipole radio-frequency trap consisting of four equal cylindrical electrodes, where all of the bars have an in-phase applied voltage. An effective potential, which describes three additional stable quasi-equilibrium points, is obtained, and an electrostatic distribution is calculated using the method of image charges. We construct an experimental setup and localize a group of charged silicate microspheres at normal pressure. The experimental results agree well with the proposed analytical model. A strong dependence on modulation of the radio-frequency field and effective potential is confirmed.},
	journal = {AIP Adv.},
	volume = {10},
	number = {8},
	pages = {085016},
	numpages = {6},
	year = {2020},
	month = {Aug},
	date = {13},
	publisher = {AIP Publishing}, 
	doi = {10.1063/5.0013810},
	**url = {https://doi.org/10.1063/5.0013810}**
}

@article{Mehta20,
	author = {Mehta, Karan K. and Zhang, Chi and Malinowski, Maciej and Nguyen, Thanh-Long and Stadler, Martin and Home, Jonathan P.},
	title = {Integrated optical multi-ion quantum logic},
	abstract = {Practical and useful quantum information processing requires substantial improvements with respect to current systems, both in the error rates of basic operations and in scale. The fundamental qualities of individual trapped-ion1 qubits are promising for long-term systems2, but the optics involved in their precise control are a barrier to scaling3. Planar-fabricated optics integrated within ion-trap devices can make such systems simultaneously more robust and parallelizable, as suggested by previous work with single ions. Here we use scalable optics co-fabricated with a surface-electrode ion trap to achieve high-fidelity multi-ion quantum logic gates, which are often the limiting elements in building up the precise, large-scale entanglement that is essential to quantum computation. Light is efficiently delivered to a trap chip in a cryogenic environment via direct fibre coupling on multiple channels, eliminating the need for beam alignment into vacuum systems and cryostats and lending robustness to vibrations and beam-pointing drifts. This allows us to perform ground-state laser cooling of ion motion and to implement gates generating two-ion entangled states with fidelities greater than 99.3(2) per cent. This work demonstrates hardware that reduces noise and drifts in sensitive quantum logic, and simultaneously offers a route to practical parallelization for high-fidelity quantum processors. Similar devices may also find applications in atom- and ion-based quantum sensing and timekeeping.},
	journal = {Nature},
	volume = {586},
	number = {7830},
	pages = {533 -- 537},
	keywords = {},
	issn = {1476-4687},
	year = {2020},
	month = {Oct},
	date = {22},
	doi = {10.1038/s41586-020-2823-6},
	**url = {https://doi.org/10.1038/s41586-020-2823-6}**,
}

@article{Joshi20,
	title = {Polarization-gradient cooling of 1D and 2D ion Coulomb crystals},
	author = {Joshi, M. K. and Fabre, A. and Maier, C. and Brydges, T. and Kiesenhofer, D. and Hainzer, H. and Blatt, R. and Roos, C. F.},
	abstract = {We present experiments on polarization gradient cooling of Ca$^+$ multi-ion Coulomb crystals in a linear Paul trap. Polarization gradient cooling of the collective modes of motion whose eigenvectors have overlap with the symmetry axis of the trap is achieved by two counter-propagating laser beams with mutually orthogonal linear polarizations that are blue-detuned from the $S_{1/2} \leftrightarrow P_{1/2}$ transition. We demonstrate cooling of linear chains of up to 51 ions and 2D-crystals in zig-zag configuration with 22 ions. The cooling results are compared with numerical simulations and the predictions of a simple model of cooling in a moving polarization gradient.},
	journal = {New J. Phys.},
	volume = {22},
	number = {10},
	pages = {103013},
	numpages = {15},
	year = {2020},
	month = {Oct},
	publisher = {IOP Publishing},
	doi = {10.1088/1367-2630/abb912},
	**url = {https://doi.org/10.1088/1367-2630/abb912}**
}

@article{Schmi20b,
	title = {Mass-selective removal of ions from Paul traps using parametric excitation},
	author = {Schmidt, Julian and H{\"o}nig, Daniel and Weckesser, Pascal and Thielemann, Fabian and Schaetz, Tobias and Karpa, Leon},
	journal = {Appl. Phys.B},
	volume = {126},
	number = {},
	pages = {176},
	numpages = {7},
	year = {2020},
	month = {Oct},
	publisher = {Springer},
	doi = {10.1007/s00340-020-07491-8},
	**url = {https://doi.org/10.1007/s00340-020-07491-8}**,
	eprint = {atom-ph/2005.02675}   
}

@article{Tiba20a,
	author = {Tibaduiza, Daniel M. and Pires, Luis and Rego, Andreson L. C. and Szilard, Daniela and Zarro, Carlos and Farina, Carlos},
	title = {Efficient algebraic solution for a time-dependent quantum harmonic oscillator},
	journal = {Phys. Scr.},
	abstract = {Using operator ordering techniques based on Baker-Campbell-Hausdorff (BCH) relations of the su(1,1) Lie algebra and a time-splitting approach, we present an alternative method of solving the dynamics of a time-dependent quantum harmonic oscillator for any initial state. We find an iterative analytical solution given by simple recurrence relations that are very well suited for numerical calculations. We use our solution to reproduce and analyse some results from the literature to prove the usefulness of our method. We also discuss the efficiency in squeezing by comparing the parametric resonance modulation with the so-called Janszky-Adam scheme.},
	year = {2020},
	month = {Sep},
	publisher = {IOP Publishing},
	volume = {95},
	number = {10},
	pages = {105102},
	numpages = {12},
	doi = {10.1088/1402-4896/abb254},
  **url = {https://doi.org/10.1088/1402-4896/abb254}**
}

@article{Tiba20b,
	author = {Tibaduiza, Daniel M. and Pires, L. and Szilard, D. and Zarro, C. A. D. and Farina, C. and Rego, A. L. C.},
	title = {A Time-Dependent Harmonic Oscillator with Two Frequency Jumps: an Exact Algebraic Solution},
	abstract = {We consider a harmonic oscillator (HO) with a time-dependent frequency which undergoes two successive abrupt changes. By assumption, the HO starts in its fundamental state with frequency $$\omega_0$$, then, at $t = 0$, its frequency suddenly increases to $$\omega_1$$ and, after a finite time interval $$\tau$$, it comes back to its original value $$\omega_0$$. Contrary to what one could naively think, this problem is quite a non-trivial one. Using algebraic methods, we obtain its exact analytical solution and show that at any time t > 0 the HO is in a vacuum squeezed state. We compute explicitly the corresponding squeezing parameter (SP) relative to the initial state at an arbitrary instant and show that, surprisingly, it exhibits oscillations after the first frequency jump (from $$\omega_0$$ to $$\omega_1$$), remaining constant after the second jump (from $$\omega_1$$ back to $$\omega_0$$). We also compute the time evolution of the variance of a quadrature. Last, but not least, we calculate the vacuum (fundamental state) persistence probability amplitude of the HO, as well as its transition probability amplitude for any excited state.},
	journal = {Brazilian J. Phys.},
	volume = {50},
	number = {5},
	pages = {634 -- 646},
	keywords = {},
	issn = {1678-4448},
	year = {2020},
	month = {Oct},
	date = {01},
	doi = {10.1007/s13538-020-00770-x},
	**url = {https://doi.org/10.1007/s13538-020-00770-x}**
}

@article{Nord20,
	author = {Nordmann, T. and Didier, A. and Dole\v{z}al, M. and Balling, P. and Burgermeister, T. and Mehlst\"{a}ubler, T. E.},
	title = {Sub-kelvin temperature management in ion traps for optical clocks},
	abstract = {The uncertainty of the ac Stark shift due to thermal radiation represents a major contribution to the systematic uncertainty budget of state-of-the-art optical atomic clocks. In the case of optical clocks based on trapped ions, the thermal behavior of the rf-driven ion trap must be precisely known. This determination is even more difficult when scalable linear ion traps are used. Such traps enable a more advanced control of multiple ions and have become a platform for new applications in quantum metrology, simulation, and computation. Nevertheless, their complex structure makes it more difficult to precisely determine its temperature in operation and thus the related systematic uncertainty. We present here scalable linear ion traps for optical clocks, which exhibit very low temperature rise under operation. We use a finite-element model refined with experimental measurements to determine the thermal distribution in the ion trap and the temperature at the position of the ions. The trap temperature is investigated at different rf-drive frequencies and amplitudes with an infrared camera and integrated temperature sensors. We show that for typical trapping parameters for In+, Al+, Lu+, Ca+, Sr+, or Yb+ ions, the temperature rise at the position of the ions resulting from rf heating of the trap stays below 700 mK and can be controlled with an uncertainty on the order of a few 100 mK maximum. The corresponding uncertainty of the trap-related blackbody radiation shift is in the low $10^{-19}$ and even $10^{-20}$ regime for $^{171}$Yb$^+$(E3) and $^{115}$In$^+$, respectively.},
	journal = {Rev. Sci. Instrum.},
	volume = {91},
	number = {11},
	pages = {111301},
	numpages = {16},
	year = {2020},
	month = {Nov},
	issn = {0034-6748},
	doi = {10.1063/5.0024693},
	**url = {https://doi.org/10.1063/5.0024693},**
	eprint = {https://pubs.aip.org/aip/rsi/article-pdf/doi/10.1063/5.0024693/19765279/111301\_1\_online.pdf},
}

@article{Wan20,
	author = {Wan, Yong and J{\"o}rdens, Robert and Erickson, Stephen D. and Wu, Jenny J. and Bowler, Ryan and Tan, Ting Rei and Hou, Pan-Yu and Wineland, David J. and Wilson, Andrew C. and Leibfried, Dietrich},
	title = {Ion {T}ransport and {R}eordering in a 2{D T}rap {A}rray},
	abstract = {Scaling quantum information processors is a challenging task, requiring manipulation of a large number of qubits with high fidelity and a high degree of connectivity. For trapped ions, this can be realized in a 2D array of interconnected traps in which ions are separated, transported, and recombined to carry out quantum operations on small subsets of ions. Here, functionality of a junction connecting orthogonal linear segments in a 2D trap array to reorder a two-ion crystal is demonstrated. The secular motion of the ions experiences low energy gain and the internal qubit levels maintain coherence during the reordering process, therefore demonstrating a promising method for providing all-to-all connectivity in a large-scale, 2D or 3D trapped-ion quantum information processor.},
	journal = {Adv. Quantum Technol.},
	volume = {3},
	number = {11},
	pages = {2000028},
	publisher = {Wiley-VCH},
	address = {Weinheim},
	keywords = {ion reordering, ion transport, quantum information processing, trapped ions},
	year = {2020},
	month = {Nov},
	date = {},
	doi = {10.1002/qute.202000028},
	**url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/qute.202000028}**
}

@article{Fan21,
	title = {Optical Mass Spectrometry of Cold ${\mathrm{RaOH}}^{+}$ and ${{\mathrm{RaOCH}}_{3}}^{+}$},
	author = {Fan, M. and Holliman, C. A. and Shi, X. and Zhang, H. and Straus, M. W. and Li, X. and Buechele, S. W. and Jayich, A. M.},
	abstract = {We present an all-optical mass spectrometry technique to identify trapped ions. The new method uses laser-cooled ions to determine the mass of a cotrapped dark ion with a sub-dalton resolution within a few seconds. We apply the method to identify the first controlled synthesis of cold, trapped RaOH$^+$ and RaOCH$_3^+$. These molecules are promising for their sensitivity to time and parity violations that could constrain sources of new physics beyond the standard model. The nondestructive nature of the mass spectrometry technique may help identify molecular ions or highly charged ions prior to optical spectroscopy. Unlike previous mass spectrometry techniques for small ion crystals that rely on scanning, the method uses a Fourier transform that is inherently broadband and comparatively fast. The technique’s speed provides new opportunities for studying state-resolved chemical reactions in ion traps.},
	journal = {Phys. Rev. Lett.},
	volume = {126},
	issue = {2},
	pages = {023002},
	numpages = {5},
	year = {2021},
	month = {Jan},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevLett.126.023002},
	url = {https://link.aps.org/doi/10.1103/PhysRevLett.126.023002}
}

@article{Zhu21,
	title = {Direct Observation of Ion Micromotion in a Linear Paul Trap},
	author = {Zhukas, Liudmila A. and Millican, Maverick J. and Svihra, Peter and Nomerotski, Andrei and Blinov, Boris B.},
	abstract = {In this paper, direct observation of micromotion for multiple ions in a laser-cooled trapped ion crystal is discussed along with a measurement technique for micromotion amplitude. Micromotion is directly observed using a time-resolving, single-photon-sensitive camera that provides both fluorescence and position data for each ion on the nanosecond time scale. Micromotion amplitude and phase for each ion in the crystal are measured, allowing this method to be sensitive to tilts and shifts of the ion chain from the null of the radio-frequency quadrupole potential in the linear trap. Spatial resolution makes this micromotion detection technique suitable for complex ion configurations, including two-dimensional geometries. It does not require any additional equipment or laser beams, and the modulation of the cooling lasers or trap voltages is not necessary for detection as it is in other methods.},
	journal = {Phys. Rev. A},
	volume = {103},
	number = {2},
	pages = {023105},
	numpages = {6},
	year = {2021},
	month = {Feb},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.103.023105},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.103.023105}**
}

@article{Mih21,
	title = {Investigations on Dynamical Stability in 3D Quadrupole Ion Traps},
	author = {Mihalcea, Bogdan M. and Lynch, Stephen},
	abstract = {We firstly discuss classical stability for a dynamical system of two ions levitated in a 3D Radio-Frequency (RF) trap, assimilated with two coupled oscillators. We obtain the solutions of the coupled system of equations that characterizes the associated dynamics. In addition, we supply the modes of oscillation and demonstrate the weakcoupling condition is inappropriate in practice, while for collective modes of motion (and strong coupling) only a peak of the mass can be detected. Phase portraits and power spectra are employed to illustrate how the trajectory executes quasiperiodic motion on the surface of torus, namely a Kolmogorov–Arnold–Moser (KAM) torus. In an attempt to better describe dynamical stability of the system, we introduce a model that characterizes dynamical stability and the critical points based on the Hessian matrix approach. The model is then applied to investigate quantum dynamics for many-body systems consisting of identical ions, levitated in 2D and 3D ion traps. Finally, the same model is applied to the case of a combined 3D Quadrupole Ion Trap (QIT) with axial symmetry, for which we obtain the associated Hamilton function. The ion distribution can be described by means of numerical modeling, based on the Hamilton function we assign to the system. The approach we introduce is effective to infer the parameters of distinct types of traps by applying a unitary and coherent method, and especially for identifying equilibrium configurations, of large interest for ion crystals or quantum logic.},
	journal = {Appl. Sci.},
	volume = {11},
	number = {7},
	pages = {2938},
	numpages = {29},
	year = {2021},
	month = {Mar},
	date = {25},
	publisher = {MDPI Basel},
	doi = {10.3390/app11072938},
	**url = {https://www.mdpi.com/2076-3417/11/7/2938/pdf}**
}

@article{Marche21,
	author = {Marchenay, Marylise and Pedregosa-Gutierrez, Jofre and Knoop, Martina and Houssin, Marie and Champenois, Caroline},
	title = {An analytical approach to symmetry breaking in multipole RF-traps},
	abstract = {Radio-frequency (RF) linear multipole traps have been shown to be very sensitive to mis-positioning of their electrodes, which results in a symmetry breaking and leads to extra local minima in the trapping potential as shown in Pedregosa-Gutierrez et al (2018 J. Mod. Opt. 65 529) disturbing the operation of the trap. In this work, we analytically describe the RF-potential of a realistic octupole trap by including lower order terms to the well-established equation for a perfectly symmetric octupole trap. We describe the geometry by a combination of identified defects, characterized by simple analytical expressions. A complete equation is proposed for a trap with any electrode deviation relying on a combination of the simple cases where the defects are taken individually. Our approach is validated by comparison between analytical and numerical results for defect sizes up to 4% of the trap radius. As described in Pedregosa-Gutierrez et al (2018 Rev. Sci. Instrum. 89 123101), an independent fine-tuning of the amplitude of the RF voltage applied on each electrode can be used to mitigate the geometrical defects of a realistic trap. In a different way than in Pedregosa-Gutierrez et al (2018 Rev. Sci. Instrum. 89 123101), the knowledge of an analytical equation for the potential allows to design the set of RF-voltages required for this compensation, based on the experimental measurement of the ion positions in the trap, without information concerning the exact position of each electrode, and with a small number of iterations. The requirements, performances and limitations of this protocol are discussed via comparison of numerical simulations and analytical results.},
	journal = {Quantum Sci. Technol.},
	volume = {6},
	number = {2},
	pages = {024016},
	numpages = {},
	year = {2021},
	month = {Mar},
	date = {31},
	publisher = {IOP Publishing},
	doi = {10.1088/2058-9565/abeaf6},
	url = {https://doi.org/10.1088/2058-9565/abeaf6}
}

@book{LaPie21,
	title = {Introduction to Quantum Computing},
	author = {LaPierre, Ray}, 
	editor = {},
	series = {The Materials Research Society Series},
	volume = {},
	edition = {},
	publisher = {Springer},
	address = {Cham},
	year =  {2021},
	month =  {},
	ISSN = {2730-7360},
	e-ISSN = {2730-7379},
	ISBN = {978-3-030-69317-6}, 
	e-ISBN = {978-3-030-69318-3},  
	doi = {10.1007/978-3-030-69318-3}, 
	**url = {https://doi.org/10.1007/978-3-030-69318-3}**
}

@article{Kel21,
	title = {Quantum Harmonic Oscillator Spectrum Analyzers},
	author = {Keller, Jonas and Hou, Pan-Yu and McCormick, Katherine C. and Cole, Daniel C. and Erickson, Stephen D. and Wu, Jenny J. and Wilson, Andrew C. and Leibfried, Dietrich},
	abstract = {Characterization and suppression of noise are essential for the control of harmonic oscillators in the quantum regime. We measure the noise spectrum of a quantum harmonic oscillator from low frequency to near the oscillator resonance by sensing its response to amplitude modulated periodic drives with a qubit. Using the motion of a trapped ion, we experimentally demonstrate two different implementations with combined sensitivity to noise from 500 Hz to 600 kHz. We apply our method to measure the intrinsic noise spectrum of an ion trap potential in a previously unaccessed frequency range.},
	journal = {Phys. Rev. Lett.},
	volume = {126},
	number = {25},
	pages = {250507},
	numpages = {6},
	year = {2021},
	month = {Jun},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevLett.126.250507},
	**url = {https://link.aps.org/doi/10.1103/PhysRevLett.126.250507}**
}

@article{Nira21,
	title = {Analysis of Multipolar Linear Paul Traps for Ion–Atom Ultracold Collision Experiments},
	author = {Niranjan, M. and Prakash, Anand and Rangwala, Sadiqali A.},
	abstract = {We evaluate the performance of multipole, linear Paul traps for the purpose of studying cold ion–atom collisions. A combination of numerical simulations and analysis based on the virial theorem is used to draw conclusions on the differences that result, by considering the trapping details of several multipole trap types. Starting with an analysis of how a low energy collision takes place between a fully compensated, ultracold trapped ion and an stationary atom, we show that a higher order multipole trap is, in principle, advantageous in terms of collisional heating. The virial analysis of multipole traps then follows, along with the computation of trapped ion trajectories in the quadrupole, hexapole, octopole and do-decapole radio frequency traps. A detailed analysis of the motion of trapped ions as a function of the amplitude, phase and stability of the ion’s motion is used to evaluate the experimental prospects for such traps. The present analysis has the virtue of providing definitive answers for the merits of the various configurations, using first principles.},
	journal = {Atoms},
	volume = {9},
	number = {3},
	pages = {38},
	numpages = {14},
	year = {2021},
	month = {Jun},
	publisher = {MDPI},
	doi = {10.3390/atoms9030038},
	**url = {https://doi.org/10.3390/atoms9030038}**
}

@article{Syr21b,
	title = {Cleaning dielectric surfaces by the electrical fields of the linear electrodynamic Paul trap},
	author = {R. A. Syrovatka and V. S. Filinov and L. M. Vasilyak and V. Ya.  Pecherkin and L. V. Deputatova and V. I. Vladimirov and O. S. Popel and A. B. Tarasenko},
	keywords = {Solar energy, Solar panels, Dust, Surface pollution, Surface cleaning, Electrodynamic trap},
	abstract = {A method against dust pollution of dielectric surfaces has been proposed. The dust removal is achieved applying alternating electric fields of special geometry created by the linear electrodynamic Paul traps, which polarize dust particles on the surface of the dielectric and draw them into the interelectrode space of the trap. The captured dust can be moved without contact with the solar panel surface toward the electrodes ends in a special container by an additional constant electric field. The feasibility of the approach was demonstrated by a series of experiments on the removal of aluminium oxide and sand particles of various sizes from a glass surface and solar panels. The cleaning rate for 70/90 $\mu$m particles uniformly scattered on the glass substrate inclined at 20$\degree$ was 92 \%. The capture of a single particle was analyzed using numerical simulations and the conditions necessary for its capturing and levitating were determined.},
	journal = {J. Electrostat.},
	volume = {112},
	pages = {103583},
	year = {2021},
	month = {Jul},
	issn = {0304-3886},
	doi = {10.1016/j.elstat.2021.103583},
	**url = {https://www.sciencedirect.com/science/article/pii/S0304388621000334}**
}

@inproceedings{Baro21,
	author = {Barontini, G. and Boyer, V. and Calmet, X. and Fitch, N. J. and Forgan, E. M. and Godun, R. M. and Goldwin, J. and Guarrera, V. and Hill, I. R. and Jeong, M. and Keller, M. and Kuipers, F. and Margolis, H. S. and Newman, P. and Prokhorov, L. and Rodewald, J. and Sauer, B. E. and Schioppo, M. and Sherrill, N. and Tarbutt, M. R. and Vecchio, A. and Worm, S.},
	title = {QSNET, a network of clock for measuring the stability of fundamental constants},
	abstract = {The QSNET consortium is building a UK network of next-generation atomic and molecular clocks that will achieve unprecedented sensitivity in testing variations of the fine structure constant, $\alpha$, and the electron-to-proton mass ratio, $\mu$. This in turn will provide more stringent constraints on a wide range of fundamental and phenomenological theories beyond the Standard Model and on dark matter models.},
	keywords = {network of atomic and molecular clocks, atomic, molecular and highly-charged ion clocks, networked quantum sensors, variations of fundamental constants, quantum technology for fundamental physics, multi-messenger detection, ultra-light dark matter, physics beyond the standard model},
	volume = {11881},
	booktitle = {Quantum Technology: Driving Commercialisation of an Enabling Science II},
	editor = {Padgett, Miles J. and Bongs, Kai and Fedrizzi, Alessandro and Politi, Alberto},
	organization = {International Society for Optics and Photonics},
	publisher = {SPIE},
	pages = {63 -- 66},
	year = {2021},
	month = {Oct},
	doi = {10.1117/12.2600493},
	**URL = {https://doi.org/10.1117/12.2600493}**
}

@article{Chak21,
	title = {Role of dissipation in the stability of a parametrically driven quantum harmonic oscillator},
	author = {Chaki, Subhasish and Bhattacherjee, Aranyabhuti},
	abstract = {We study the dissipative dynamics of a single quantum harmonic oscillator subjected to parametric driving within an effective Hamiltonian approach. Using the Liouville–von Neumann approach, we show that the time evolution of a parametrically driven dissipative quantum oscillator has a strong connection with the classical, damped Mathieu equation. Based on the numerical analysis of the Monodromy matrix, we demonstrate that the dynamical instability generated by the parametric driving is reduced by the effect of dissipation. Furthermore, we obtain a close relationship between the localization of the Wigner function and the stability of the damped Mathieu equation.},
	journal = { J. Korean Phy. Soc.},
	volume = {79},
	number = {7},
	pages = {600 -- 605},
	numpages = {0},
	year = {2021},
	month = {Oct},
	date = {01},
	publisher = {Korean Physical Society},
	issn = {1976-8524},
	doi = {10.1007/s40042-021-00260-6},
	**url = {https://doi.org/10.1007/s40042-021-00260-6}**
}

@article{Neug21,
	title = {Probability distributions in quadrupole ion traps},
	author = {Thomas S. Neugebauer and Thomas Drewello},
	keywords = {Probability distributions, Mathieu's equation, Hill's equation, Quadrupole ion trap, Phase-space ellipses, twiss parameters},
	abstract = {The solution to Mathieu's equation was modified and combined to only one single sine function. In this form, the cumulative probability functions for a fixed RF-phase can be determined as arcsine distributions. The probability of a normalized position u/umax or normalized velocity u˙/u˙max is relative to 1/1−(u/umax)2, u˙/u˙max respectively. This corrects reference descriptions present in the literature, which stated the probability to be proportional to 1−(u/umax)2. Resulting probability plots show that ions that are heavy on the relative mass scale of quadrupole ion traps, are more likely to be found close to the maximum of their oscillation amplitude. Light ions are more likely to be found at the center of their oscillation. The velocity distributions show that the likeliest velocity converges for low $q$-values to the mean-square velocity but splits into two likely velocity regions with increasing $q$-value. It is further emphasized, that these distributions describe the probability of a single ion and in order to describe the behavior of an ensemble of ions, it is inevitably needed to define a distribution of oscillation amplitudes $u_{max}$.},
	journal = {Int. J. Mass Spectrom.},
	volume = {468},
	pages = {116641},
	year = {2021},
	month = {Oct},
	issn = {1387-3806},
	doi = {10.1016/j.ijms.2021.116641},
  **url = {https://www.sciencedirect.com/science/article/pii/S1387380621001214}**
}

@article{Vasil21,
	author = {Vasilyev, Maxim and Rudyi, Semyon and Rozhdestvensky, Yuri},
	title ={Theoretical description of electric fields in three-dimensional multipole ion traps},
	abstract = {In this paper, the principle of forming the spatial distribution of the potential in multipole three-dimensional ion traps of a general type is considered. A matrix method for describing the electric fields in ion traps for the $n$th order of multipole is proposed. Typical electrode geometries for hexapole and octupole traps are considered.},
	journal = {Eur. J. Mass Spectrom.},
	volume = {27},
	number = {5},
	pages = {158 -- 165},
	year = {2021},
	month = {Oct},
	date = {28},
	doi = {10.1177/14690667211047918},
	**	note ={PMID: 34709079},**
	**URL = {https://doi.org/10.1177/14690667211047918}**
}

@article{Cho21,
	title = {Application of Single-Particle Mass Spectrometer to Obtain Chemical Signatures of Various Combustion Aerosols},
	author = {{Hee-joo} Cho and Joonwoo Kim and Nohhyeon Kwak and Heesung Kwak and Taewan Son and Donggeun Lee and Kihong Park},
	journal = {Int. J. Environ. Res. Pub. Health},
	volume = {18},
	number = {21},
	pages = {11580},
	numpages = {12},
	year = {2021},
	month = {Nov},
	date = {},
	publisher = {MDPI},
	address = {Basel},
	doi = {10.3390/ijerph182111580},
	**url = {https://doi.org/10.3390/ijerph182111580}**
}

@Article{Gonz21,
	author = {Gonzalez, L. Edwin and Szalwinski, Lucas J. and Marsh, Brett M. and Wells, J. Mitchell and Cooks, R. Graham},
	title  = {Immediate and sensitive detection of sporulated Bacillus subtilis by microwave release and tandem mass spectrometry of dipicolinic acid},
	abstract  = {Spore lysis of Bacillus species is achieved by brief (1 min) microwave irradiation while tandem mass spectrometry (MS/MS) allows identification of the characteristic spore marker, dipicolinic acid. This rapid measurement, made on 105–108 spores, has significant implications for biothreat recognition.},
	journal  = {Analyst},
	year  = {2021},
	month = {Dec},
	date = {07},
	volume  = {146},
	number  = {23},
	pages  = {7104 -- 7108},
	publisher  = {The Royal Society of Chemistry},
	doi  = {10.1039/D1AN01796A},
	**url  = {https://doi.org/10.1039/D1AN01796A}**
}

@article{Wolf21,
	title = {Quantum sensing of oscillating electric fields with trapped ions},
	author = {Wolf, Fabian and Schmidt, Piet O.},
	abstract = {Quantum noise is a fundamental limitation for quantum sensors and results in the so-called shot-noise limit. Nowadays, several systems such as optical clocks or gravitational wave detectors approach measurement sensitivities where this limitation poses a major contribution to the total statistical uncertainty. It is known that this limit can be overcome by preparing the probe in a non-classical state. We will give an overview over the different non-classical states that have been implemented in the motion of single trapped ions and discuss their individual advantages and limitations in metrology. Possible applications for the presented experiments are the measurement of small oscillating electric fields and trapping frequencies. The Focus will be on our experimental work on Fock states, where quantum-enhanced sensing in both scenarios is possible with the same quantum state.},
	journal = {Measurement: Sensors},
	volume = {18},
	pages = {100271},
	year = {2021},
	month = {Dec},
	issn = {2665-9174},
	doi = {10.1016/j.measen.2021.100271},
	**url = {https://www.sciencedirect.com/science/article/pii/S2665917421002348}**
}

@article{Hass22,
	title = {Quantum dynamics for general time-dependent three coupled oscillators based on an exact decoupling},
	author = {Hassoul, Sara and Menouar, Salah and Benseridi, Hamid and Choi, Jeong Ryeol},
	keywords = {Time-dependent three coupled oscillators, Unitary transformation, Euler angles, Hamiltonian, Wave function},
	abstract = {Quantum dynamics of general time-dependent three coupled oscillators is investigated through an alternative approach based on decoupling of them using the unitary transformation method. From a first unitary transformation, the quantal Hamiltonian of the complicated original system is transformed to an equal but a simple one associated with the three coupled oscillators of which masses are unity. To diagonalize the transformed Hamiltonian, we transform the Hamiltonian once again by introducing a new unitary operator. This transformation corresponds to a three-dimensional rotation parameterized by Euler angles. Through these procedures, the coupled oscillatory subsystems are completely decoupled. The importance of this decouplement is that it enables us to develop exact theory for mechanical treatment of the originally-coupled systems without any restriction in the form of time-varying parameters.},
	journal = {Phys. A: Stat. Mech. Appl.},
	volume = {604},
	pages = {127755},
	year = {2022},
	month = {Jun},
	issn = {0378-4371},
	doi = {10.1016/j.physa.2022.127755},
	**url = {https://www.sciencedirect.com/science/article/pii/S0378437122005039}**
}

@book{Kais22,
	title   = {Ultra-Cold Atoms, Ions, Molecules and Quantum Technologies},
	author  = {},
	editor  = {Kaiser, Robin and Leduc, Mich\`{e}le and Perrin, H\'{e}l\`{e}ne},
	series = {},
	publisher = {EDP Sciences},
	address   = {91944 Les Ulis Cedex A, France},
	pages   = {},
	year    = {2022},
	month = {},
	date = {},
	isbn = {978-2-7598-2745-9},
	e-isbn = {978-2-7598-2746-6},
	doi = {10.1051/978-2-7598-2745-9},
	**url = {}**
}

@article{Kaur22,
	author = {Kaur Kohli, Ravleen and Van Berkel, Gary J. and Davies, James F.},
	title = {An Open Port Sampling Interface for the Chemical Characterization of Levitated Microparticles},
	abstract = {Several studies have reported ionization methods to classify the chemical composition of levitated particles held in an electrodynamic balance using mass spectrometry (MS). These methods include electrospray-based paper spray (PS) ionization, plasma discharge ionization, and direct analysis in real-time (DART) ionization, with each showing advantages and disadvantages. Our recent work demonstrated that PS ionization could yield accurate data for the chemical evolution of mixed component particles undergoing evaporation. However, measurements were performed using an internal standard to account for and correct the inherent variability in the PS ionization source. Here, we explore a new electrospray-based method coupled to particle levitation–the Open Port Sampling Interface (OPSI), which provides many advantages over the PS method, with few disadvantages. In this application note we report experiments in which micron-sized particles, containing analytes such as citric acid, maleic acid, and tetraethylene glycol, were levitated and optically probed to determine their size and mass. Subsequent transfer of individual levitated particles into the OPSI allowed for the ionization and mass spectrometry analysis of these particles. We discuss the stability and reproducibility of MS measurements, demonstrate effective quantitation in both positive and negative mode, and determine the sensitivity of the OPSI to a range of analyte mass present in levitated particles. Importantly, we show stability of the OPSI over $> 6$ h without the need for normalizing signal variations with an internal standard in the sample, demonstrating robust application of the OPSI to measurements over extended periods of time.},
	journal = {Anal. Chem.},
	volume = {94},
	number = {8},
	pages = {3441 -- 3445},
	year  = {2022},
	month = {Feb},
	date = {15},
	publisher = {American Chemical Society},
	issn = {0003-2700},
	doi = {10.1021/acs.analchem.1c05550},
	**URL = {https://doi.org/10.1021/acs.analchem.1c05550}**,
}  

@article{Kato22,
	title = {Two-tone Doppler cooling of radial two-dimensional crystals in a radio-frequency ion trap},
	author = {Kato, Alexander and Goel, Apurva and Lee, Raymond and Ye, Zeyu and Karki, Samip and Liu, Jian Jun and Nomerotski, Andrei and Blinov, Boris B.},
	abstract = {We study the Doppler-cooling of radial two-dimensional (2D) Coulomb crystals of trapped barium ions in a radiofrequency trap. Ions in radial 2D crystals experience micromotion of an amplitude that increases linearly with the distance from the trap center, leading to a position-dependent frequency modulation of laser light in each ion's rest frame. We use two tones of Doppler-cooling laser light separated by approximately 100 MHz to efficiently cool distinct regions in the crystals with differing amplitudes of micromotion. This technique allows us to trap and cool more than 50 ions populating four shells in a radial 2D crystal, where with a single tone of Doppler cooling light we are limited to 30 ions in three shells. We also individually characterize the micromotion of all ions within the crystals and use this information to locate the center of the trap and to determine the Matthieu parameters $q_x$ and $q_y$.},
	journal = {Phys. Rev. A},
	volume = {105},
	number = {2},
	pages = {023101},
	numpages = {8},
	year = {2022},
	month = {Feb},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.105.023101},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.105.023101}**
}

@article{Spiv22,
	title = {High-Stability Cryogenic System for Quantum Computing With Compact Packaged Ion Traps},
	author = {Spivey, Robert Fulton and Inlek, Ismail Volkan and Jia, Zhubing and Crain, Stephen and Sun, Ke and Kim, Junki and Vrijsen, Geert and Fang, Chao and Fitzgerald, Colin and Kross, Steffen and Noel, Tom and Kim, Jungsang},
	abstract = {Cryogenic environments benefit ion trapping experiments by offering lower motional heating rates, collision energies, and an ultrahigh vacuum (UHV) environment for maintaining long ion chains for extended periods of time. Mechanical vibrations caused by compressors in closed-cycle cryostats can introduce relative motion between the ion and the wavefronts of lasers used to manipulate the ions. Here, we present a novel ion trapping system where a commercial low-vibration closed-cycle cryostat is used in a custom monolithic enclosure. We measure mechanical vibrations of the sample stage using an optical interferometer, and observe a root-mean-square relative displacement of 2.4 nm and a peak-to-peak displacement of 17 nm between free-space beams and the trapping location. We packaged a surface ion trap in a cryopackage assembly that enables easy handling while creating a UHV environment for the ions. The trap cryopackage contains activated carbon getter material for enhanced sorption pumping near the trapping location, and source material for ablation loading. Using $^{171}$Yb$^+$ as our ion, we estimate the operating pressure of the trap as a function of package temperature using phase transitions of zig-zag ion chains as a probe. We measured the radial mode heating rate of a single ion to be 13 quanta/s on average. The Ramsey coherence measurements yield 330-ms coherence time for counter-propagating Raman carrier transitions using a 355-nm mode-locked pulse laser, demonstrating the high optical stability.},
	journal = {IEEE Trans. Quant. Eng.},
	volume = {3},
	number = {},
	pages = {1 -- 11},
	numpages = {0},
	year = {2022},
	month = {Feb},
	publisher = {IEEE},
	e-issn = {2689-1808},
	doi = {10.1109/TQE.2021.3125926},
	**url = {https://ieeexplore.ieee.org/document/9606562}**
}

@article{Suth22,
	title = {One- and two-qubit gate infidelities due to motional errors in trapped ions and electrons},
	author = {Sutherland, R. Tyler and Yu, Qian and Beck, Kristin M. and H\"affner, Hartmut},
	abstract = {In this work, we derive analytic formulas that determine the effect of error mechanisms on one- and two-qubit gates in trapped ions and electrons. First, we analyze and derive expressions for the effect of driving field inhomogeneities on one-qubit gate fidelities. Second, we derive expressions for two-qubit gate errors, including static motional frequency shifts, trap anharmonicities, field inhomogeneities, heating, and motional dephasing. We show that, for small errors, each of our expressions for infidelity converges to its respective numerical simulation; this shows that our formulas are sufficient for determining error budgets for high-fidelity gates, obviating numerical simulations in future projects. All of the derivations are general to any internal qubit state, and any {\emph {mixed}} state of the ion crystal's motion that is diagonal in the Fock state basis. Our treatment of static motional frequency shifts, trap anharmonicities, heating, and motional dephasing apply to both laser-based and laser-free gates, while our treatment of field inhomogeneities applies to laser-free systems.},
	journal = {Phys. Rev. A},
	volume = {105},
	number = {2},
	pages = {022437},
	numpages = {13},
	year = {2022},
	month = {Feb},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.105.022437},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.105.022437}**
}

@inproceedings{Pat22,
	title = {Environmental Analysis of the Bounded Lunar Exosphere (ENABLE): Lessons in Gas Sources from Apollos 11 to 17},
	author = {Patrick, E. L. and Blase, R. C. and Libardoni, M. J. and Poston, M. J.},
	booktitle = {Proc. of the 53rd Lunar and Planetary Science Conference (LPSC)},     
	series = {},
	volume = {},
	edition = {},
	editor = {},
	pages = {},
	year = {2022},
	month = {Mar},
	date = {7 -- 11},
	publisher = {},
	address = {}, 
	organization = {Lunar and Planetary Institute \& NASA},
	url = {https://www.hou.usra.edu/meetings/lpsc2022/pdf/2731.pdf}
}

@article{Huang22,
	title = {Liquid-Nitrogen-Cooled $\mathrm{Ca}{}^{+}$ Optical Clock with Systematic Uncertainty of $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}$},
	author = {Huang, Yao and Zhang, Baolin and Zeng, Mengyan and Hao, Yanmei and Ma, Zixiao and Zhang, Huaqing and Guan, Hua and Chen, Zheng and Wang, Miao and Gao, Kelin},
	abstract = {We present a liquid-nitrogen-cooled Ca$^+$ optical clock with an overall systematic uncertainty of $3.0 \times 10^{−18}$. In contrast to the room-temperature Ca$^+$ optical clock that we have reported previously, the cryogenic black-body radiation (BBR) shield in vacuum is cooled to $82 \pm 5$ K using liquid nitrogen. We also implement an ion trap with a reduced heating rate and improved laser cooling. This allows the ion temperature to fall to the Doppler-cooling limit during the clock operation and the systematic uncertainty associated with the secular (thermal) motion of the ion is reduced to $ < 1 \times 10^{−18}$. The uncertainty arising from the probe laser light shift and the servo error is also reduced to $< 1 \times 10^{−19}$ and $4 \times 10^{−19}$ with the hyper-Ramsey method and the higher-order servo algorithm, respectively. By comparing the output frequency of the cryogenic clock to that of a room-temperature clock, the differential BBR shift between the two is determined with a fractional statistical uncertainty of $7 \times 10^{-18}$. The differential BBR shift is used to calculate the static differential polarizability and the result is found to be in excellent agreement with our previous measurement using a different method. This work suggests that the BBR shift of optical clocks can be suppressed well in a liquid-nitrogen environment. Systems similar to what is presented here can also be used to suppress the BBR shift significantly in other types of optical clocks, such as Yb$^+$, Sr$^+$, Yb, Sr, etc.},
	journal = {Phys. Rev. Appl.},
	volume = {17},
	number = {3},
	pages = {034041},
	numpages = {11},
	year = {2022},
	month = {Mar},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevApplied.17.034041},
	**url = {https://link.aps.org/doi/10.1103/PhysRevApplied.17.034041}**
}

@article{Colo22,
	author = {Colombo, Simone and Pedrozo-Pe\~{n}afiel, Edwin and Adiyatullin, Albert F. and Li, Zeyang and Mendez, Enrique and Shu, Chi and Vuleti\'{c}, Vladan},
	title = {Time-reversal-based quantum metrology with many-body entangled states},
	abstract = {Linear quantum measurements with independent particles are bounded by the standard quantum limit, which limits the precision achievable in estimating unknown phase parameters. The standard quantum limit can be overcome by entangling the particles, but the sensitivity is often limited by the final state readout, especially for complex entangled many-body states with non-Gaussian probability distributions. Here, by implementing an effective time-reversal protocol in an optically engineered many-body spin Hamiltonian, we demonstrate a quantum measurement with non-Gaussian states with performance beyond the limit of the readout scheme. This signal amplification through a time-reversed interaction achieves the greatest phase sensitivity improvement beyond the standard quantum limit demonstrated to date in any full Ramsey interferometer. These results open the field of robust time-reversal-based measurement protocols offering precision not too far from the Heisenberg limit. Potential applications include quantum sensors that operate at finite bandwidth, and the principle we demonstrate may also advance areas such as quantum engineering, quantum measurements and the search for new physics using optical-transition atomic clocks.},
	journal = {Nat. Phys.},
	volume = {18},
	number = {8},
	pages = {925 -- 930},
	numpages = {0},
	year = {2022},
	month = {Jul},
	date = {14},
	publisher = {Nature},
	issn = {1745-2481},
	doi = {10.1038/s41567-022-01653-5},
	**url = {https://doi.org/10.1038/s41567-022-01653-5}**
}

@article{Dere22,
	author = {Derevianko, Andrei and Gibble, Kurt and Hollberg, Leo and Newbury, Nathan R. and Oates, Chris and Safronova, Marianna S. and Sinclair, Laura C. and Yu, Nan},
	title = {Fundamental physics with a state-of-the-art optical clock in space},
	journal = {Quantum Sci. Technol.},
	abstract = {Recent advances in optical atomic clocks and optical time transfer have enabled new possibilities in precision metrology for both tests of fundamental physics and timing applications. Here we describe a space mission concept that would place a state-of-the-art optical atomic clock in an eccentric orbit around Earth. A high stability laser link would connect the relative time, range, and velocity of the orbiting spacecraft to earthbound stations. The primary goal for this mission would be to test the gravitational redshift, a classical test of general relativity, with a sensitivity 30 000 times beyond current limits. Additional science objectives include other tests of relativity, enhanced searches for dark matter and drifts in fundamental constants, and establishing a high accuracy international time/geodesic reference.},
	volume = {7},
	number = {4},
	pages = {044002},
	numpages = {},
	year = {2022},
	month = {Jul},
	publisher = {IOP Publishing},
	doi = {10.1088/2058-9565/ac7df9},
	url = {https://doi.org/10.1088/2058-9565/ac7df9},
}

@article{Mih22,
	title = {Quasienergy operators and generalized squeezed states for systems of trapped ions},
	author = {Mihalcea, Bogdan M.},
	abstract = {Collective many-body dynamics for time-dependent quantum Hamiltonian functions is investigated for a dynamical system that exhibits multiple degrees of freedom, in this case a combined (Paul and Penning) trap. Quantum stability is characterized by a discrete quasienergy spectrum, while the quasienergy states are symplectic coherent states. We introduce the generators of the Lie algebra of the symplectic group ${\mathcal {SL}}(2, \mathbb R)$, which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO) to which we associate the quantum Hamilton function. We obtain the kinetic and potential energy operators as functions of the Lie algebra generators and supply the expressions for the classical coordinate, momentum, kinetic and potential energy, along with the total energy. Moreover, we also infer the dispersions for the coordinate and momentum, together with the asymmetry and the flatness parameter for the distribution. The system interaction with laser radiation is also examined for a system of identical two-level atoms. The Hamilton function for the Dicke model is derived. The optical system is modelled as a HO (trapped ion) that undergoes interaction with an external laser field and we use it to engineer a squeezed state of the electromagnetic (EM) field. We consider coherent and squeezed states associated to both ion dynamics and to the EM field. The approach used enables one to build CS in a compact and smart manner by use of the group theory.},
	journal = {Ann. Phys. (N. Y.)},
	volume = {442},
	number = {},
	pages = {169826},
	numpages = {26},
	year = {2022},
	month = {Jul},
	date = {},
	publisher = {Elsevier},
	issn = {0003-4916},
	doi = {10.1016/j.aop.2022.168926},
	**url = {https://www.sciencedirect.com/science/article/pii/S0003491622001038}**
}

@article{Fakk22,
	title = {Classical and quantum integrability of the three-dimensional generalized trapped ion Hamiltonian},
	author = {{El Fakkousy}, Idriss and Zouhairi, Bouchta and Benmalek, Mohammed and Kharbach, Jaouad and Rezzouk, Abdellah and Ouazzani-Jamil, Mohammed},
	keywords = {Painlevé analysis, Dynamic system, Hamiltonian system, Trapped ion, Three-dimensional integrability, Classical and quantum integrability},
	abstract = {In this paper, we study the trapped ion Hamiltonian in three-dimensional (3D) with the generalized potential in the quadrupole field with superposition of the hexapole and octopole fields. We determine new integrable cases by using the Painlevé analysis and find the second and third classical invariants for each P-case. Moreover, we perturb this Hamiltonian by an inverse square potential and we prove that the 3D perturbed Hamiltonian is completely integrable in the sense of Liouville for the special conditions. Quantum invariants are obtained by adding deformation terms, computed using Moyal's bracket, to the corresponding classical counterparts. Furthermore, we use Python programming language to plot the third classical invariant, the deformation and the third quantum invariant in phase space for each quantum integrable case in order to confirm the analytical results.},
	journal = {Chaos Solit. Fractals},
	volume = {161},
	pages = {112361},
	year = {2022},
	month = {Aug},
	issn = {0960-0779},
	doi = {10.1016/j.chaos.2022.112361},
	**url = {https://www.sciencedirect.com/science/article/pii/S0960077922005719}**
}

@article{Mart22,
	title = {Ab initio quantum theory of mass defect and time dilation in trapped-ion optical clocks},
	author = {Mart\'{\i}nez-Lahuerta, V. J. and Eilers, S. and Mehlst\"aubler, T. E. and Schmidt, P. O. and Hammerer, K.},
	abstract = {We derive a Hamiltonian for the external and internal dynamics of an electromagnetically bound, charged two-particle system in external electromagnetic and gravitational fields, including leading-order relativistic corrections. We apply this Hamiltonian to describe the relativistic coupling of the external and internal dynamics of cold ions in Paul traps, including the effects of micromotion, excess micromotion, and trap imperfections. This provides a systematic and fully quantum-mechanical treatment of relativistic frequency shifts in atomic clocks based on single trapped ions. Our approach reproduces well-known formulas for the second-order Doppler shift for thermal states, which were previously derived on the basis of semiclassical arguments. We complement and clarify recent discussions in the literature of the role of time dilation and mass defect in ion clocks.},
	journal = {Phys. Rev. A},
	volume = {106},
	number = {3},
	pages = {032803},
	numpages = {12},
	year = {2022},
	month = {Sep},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.106.032803},
	**url = {https://link.aps.org/doi/10.1103/PhysRevA.106.032803}**
}

@article{Lind22,
	author = {Lindvall, T. and Hanhijärvi, K. J. and Fordell, T. and Wallin, A. E.},
	title = {High-accuracy determination of Paul-trap stability parameters for electric-quadrupole-shift prediction},
	abstract = {The motion of an ion in a radiofrequency (rf) Paul trap is described by the Mathieu equation and the associated stability parameters that are proportional to the rf and dc electric field gradients. Here, a higher-order, iterative method to accurately solve the stability parameters from measured secular frequencies is presented. It is then used to characterize an endcap trap by showing that the trap’s radial asymmetry is dominated by the dc field gradients and by measuring the relation between the applied voltages and the gradients. The results are shown to be in good agreement with an electrostatic finite-element-method simulation of the trap. Furthermore, a method to determine the direction of the radial trap axes using a “tickler” voltage is presented, and the temperature dependence of the rf voltage is discussed. As an application for optical ion clocks, the method is used to predict and minimize the electric quadrupole shift (EQS) using the applied dc voltages. Finally, a lower limit of 1070 for the cancellation factor of the Zeeman-averaging EQS cancellation method is determined in an interleaved low-/high-EQS clock measurement. This reduces the EQS uncertainty of our $^{88}$Sr$^+$ optical clock to $ \leq 1\times 10^{-19}$ in fractional frequency units.},
	journal = {J. Appl. Phys.},
	volume = {132},
	number = {12},
	pages = {124401},
	year = {2022},
	month = {09},
	issn = {0021-8979},
	doi = {10.1063/5.0106633},
	**url = {https://doi.org/10.1063/5.0106633}**
}

@article{Rud22a,
	title = {Stability problem in 3D multipole ion traps},
	author = {Semyon Rudyi and Maxim Vasilyev and Vadim Rybin and Yuri Rozhdestvensky},
	keywords = {Ion Traps, Multipole traps, 3DQIT, Paul Trap, Nonlinear Dynamics, Stability Problem},
	abstract = {We suggest a concept of ion trapping in three-dimensional multipole electric fields. In this work, we propose a vibromechanical interpretation of an ion localization stability problem in a three-dimensional octupole radio-frequency electric fields and have made a transition to an equivalent autonomous dynamical system. The conditions for stable ion localization in three-dimensional octupole trap in radio-frequency-only-mode and mixed-mode are determined. The ion's effective potential for the mixed-mode is calculated and the coordinates of the stable quasi-equilibrium points are obtained. The sensitivity of the effective potential configuration to the polarity of the DC component is estimated. The ion's dynamics in an original and autonomous systems is considered, and a method for optimizing the shape of the trap electrodes is proposed.},
	journal = {Int. J. Mass Spectrom.},
	volume = {479},
	pages = {116894},
	year = {2022},
	month = {Sep},
	issn = {1387-3806},
	doi = {10.1016/j.ijms.2022.116894},
	**url = {https://www.sciencedirect.com/science/article/pii/S1387380622000999}**
}

@article{Shen22,
	author = {Shen, Qi and Guan, Jian-Yu and Ren, Ji-Gang and Zeng, Ting and Hou, Lei and Li, Min and Cao, Yuan and Han, Jin-Jian and Lian, Meng-Zhe and Chen, Yan-Wei and Peng, Xin-Xin and Wang, Shao-Mao and Zhu, Dan-Yang and Shi, Xi-Ping and Wang, Zheng-Guo and Li, Ye and Liu, Wei-Yue and Pan, Ge-Sheng and Wang, Yong and Li, Zhao-Hui and Wu, Jin-Cai and Zhang, Yan-Yan and Chen, Fa-Xi and Lu, Chao-Yang and Liao, Sheng-Kai and Yin, Juan and Jia, Jian-Jun and Peng, Cheng-Zhi and Jiang, Hai-Feng and Zhang, Qiang and Pan, Jian-Wei},
	title = {Free-space dissemination of time and frequency with ${10}^{\ensuremath{-}19}$ instability over 113 km},
	journal = {Nature},
	abstract = {Networks of optical clocks find applications in precise navigation in efforts to redefine the fundamental unit of the ‘second’ and in gravitational tests. As the frequency instability for state-of-the-art optical clocks has reached the $10^{-19}$ level, the vision of a global-scale optical network that achieves comparable performances requires the dissemination of time and frequency over a long-distance free-space link with a similar instability of $10^{-19}$. However, previous attempts at free-space dissemination of time and frequency at high precision did not extend beyond dozens of kilometres. Here we report time–frequency dissemination with an offset of $6.3 \times 10^{-20} \pm 3.4 \times 10^{-19}$ and an instability of less than $4 \times 10^{-19}$ at 10,000 s through a free-space link of 113 km. Key technologies essential to this achievement include the deployment of high-power frequency combs, high-stability and high-efficiency optical transceiver systems and efficient linear optical sampling. We observe that the stability we have reached is retained for channel losses up to 89 dB. The technique we report can not only be directly used in ground-based applications, but could also lay the groundwork for future satellite time–frequency dissemination.},
	volume = {610},
	number = {7933},
	pages = {661 -- 666},
	year = 2022,
	month = {oct},
	date = {18},
	publisher = {Nature},
	issn = {1476-4687},
	doi = {10.1038/s41586-022-05228-5},
	**url = {https://doi.org/10.1038/s41586-022-05228-5}**
}

@article{Schko22,
	author = {Vladimir Schkolnik and Dmitry Budker and Oliver Fartmann and Victor Flambaum and Leo Hollberg and Tigran Kalaydzhyan and Shimon Kolkowitz and Markus Krutzik and Andrew Ludlow and Nathan Newbury and Christoph Pyrlik and Laura Sinclair and Yevgeny Stadnik and Ingmari Tietje and Jun Ye and Jason Williams},
	title = {Optical atomic clock aboard an Earth-orbiting space station ({OACESS}): enhancing searches for physics beyond the standard model in space},
	journal = {Quantum Sci. Technol.},
	abstract = {We present a concept for a high-precision optical atomic clock (OAC) operating on an Earth-orbiting space station. This pathfinder science mission will compare the space-based OAC with one or more ultra-stable terrestrial OACs to search for space-time-dependent signatures of dark scalar fields that manifest as anomalies in the relative frequencies of station-based and ground-based clocks. This opens the possibility of probing models of new physics that are inaccessible to purely ground-based OAC experiments where a dark scalar field may potentially be strongly screened near Earth’s surface. This unique enhancement of sensitivity to potential dark matter candidates harnesses the potential of space-based OACs.},
	volume = {8},
	number = {1},
	pages = {014003},
	year = {2022},
	month = {nov},
	publisher = {IOP Publishing},
	doi = {10.1088/2058-9565/ac9f2b},
	**url = {https://doi.org/10.1088/2058-9565/ac9f2b}**
} 

@article{Ryb22b,
	title = {Nano- and microparticle nonlinear damping identification in quadrupole trap},
	author = {Rybin, Vadim and Rudyi, Semyon and Rozhdestvensky, Yuri},
	keywords = {Ion traps, Nonlinear dynamics, Nonlinear damping identification, Paul trap, Nanostructures},
	abstract = {We propose a Nonlinear Damping Identification method for charged nano- and microparticles in a linear quadrupole Paul trap under nonlinear damping forces. We analytically and numerically evaluate conditions for the formation of plane limit cycle trajectories, i.e. “extended orbits” in spherical particle approximation. We introduce numerical metrics for such orbits and define a functional relation between the metrics and physical parameters of a trapped particle: size, mass, and charge. The method proposed allows non-destructive, simultaneous and separate determination of the trapped particle parameters in a wide range of sizes.},
	journal = {Int. J. Non Linear Mech.},
	volume = {147},
	pages = {104227},
	numpages = {12},
	year = {2022},
	month = {Dec},
	issn = {0020-7462},
	doi = {10.1016/j.ijnonlinmec.2022.104227},
	**url = {https://www.sciencedirect.com/science/article/pii/S0020746222001974}**
}

@Article{Coel22,
	AUTHOR = {Coelho, Stanley S. and Queiroz, Lucas and Alves, Danilo T.},
	TITLE = {Exact Solution of a Time-Dependent Quantum Harmonic Oscillator with Two Frequency Jumps via the Lewis-Riesenfeld Dynamical Invariant Method},
	JOURNAL = {Entropy},
	VOLUME = {24},
	YEAR = {2022},
	MONTH = {Dec},
	NUMBER = {12},
	ARTICLE-NUMBER = {1851},
	PubMedID = {36554256},
	ISSN = {1099-4300},
	ABSTRACT = {Harmonic oscillators with multiple abrupt jumps in their frequencies have been investigated by several authors during the last decades. We investigate the dynamics of a quantum harmonic oscillator with initial frequency ω0, which undergoes a sudden jump to a frequency ω1 and, after a certain time interval, suddenly returns to its initial frequency. Using the Lewis–Riesenfeld method of dynamical invariants, we present expressions for the mean energy value, the mean number of excitations, and the transition probabilities, considering the initial state different from the fundamental. We show that the mean energy of the oscillator, after the jumps, is equal or greater than the one before the jumps, even when ω1<ω0. We also show that, for particular values of the time interval between the jumps, the oscillator returns to the same initial state.},
	DOI = {10.3390/e24121851},
  **URL = {https://www.mdpi.com/1099-4300/24/12/1851}**
}

@article{Hong22,
	title = {Numerical investigation of a segmented-blade ion trap with biasing rods},
	author = {Hong, Jungsoo and Kim, Myunghun and Lee, Hyegoo and Lee, Moonjoo},
	abstract = {We report a numerical study of a linear ion trap that has segmented blades and biasing rods. Our system consists of radio frequency (rf) blades, dc blades with ten separate electrodes, and two biasing rods for compensating the ions’ micromotion. After calculating the optical access for the ions, we find rf and dc voltages that result in a stable trapping configuration of $^{171}$Yb$^{+}$ ions. We also explore the micromotion compensation with the biasing rods, and calculate the influence of blade misalignment to the trap potential. Our work offers quantitative understanding of the trap architecture, assisting reliable operation of an ion-trap quantum computer.},
	journal = {Appl. Phys. B},
	volume = {129},
	number = {1},
	pages = {16},
	numpages = {11},
	year = {2022},
	month = {Dec},
	date = {26},
	publisher = {Springer},
	issn = {1432-0649},
	doi = {10.1007/s00340-022-07955-z},
	**url = {https://doi.org/10.1007/s00340-022-07955-z}**
}

@book{Kaji22,
	author = {Kajita, Masatoshi},
	title = {Ion Traps},
	abstract = {This engaging book presents the fundamentals of ion traps and their use in physics, chemistry and their technological applications. Following an overview of the types of traps and their operation, the book explores their key areas of application for the development of new physics, chemistry, or engineering at a level accessible by students. The introductory nature and broad coverage will also make the book essential reading for scientists who wish to understand and explore the use of ion traps in their research. Embracing optical manipulation, entanglement and exploitation in quantum computing, chemical reactivity, atomic clocks and testing fundamental physics this book provides a broad and accessible introduction to the world of ion traps and how our understanding and exploitation of trapped ions is furthering modern science and technology.Key features• An accessible overview of ion traps and their applications.• Extensive coverage includes relevant physics and applications in physics and chemistry.• Introduces the main areas of application in modern scientific research.• Appendices feature mathematical topics and descriptions for advanced readers.},
	publisher = {IOP Publishing},
	year = {2022},
	month = {Dec},
	issn = {2053-2563},
	isbn = {978-0-7503-5472-1},
	doi = {10.1088/978-0-7503-5472-1},
	**url = {https://doi.org/10.1088/978-0-7503-5472-1}**
}

@article{Kim23,
	title = {Improved interspecies optical clock comparisons through differential spectroscopy},
	author = {Kim, May E. and McGrew, William F. and Nardelli, Nicholas V. and Clements, Ethan R. and Hassan, Youssef S. and Zhang, Xiaogang and Valencia, Jose L. and Leopardi, Holly and Hume, David B. and Fortier, Tara M. and Ludlow, Andrew D. and Leibrandt, David R.},
	abstract = {Comparisons of high-accuracy optical atomic clocks1 are essential for precision tests of fundamental physics2, relativistic geodesy3–5 and the anticipated redefinition of the second by the International System of Units6. The scientific reach of these applications is restricted by the statistical precision of comparison measurements between clocks realized with different atomic species. The instability of individual clocks is limited by the finite coherence time of the optical local oscillator, which bounds the maximum atomic interrogation time. Here we experimentally demonstrate differential spectroscopy7, a comparison protocol that enables interrogating times beyond the optical local oscillator coherence time. By phase coherently linking a zero-dead-time8 Yb optical lattice clock with an Al$^+$ single-ion clock via an optical frequency comb and performing synchronized Ramsey spectroscopy, we show an improvement in comparison instability relative to previous results9 of nearly an order of magnitude. This result represents one of the most stable interspecies clock comparisons to date.},
	journal = {Nat. Phys.},
	volume = {19},
	number = {1},
	pages = {25 -- 29},
	numpages = {0},
	year = {2023},
	month = {Jan},
	publisher = {Nature},
	issn = {1745-2481},
	doi = {10.1038/s41567-022-01794-7},
	**url = {https://doi.org/10.1038/s41567-022-01794-7}**
}

@article{Tsai23,
	author = {Tsai, Yu-Dai and Eby, Joshua and Safronova, Marianna S.},
	title = {Direct detection of ultralight dark matter bound to the Sun with space quantum sensors},
	abstract = {Recent advances in quantum sensors, including atomic clocks, enable searches for a broad range of dark matter candidates. The question of the dark matter distribution in the Solar system critically affects the reach of dark matter direct detection experiments. Partly motivated by the NASA Deep Space Atomic Clock and the Parker Solar Probe, we show that space quantum sensors present new opportunities for ultralight dark matter searches, especially for dark matter states bound to the Sun. We show that space quantum sensors can probe unexplored parameter space of ultralight dark matter, covering theoretical relaxion targets motivated by naturalness and Higgs mixing. If a two-clock system were able to make measurements on the interior of the solar system, it could probe this highly sensitive region directly and set very strong constraints on the existence of such a bound-state halo in our solar system. We present sensitivity projections for space-based probes of ultralight dark matter, which couples to electron, photon and gluon fields, based on current and future atomic, molecular and nuclear clocks.},
	journal = {Nat. Astron.},
	volume = {7},
	number = {1},
	pages = {113 -- 121},
	numpages = {0},
	year = {2023},
	month = {Jan},
	date = {1},
	publisher = {Nature},
	issn = {2397-3366},
	doi = {10.1038/s41550-022-01833-6},
	**url = {https://doi.org/10.1038/s41550-022-01833-6}**
}

@article{Affo23,
	title = {Toward improved quantum simulations and sensing with trapped two-dimensional ion crystals via parametric amplification},
	author = {Affolter, M. and Ge, W. and Bullock, B. and Burd, S. C. and Gilmore, K. A. and Lilieholm, J. F. and Carter, A. L. and Bollinger, J. J.},
	abstract = {Improving coherence is a fundamental challenge in quantum simulation and sensing experiments with trapped ions. Here we discuss, experimentally demonstrate, and estimate the potential impacts of two different protocols that enhance, through motional parametric excitation, the coherent spin-motion coupling of ions obtained with a spin-dependent force. The experiments are performed on two-dimensional crystal arrays of approximately 100 $^9$Be$^+$ ions confined in a Penning trap. By modulating the trapping potential at close to twice the center-of-mass mode frequency, we squeeze the motional mode and enhance the spin-motion coupling while maintaining spin coherence. With a stroboscopic protocol, we measure 5.4±0.9dB of motional squeezing below the ground-state motion, from which theory predicts a 10-dB enhancement in the sensitivity for measuring small displacements using a recently demonstrated protocol [K. A. Gilmore et al., Science 373, 673 (2021)]. With a continuous squeezing protocol, we measure and accurately calibrate the parametric coupling strength. Theory suggests this protocol can be used to improve quantum spin squeezing, limited in our system by off-resonant light scatter. We illustrate numerically the trade-offs between strong parametric amplification and motional dephasing in the form of center-of-mass frequency fluctuations for improving quantum spin squeezing in our setup.},
	journal = {Phys. Rev. A},
	volume = {107},
	issue = {3},
	pages = {032425},
	numpages = {11},
	year = {2023},
	month = {Mar},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevA.107.032425},
	url = {https://link.aps.org/doi/10.1103/PhysRevA.107.032425}
}

@article{Mih23a,
	title = {The physics and applications of strongly coupled Coulomb systems (plasmas) levitated in electrodynamic traps},
	author = {Mihalcea, Bogdan M. and Filinov, Vladimir S. and Syrovatka, Roman A. and Vasilyak, Leonid M.},
	keywords = {Paul (radiofrequency) trap, Nonlinear dynamics, Strongly coupled Coulomb system (SCCS), Solitary wave, Many-body system, Laser plasma accelerated particle physics},
	abstract = {Charged microparticles confined in electrodynamic traps evolve into strongly coupled Coulomb systems (SCCS) which are the subject of current investigation. Recent results with respect to particle dynamics in linear and nonlinear Paul traps are reviewed, including the case of a confined microparticle in presence of an acoustic wave. An analytical model is used to discuss dynamical stability for a system of two coupled ions confined in a Paul trap. The model is then extended to discuss quantum stability for many-body systems of trapped ions. Dynamical stability for many-body systems of identical ions confined in 3D quadrupole ion traps (QIT) is studied locally, in the neighbourhood of minimum configurations that characterize ordered structures. The analytical model is particularized to the case of a combined trap. It is demonstrated that Paul (ion) traps are versatile instruments to investigate one-component strongly coupled Coulomb systems (microplasmas). Exciting physical phenomena associated to Coulomb systems are reported such as autowave generation, phase transitions, defect formation, system self-locking at the edges of a linear Paul trap, self-organization in layers, or pattern formation and scaling. The dynamics of ordered structures consisting of highly nonideal similarly charged solid particles with coupling parameter of the order $\Gamma = 10^8$ is explored. The approach used enables one to explore the interaction of microparticle structures in presence and in absence of the neutralizing plasma background, as well as to investigate various types of phenomena and physical forces experienced by these patterns. Brownian dynamics (BD) is used to characterize charged particle evolution in time and thus identify regions of stable trapping. Analytical models are used to explain the experimental results. Numerical modelling considers stochastic forces of random collisions with neutral particles, viscosity of the gas medium, regular forces produced by the a.c. trapping voltage, and gravitational force. Microparticle dynamics is characterized by a stochastic Langevin differential equation. Laser plasma acceleration of charged particles is also discussed, with an emphasis on Paul traps employed to investigate collective effects in space-charge-dominated (relativistic) beams and for target micropositioning. This review paper is both an add-on as well as an update on late progress in SCCS confined in electrodynamic traps.},
	journal = {Phys. Rep.},
	volume = {1016},
	pages = {1 -- 103},
	year = {2023},
	month = {May},
	date = {15},
	publisher = {Elsevier},
	issn = {0370-1573},
	doi = {10.1016/j.physrep.2023.03.004},
	**url = {https://www.sciencedirect.com/science/article/pii/S0370157323001254}**
}

@inbook{Mih23b,
	author = {Mihalcea, B.},
	title = {Coherent States for Trapped Ions: Applications in Quantum Optics and Precision Measurements},
	abstract = {The evolution of squeezed coherent states of motion for trapped ions is investigated by applying the time-dependent variational principle for the Schrödinger equation. The method is applied in case of Paul and combined traps, for which the classical Hamiltonian and equations of motion are derived. Hence, coherent states provide a natural framework to: (a) engineer quantum correlated states for trapped ions intended for ultraprecise measurements, (b) explore the mechanisms responsible for decoherence, and (c) investigate the quantum–classical transition.},
	booktitle = {CPT and Lorentz Symmetry},
	chapter = {},
	pages = {192 -- 195},
	year = {2023},
	month = {Jun},
	publisher = {World Scientific},
	address = {Singapore},
	isbn = {978-981-127-531-1},
	e-isbn = {978-981-127-538-8},
	doi = {10.1142/9789811275388_0043},
	**URL = {https://www.worldscientific.com/doi/abs/10.1142/9789811275388_0043}**
}

@article{Onah23,
	author = {Onah, F. E. and Garcí\'{\i}a Herrera, E. and  Ruelas-Galv\'{a}n, J. A. and Ju\'{a}rez Rangel, G. and Real Norzagaray, E. and Rodr\'{\i}guez-Lara, B. M.},
	title = {A quadratic time-dependent quantum harmonic oscillator},
	abstract = {We present a Lie algebraic approach to a Hamiltonian class covering driven, parametric quantum harmonic oscillators where the parameter set—mass, frequency, driving strength, and parametric pumping—is time-dependent. Our unitary-transformation-based approach provides a solution to our general quadratic time-dependent quantum harmonic model. As an example, we show an analytic solution to the periodically driven quantum harmonic oscillator without the rotating wave approximation; it works for any given detuning and coupling strength regime. For the sake of validation, we provide an analytic solution to the historical Caldirola–Kanai quantum harmonic oscillator and show that there exists a unitary transformation within our framework that takes a generalized version of it onto the Paul trap Hamiltonian. In addition, we show how our approach provides the dynamics of generalized models whose Schr\"{o}dinger equation becomes numerically unstable in the laboratory frame.},
	journal = {Sci. Rep.},
	volume = {13},
	number = {1},
	pages = {8312},
	numpages = {}, 
	year = {2023},
	month = {May},
	date = {23},
	publisher = {Nature},
	issn = {2045-2322},
	doi = {10.1038/s41598-023-34703-w},
	**url = {https://doi.org/10.1038/s41598-023-34703-w}**
}

@incollection{Peik23,
	author = {Peik, Ekkehard},
	title = {Optical Atomic Clocks},
	booktitle = {Photonic Quantum Technologies},
	chapter = {14},
	pages = {333 -- 348},
	editor = {Benyoucef, Mohammed},
	keywords = {atomic clock, laser cooling and trapping, optical frequency comb, unit of time},
	abstract = {Summary Based on the development of atomic clocks and the definition of the unit of time via an atomic resonance frequency, time and frequencies are the most precisely measurable physical quantities. Over the past 30 years, the development of methods of cooling and trapping of atoms and ions and the development of low-noise laser oscillators and the invention of the optical frequency comb that establishes a link between microwave and optical frequencies with nearly perfect fidelity have led to impressive progress in the field. Optical clocks based on laser-cooled and trapped atoms and ions reach a systematic uncertainty in the low $10^{-18}$ range and below. This chapter presents the principles and methods that have enabled these performances.},
	year = {2023},
	month = {May},
	publisher = {Wiley},
	isbn = {9783527837427},
	doi = {10.1002/9783527837427.ch14},
	**url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/9783527837427.ch14}**
}

@article{Zhiqi23,
	title = {$^{176}{\mathrm{Lu}}^{+}$ clock comparison at the $10^{-18}$ level via correlation spectroscopy},
	author = {Zhiqiang, Zhang and Arnold, Kyle J. and Kaewuam, Rattakorn and Barrett, Murray D.},
	abstract = {The extreme precision of optical atomic clocks has led to an anticipated redefinition of the second by the International System of Units. Furthermore, accuracies pushing the boundary of 1 part in $10^{18}$ and beyond will enable new applications, such as in geodesy and tests of fundamental physics. The $^1S_0$ to $^3D_1$ optical transition in $^{176}$Lu$^+$ has exceptionally low sensitivity to external perturbations, making it suitable for practical clock implementations with inaccuracy at or below $10^{-18}$. Here, we perform high-accuracy comparisons between two $^{176}$Lu$^+$ references using correlation spectroscopy. A comparison at different magnetic fields is used to obtain a quadratic Zeeman coefficient of $-4.89264(88)$ Hz/mT for the reference frequency. With a subsequent comparison at low field, we demonstrate agreement at the low $10^{-18}$ level, statistically limited by the averaging time of 42 hours. The evaluated uncertainty in the frequency difference is $9 \times 10^{-19}$ and the lowest reported in comparing independent optical references. Lutetium atomic frequency reference reaches below $10^{-18}$ inaccuracy on its primary clock transition.},
	journal = {Sci. Adv.},
	volume = {9},
	number = {18},
	pages = {eadg1971},
	year = {2023},
	month = {May},
	date = {3},
	doi = {10.1126/sciadv.adg1971},
	**URL = {https://doi.org/10.1126/sciadv.adg1971}**
	eprint = {https://www.science.org/doi/pdf/10.1126/sciadv.adg1971},
}

@ARTICLE{Ghosh23,
	author={Ghosh, Ipshit and Saxena, Varun and Krishnamachari, Annangarachari},
	journal={IEEE Trans. Plasma Sci.}, 
	title={Resonance Curves and Jump Frequencies in a Dual-Frequency Paul Trap on Account of Octopole Field Imperfection}, 
	abstract = {Nonlinear phenomena inside a dual-frequency Paul trap are investigated. The dynamics inside the trap under the influence of an external sinusoidal driving force, collisional damping, and even-order octopole field imperfection can be encapsulated by a damped driven Duffing-type equation. Harmonic balance technique is applied to analyze the dynamics around the primary resonance. The amplitude of the resonance curves, the inflection frequency at which the jump phenomena occurs, and transitional frequencies often termed as jump-up and jump-down frequencies are found to depend on the voltage and frequency ratios of the dual-frequency Paul trap},
	year={2023},
	month = {Jun},
	date = {29},
	volume={51},
	number={7},
	pages={1924-1931},
	publisher = {IEEE},
	issn = {0093-3813},
	doi={10.1109/TPS.2023.3285260}
}

@article{Rud23a,
	title = {Numerical analysis of phase transitions in ion Coulomb crystals},
	author = {Rudyi, Semyon S. and Romanova, Anna V. and Rozhdestvensky, Yu. V.},
	abstract = {The present work suggests numerical approach to determine phase transitions in small-ion Coulomb crystals in ion traps. The proposed method is based on the analysis of the trapped Coulomb crystal geometrical parameters a cross size $\rho _i$, relative polar angle between neighboring particles $\Delta \phi _i$, the surface area $S_A$ and the volume $V_p$ of a circumscribed polyhedron of a Coulomb crystal. We demonstrate an analysis procedure for the numerical determination of extremes of interpolated geometrical parameters functions. Phase transitions points are defined as the extremes of the proposed interpolated functions.},
	journal = {Comput. Part. Mech.},
	volume = {},
	number = {},
	pages = {},
	numpages = {},
	year = {2023},
	month = {Jun},
	date = {26},
	publisher = {Springer},
	issn = {2196-4386},
	doi = {10.1007/s40571-023-00625-7},
	**url = {https://doi.org/10.1007/s40571-023-00625-7}**
}

@Article{Rud23b,
	AUTHOR = {Rudyi, Semyon and Ivanov, Andrei and Shcherbinin, Dmitrii},
	TITLE = {Fractal Quasi-Coulomb Crystals in Ion Trap with Cantor Dust Electrode Configuration},
	JOURNAL = {Fractal Fract.},
	VOLUME = {7},
	NUMBER = {9},
	ARTICLE-NUMBER = {686},
	YEAR = {2023},
	MONTH = {Sep},
	DATE = {15},
	PUBLISHER = {MDPI},
	ISSN = {2504-3110},
	ABSTRACT = {We propose a new concept of fractal quasi-Coulomb crystals. We have shown that self-similar quasi-Coulomb crystals can be formed in surface electrodynamic traps with the Cantor Dust electrode configuration. Quasi-Coulomb crystal fractal dimension appears to depend on the electrode parameters. We have identified the conditions for transforming trivial quasi-Coulomb crystals into self-similar crystals and described the features of forming 25 Ca$^+$ self-similar quasi-Coulomb crystals. The local potential well depth and width have been shown to take a discrete value dependent on the distance from the electrode surface. Ions inside the crystals studied possess varied translational secular frequencies. We believe that the extraordinary properties of self-similar quasi-Coulomb crystals may contribute to the new prospects within levitated optomechanics, quantum computing and simulation.},
	DOI = {10.3390/fractalfract7090686},
	**URL = {https://www.mdpi.com/2504-3110/7/9/686}**
}

@article{Wen23,
	title = {Generalized Kibble-Zurek mechanism for defects formation in trapped ions},
	author = {Wen, Wei and Zhu, Shanhua and Xie, Yi and Ou, Baoquan and Wu, Wei and Chen, Pingxing and Gong, Ming and Guo, Guangcan},
	abstract = {The Kibble-Zurek (KZ) mechanism has played a fundamental role in defect formation with universal scaling laws in nonequilibrium phase transitions. However, this theory may not accurately predict the scaling laws in inhomogeneous systems and slow quenching processes. Here, we present a generalized KZ mechanism for the defect formation in trapped ions with the freeze-out condition $g\hat t = {b_0}\tau (\hat t)$, where $g$ is a universal quenching velocity function and $b_0$ is a constant. We derived a differential equation $\varphi(x, t)$ to account for the frozen correlation length of a kink in an inhomogeneous system and demonstrated a smooth crossover from a fast quenching process to a slow quenching process, which agrees well with the experiments performed by Ulm et al. [Nat. Commun. 4, 2290 (2013)] and Pyka et al. [Nat. Commun. 4, 2291 (2013)]. Furthermore, we confirmed our theoretical model using molecular dynamics simulation by solving the stochastic differential equation, showing excellent agreement with the results from the differential equation. Our theory provides a general theoretical framework for studying KZ physics in inhomogeneous systems, which has applications in other nonequilibrium platforms studied experimentally.},
	journal = {Sci. China: Phys. Mech. Astron.},
	volume = {66},
	number = {8},
	pages = {280311},
	numpages = {6},
	year = {2023},
	month = {Jul},
	date = {06},
	publisher = {Springer},
	issn = {1869-1927},
	doi = {10.1007/s11433-023-2119-8},
	**url = {https://doi.org/10.1007/s11433-023-2119-8}**
}

@article{Duca23,
	title = {Orientational Melting in a Mesoscopic System of Charged Particles},
	author = {Duca, Lucia and Mizukami, Naoto and Perego, Elia and Inguscio, Massimo and Sias, Carlo},
	abstract = {A mesoscopic system of a few particles can undergo changes of configuration that resemble phase transitions but with a nonuniversal behavior. A notable example is orientational melting, in which localized particles with long-range repulsive interactions forming a two-dimensional crystal become delocalized in common closed trajectories. Here we report the observation of orientational melting occurring in a two-dimensional crystal of up to 15 ions. We measure density-density correlations to quantitatively characterize the occurrence of melting, and use a Monte Carlo simulation to extract the angular kinetic energy of the ions. By adding a pinning impurity, we demonstrate the nonuniversality of orientational melting and create novel configurations in which localized and delocalized particles coexist. Our system realizes an experimental testbed for studying changes of configurations in two-dimensional mesoscopic systems, and our results pave the way for the study of quantum phenomena in ensembles of delocalized ions.},
	journal = {Phys. Rev. Lett.},
	volume = {131},
	number = {8},
	pages = {083602},
	numpages = {7},
	year = {2023},
	month = {Aug},
	publisher = {American Physical Society},
	doi = {10.1103/PhysRevLett.131.083602},
	**url = {https://doi.org/10.1103/PhysRevLett.131.083602}**
}

@www{Mero,
	author = {},
	title = {Meromorphic Function},
	url = {https://mathworld.wolfram.com/MeromorphicFunction.html},
	urldate = {02. 05. 2024}
}

@www{Jones,
	author = {Jones, Timothy},
	title = {Mathieu's Equations and the ideal RF-Paul Trap},
	url = {http://einstein.drexel.edu/~tim/open/mat/mat.pdf},
	urldate = {21. 05. 2024}
}

@article{Katz23,
	title = {Programmable $N$-Body Interactions with Trapped Ions},
	author = {Katz, Or and Cetina, Marko and Monroe, Christopher},
	abstract = {Trapped atomic ion qubits or effective spins are a powerful quantum platform for quantum computation and simulation, featuring densely connected and efficiently programmable interactions between the spins. While native interactions between trapped-ion spins are typically pairwise, many quantum algorithms and quantum spin models naturally feature couplings between triplets, quartets, or higher orders of spins. Here, we formulate and analyze a mechanism that extends the standard M\o{}lmer-S\o{}rensen pairwise entangling gate and generates a controllable and programmable coupling between $N$ spins of trapped ions. We show that spin-dependent optical parametric drives applied at twice the motional frequency generate a coordinate transformation of the collective ion motion in phase space, rendering displacement forces that are nonlinear in the spin operators. We formulate a simple framework that enables a systematic and faithful construction of high-order spin Hamiltonians and gates, including the effect of multiple modes of motion, and characterize the performance of such operations under realistic conditions.},
	journal = {PRX Quantum},
	volume = {4},
	issue = {3},
	pages = {030311},
	numpages = {18},
	year = {2023},
	month = {Jul},
	publisher = {American Physical Society},
	doi = {10.1103/PRXQuantum.4.030311},
  **url = {https://link.aps.org/doi/10.1103/PRXQuantum.4.030311}**
}


@article{Land23,
	author = {Landau, Arie and Eduardus and Behar, Doron and Wallach, Eliana Ruth and Pa\v{s}teka, Luk\'{a}\v{s} F. and Faraji, Shirin and Borschevsky, Anastasia and Shagam, Yuval},
	title = {Chiral molecule candidates for trapped ion spectroscopy by ab initio calculations: From state preparation to parity violation},
	abstract = {Parity non-conservation (PNC) due to the weak interaction is predicted to give rise to enantiomer dependent vibrational constants in chiral molecules, but the phenomenon has so far eluded experimental observation. The enhanced sensitivity of molecules to physics beyond the Standard Model (BSM) has led to substantial advances in molecular precision spectroscopy, and these may be applied to PNC searches as well. Specifically, trapped molecular ion experiments leverage the universality of trapping charged particles to optimize the molecular ion species studied toward BSM searches, but in searches for PNC, only a few chiral molecular ion candidates have been proposed so far. Importantly, viable candidates need to be internally cold, and their internal state populations should be detectable with high quantum efficiency. To this end, we focus on molecular ions that can be created by near threshold resonant two-photon ionization and detected via state-selective photo-dissociation. Such candidates need to be stable in both charged and neutral chiral versions to be amenable to these methods. Here, we present a collection of suitable chiral molecular ion candidates we have found, including CHDBrI$^+$ and CHCaBrI$^+$, that fulfill these conditions according to our ab initio calculations. We find that organo-metallic species have low ionization energy as neutrals and relatively high dissociation thresholds. Finally, we compute the magnitude of the PNC values for vibrational transitions for some of these candidates. An experimental demonstration of state preparation and readout for these candidates will be an important milestone toward measuring PNC in chiral molecules for the first time.},
	journal = {J. Chem. Phys.},
	volume = {159},
	number = {11},
	pages = {114307},
	numpages = {},
	year = {2023},
	month = {09},
	issn = {0021-9606},
	doi = {10.1063/5.0163641},
	**url = {https://doi.org/10.1063/5.0163641}**
}

@article{Rud23c,
	author = {Rudyi, S. S. and Rybin, V. V. and Semynin, M. S. and Shcherbinin, D. P. and Rozhdestvensky, Yu. V. and Ivanov, A. V.},
	title = {Period-doubling bifurcation in surface radio-frequency trap: Transition to chaos through Feigenbaum scenario},
	abstract = {We have numerically investigated the dynamics of charged microparticles in a “five-wire” surface radio-frequency trap. The period-doubling bifurcation conditions have been shown to depend on the particle, the trap, and the alternating voltage parameters. For a comprehensive study of the dynamics chaotization through a cascade of period doubling, we have used Fourier analysis of a particle trajectory as well as the calculations of a non-trivial Lyapunov exponent map. We have demonstrated that the period-doubling bifurcation is consistent with a Feigenbaum scenario. A new approach to particle property determination can, thus, be based on observing a period-doubling bifurcation.},
	journal = {Chaos},
	volume = {33},
	number = {9},
	pages = {093133},
	numpages = {8},
	year = {2023},
	month = {Sep},
	date = {22},
	publisher = {American Institute of Physics},
	issn = {1054-1500},
	doi = {10.1063/5.0157397},
	**url = {https://doi.org/10.1063/5.0157397}**
}

@article{Ryb23b,
	title = {Novel nonlinear damping identification method: Simultaneous size, mass, charge and density measurements of a microparticle in quadrupole trap},
	author = {Vadim Rybin and Dmitrii Shcherbinin and Maxim Semynin and Anton Gavenchuk and Viktor Zakharov and Andrei Ivanov and Yuri Rozhdestvensky and Semyon Rudyi},
	keywords = {NDI methods, Paul trap, Quadrupole trap, Comprehensive analysis},
	abstract = {In this paper, we present the nonlinear damping identification method for the fast and comprehensive study of individual microparticles localized in a quadrupole electrodynamic Paul trap. The measurement procedure is discussed in detail. The size, mass and charge of individual silica microspheres from the studied sample are determined simultaneously and non- destructively. Experimental results agree well with the results of independent microscopic examination and density reference values. The further development of the method are outlined.},
	journal = {Powder Technol.},
	volume = {427},
	pages = {118717},
	year = {2023},
	month = {Sep},
	date = {1},
	issn = {0032-5910},
	doi = {https://doi.org/10.1016/j.powtec.2023.118717},
	**url = {https://www.sciencedirect.com/science/article/pii/S0032591023005016}**
}

@article{Tand23,
	author = {Tandel, D. D. and Chatterjee, Anindya and Mohanty, Atanu K.},
	title = {Quadrupole ion trap with dipolar DC excitation: motivation, nonlinear dynamics, and simple formulas},
	abstract = {We study the dynamics of a trapped ion in a mass spectrometer under the action of both the usual quadrupolar RF and dipolar DC excitation. The relevant governing equation, derived from electrostatic field calculations for realistic geometries, is a classical Mathieu equation perturbed with a constant inhomogeneous term and a small quadratic nonlinearity. An early paper by Plass examined the case without the quadratic term using variation of parameters. Here, we note a significantly simpler particular solution than used by Plass, include the quadratic term, and develop a second-order averaging-based approximation. The averaging results show that a particular underlying simple periodic solution is stable. We then show that a two-frequency approximation matches that solution well for practical purposes. Finally, we present and validate an easy iterative calculation for obtaining that two-frequency solution and quantify the effect of the quadratic nonlinearity.},
	journal = {Nonlin. Dyn.},
	volume = {111},
	number = {17},
	pages = {15837 -- 15852},
	numpages = {0},
	year = {2023},
	month = {09},
	date = {01},
	issn = {1573-269X},
	doi = {10.1007/s11071-023-08706-1},
	**url = {https://doi.org/10.1007/s11071-023-08706-1}**
}

@article{Wang23,
	author = {Ying-Xiang Wang and Sheng-Chen Liu and Lin Cheng and Liang-You Peng},
	title = {Systematic investigations on ion dynamics with noises in Paul trap},
	abstract = {Ions confined in a Paul trap serve as crucial platforms in various research fields, including quantum computing and precision spectroscopy. However, the ion dynamics is inevitably influenced by different types of noise, which require accurate computations and general analytical analysis to facilitate diverse applications based on trapped ions with white or colored noise. In the present work, we investigate the motion of ions in a Paul trap via the Langevin equation using both analytical and numerical methods, systematically studying three different types of noise: the white noise, the colored noise via the Ornstein–Uhlenbeck process and the Wiener process. For the white noise of the case, we provide a recursion method to calculate ion motion for a wide range of parameters. Furthermore, we present an analytical solution to the more realistic stochastic process associated with the colored noise, verified by the Monte Carlo simulation. By comparing the results of the colored noise with those of the white noise, and additionally considering another limit of noise parameters corresponding to the Wiener process, we summarize the effects of different noise types on the ion dynamics.},
	journal = {J. Phys. A: Math. Theor.},
	volume = {56},
	number = {46},
	pages = {465302},
	numpages = {},
	year = {2023},
	month = {oct},
	publisher = {IOP Publishing},
	doi = {10.1088/1751-8121/ad0348},
	**url = {https://dx.doi.org/10.1088/1751-8121/ad0348}**
}

@article{Garc23,
	author = {E. {Garc\'{\i}a Herrera} and F. Torres-Leal and B. M. Rodr\'{\i}guez-Lara},
	title = {Continuous-time quantum harmonic oscillator state engineering},
	journal = {New J. Phys.},
	abstract = {The center of mass motion of trapped ions and neutral atoms is suitable for approximation by a time-dependent driven quantum harmonic oscillator whose frequency and driving strength may be controlled with high precision. We show the time evolution for these systems with continuous differentiable time-dependent parameters in terms of the three basic operations provided by its underlying symmetry, rotation, displacement, and squeezing, using a Lie algebraic approach. Our factorization of the dynamics allows for the intuitive construction of protocols for state engineering, for example, creating and removing displacement and squeezing, as well as their combinations, optimizing squeezing, or more complex protocols that work for slow and fast rates of change in the oscillator parameters.},
	volume = {25},
	number = {12},
	pages = {123045},
	numpages = {11},
	year = {2023},
	month = {dec},
	publisher = {IOP Publishing},
	doi = {10.1088/1367-2630/ad149c},
**url = {https://doi.org/10.1088/1367-2630/ad149c}**
}

@article{Dib24,
	title = {An innovative efficient approach to solving damped Mathieu–Duffing equation with the non-perturbative technique},
	author = {Yusry O. El-Dib},
	keywords = {Damping nonlinear oscillator, Non-perturbative technique, Analytical solution, Mathieu–Duffing equation, Bifurcation},
	abstract = {The effectiveness of the non-perturbative technique is further proven in this paper for the parametric nonlinear oscillatory system. The method's efficiency and practicality in obtaining the frequency amplitude of parametric nonlinear issues have been effectively shown. The obtained approximate solutions are not based on a series expansion. Our interest in this work is to begin to obtain approximate solutions without restraints to a small amplitude of the parametric coefficients far from the perturbation methods. As is explicitly shown, the accuracy obtained is independent of the value of the parametric coefficients. In addition, the analysis is extended to establish accurate solutions for the large amplitude of nonlinear oscillation. The most important property is a quick estimate of the frequency–amplitude relationship to obtain successive approximate solutions for the parametric nonlinear oscillation. The stability behavior and the bifurcation are explored. It has been proven that the current technique is quite accurate. The method is uncomplicated in its fundamentals, simple to apply, appropriate, and provides very good numerical accuracy. It is also useful as a mathematical tool for dealing with nonlinear parametric problems because it avoids any mathematical intricacy.},
	journal = {Commun. Nonlinear Sci. Numer. Simul.},
	volume = {128},
	pages = {107590},
	numpages = {14},
	year = {2024},
	month = {Jan},
	issn = {1007-5704},
	doi = {10.1016/j.cnsns.2023.107590},
   **url = {https://www.sciencedirect.com/science/article/pii/S1007570423005117}**
}

@article{Dima24,
	author = {Dimarcq, N. and Gertsvolf, M. and Mileti, G. and Bize, S. and C. W. Oates and Peik, E. and Calonico, D. and Ido, T. and Tavella, P. and Meynadier, F. and Petit, G. and Panfilo, G. and Bartholomew, J. and Defraigne, P. and Donley, E. A. and Hedekvist, P. O. and Sesia, I. and Wouters, M. and Dub\'{e}, P. and Fang, F. and Levi, F. and Lodewyck, J. and  Margolis, H. S. and Newell, D. and Slyusarev, S. and Weyers, S. and Uzan, J.-P. and Yasuda, M. and Yu, D.-H. and Rieck, C. and Schnatz, H. and Hanado, Y. and Fujieda, M. and Pottie, P.-E. and Hanssen, J. and Malimon, A. and Ashby, N.},
	title = {Roadmap towards the redefinition of the second},
	journal = {Metrologia},
	abstract = {This paper outlines the roadmap towards the redefinition of the second, which was recently updated by the CCTF Task Force created by the CCTF in 2020. The main achievements of optical frequency standards (OFS) call for reflection on the redefinition of the second, but open new challenges related to the performance of the OFS, their contribution to time scales and UTC, the possibility of their comparison, and the knowledge of the Earth’s gravitational potential to ensure a robust and accurate capacity to realize a new definition at the level of $10^{-18}$ uncertainty. The mandatory criteria to be achieved before redefinition have been defined and their current fulfilment level is estimated showing the fields that still needed improvement. The possibility to base the redefinition on a single or on a set of transitions has also been evaluated. The roadmap indicates the steps to be followed in the next years to be ready for a sound and successful redefinition.},
	volume = {61},
	number = {1},
	pages = {012001},
	numpages = {19},
	year = {2024},
	month = {jan},
	publisher = {IOP Publishing},
	doi = {10.1088/1681-7575/ad17d2},
	**url = {https://doi.org/10.1088/1681-7575/ad17d2}**
}

@article{Cald24,
	author = {Caldwell, Emily D. and Sinclair, Laura C. and Deschenes, Jean-Daniel and Giorgetta, Fabrizio and Newbury, Nathan R.},
	title = {Application of quantum-limited optical time transfer to space-based optical clock comparisons and coherent networks},
	abstract = {With the demonstration of quantum-limited optical time transfer capable of tolerating the losses associated with long ground-to-space links, two future applications of free-space time transfer have emerged: intercontinental clock comparisons for time dissemination and coherence transfer for future distributed sensing in the mm-wave region. In this paper, we estimated the projected performance of these two applications using quantum-limited optical time transfer and assuming existing low-size, low-weight, and low-power hardware. In both cases, we limit the discussion to the simplest case of a single geosynchronous satellite linked to either one or two ground stations. One important consideration for such future space-based operations is the choice of reference oscillator onboard the satellite. We find that with a modestly performing optical reference oscillator and low-power fiber-based frequency combs, quantum-limited time transfer could support intercontinental clock comparisons through a common-view node in geostationary orbit with a modified Allan deviation at the $10^{-16}$ level at 10-s averaging time, limited primarily by residual turbulence piston noise. In the second application of coherence transfer from ground-to-geosynchronous orbit, we find the system should support high short-term coherence with $\sim 10$ millirad phase noise on a 300 GHz carrier at essentially unlimited integration times.},
	journal = {APL Photonics},
	volume = {9},
	number = {1},
	pages = {016112},
	numpages = {},
	year = {2024},
	month = {01},
	issn = {2378-0967},
	doi = {10.1063/5.0170107},
	**url = {https://doi.org/10.1063/5.0170107}**
}

@article{Spam24,
	author = {Spampinato, Alessio and Stacey, Jonathan and Mulholland, Sean and Robertson, Billy I. and Klein, Hugh A. and Huang, Guilong and Barwood, Geoffrey P. and Gill, Patrick },
	title = {An ion trap design for a space-deployable strontium-ion optical clock},
	abstract = {Optical atomic clocks demonstrate a better stability and lower systematic uncertainty than the highest performance microwave atomic clocks. However, the best performing optical clocks have a large footprint in a laboratory environment and require specialist skills to maintain continuous operation. Growing and evolving needs across several sectors are increasing the demand for compact robust and portable devices at this capability level. In this paper we discuss the design of a physics package for a compact laser-cooled $^{88}$Sr$^+$ optical clock that would, with further development, be suitable for space deployment. We review the design parameters to target a relative frequency uncertainty at the low parts in 1018 with this system. We then explain the results of finite-element modelling to simulate the response of the ion trap and vacuum chamber to vibration, shock and thermal conditions expected during launch and space deployment. Additionally, an electrostatic model has been developed to investigate the relationship between the ion trap geometrical tolerances and the trapping efficiency. We present the results from these analyses that have led to the design of a more robust prototype ready for experimental testing.},
	keywords = {optical atomic clock, single ion trap, space clock, laser cooling, strontium ion, finite element analysis},
	journal = {Proc. R. Soc. A},
	volume = {480},
	number = {2284},
	pages = {20230593},
	numpages = {25},
	year = {2024},
	month = {Feb},
	date = {21},
	publisher = {Royal Society},
	issn = {1364-5021},
	e-issn = {1471-2946},
	doi = {10.1098/rspa.2023.0593},
	**URL = {https://doi.org/10.1098/rspa.2023.0593}**
}

@misc{Haus24,
	title = {An $^{115}$In$^+$--$^{172}$Yb$^+$ Coulomb crystal clock with $2.5\times 10^{-18}$ systematic uncertainty}, 
	author = {H. N. Hausser and J. Keller and T. Nordmann and N. M. Bhatt and J. Kiethe and H. Liu and M. von Boehn and J. Rahm and S. Weyers and E. Benkler and B. Lipphardt and S. Doerscher and K. Stahl and J. Klose and C. Lisdat and M. Filzinger and N. Huntemann and E. Peik and T. E. Mehlst\"{a}ubler},
	abstract = {We present a scalable mixed-species Coulomb crystal clock based on the $^1S_0 \leftrightarrow ^3P_0$ transition in $^{115}$In$^+$. $^{172}$Yb$^+$ ions are co-trapped and used for sympathetic cooling. Reproducible interrogation conditions for mixed-species Coulomb crystals are ensured by a conditional preparation sequence with permutation control. We demonstrate clock operation with a $1$In$^+$-$3$Yb$^+$ crystal, achieving a relative systematic uncertainty of $2.5 \times 10^{-18}$ and a relative frequency instability of $1.6 \times 10^{-15}/{\sqrt {\tau/1 s}}. We report on an absolute frequency measurement with an uncertainty of $1.3\times $10^{-16}$ and optical frequency ratios relative to the $^{171}$Yb$^+$ (E3) and $^{87}$Sr clock transitions with fractional uncertainties of 4.4 and 4.7 parts in $10^{18}$, respectively. The latter are among the most precise measurements of frequency ratios to date and improve upon the previous uncertainty of the $^{115}$In$^+$/$^{87}$Sr ratio by two orders of magnitude. We also demonstrate operation with four $^{115}$In$^+$ clock ions, which reduces the instability to $9.2 \times 10^{-16}/{\sqrt {τ/1 s}}$.},
	year = {2024},
	month = {Feb},
	eprint = {2402.16807},
	archivePrefix = {arXiv},
	primaryClass = {physics.atom-ph},
    doi = {10.48550/arXiv.2402.16807}
}

 @article{Chim24,
	author = {Chimwal, Deepak and Kumar, Sugam and Joshi, Yash and Lal, Aditya Aryan and Nair, Lekha and Quint, Wolfgang and Vogel, Manuel},
	title = {Electrostatic anharmonicity in cylindrical Penning traps induced by radial holes to the trap center},
	abstract = {We present a systematic and quantitative investigation of the electrostatic anharmonicity in cylindrical Penning traps that is induced by the presence of radial holes for access to the trap center. The expected distortion of the electrostatic potential is studied as a function of the relative hole size, shape, arrangement, and number.},
	journal = {Phys. Scr.},
	volume = {99},
	number = {5},
	pages = {055404},
	year={2024},
	month = {Apr},
	date = {9},
	publisher = {IOP Publishing},
	doi = {10.1088/1402-4896/ad38e7},
	**url = {http://iopscience.iop.org/article/10.1088/1402-4896/ad38e7}**
}

@article{Mand24,
	title = {Non-degenerate dodecapole resonances in an asymmetric linear ion trap of round rod geometry},
	author = {Mandal, Pintu and Mukherjee, Manas},
	keywords = {Linear ion trap, Nonlinear resonances, Dodecapole potential, Mass spectrometry},
	abstract = {Linear ion traps and quadrupole mass spectrometers play vital roles in quantum technologies and classical mass spectrometry respectively. Therefore, such systems with nearly same operational principles, are very well studied both theoretically and experimentally. Despite such vast knowledge base, linear ion traps continue to be an important research topic, particularly the effect of small perturbations from ideal conditions. Here, non-degenerate nonlinear resonances of the ion dynamics in a linear three-segmented round-rod radio-frequency ion trap have been studied. Purposefully designed perturbation from an ideal 4-rod symmetric structure results in additional instabilities in the dynamics of the trapped ions. The weightage of the dodecapole potential increases by an order of magnitude in such an asymmetric setup compared to that present in usual symmetric setup of nearly equivalent geometry. The experimental results are corroborated by ion dynamics simulations with commercial software SIMION. Increased dodecapole potential in the asymmetric structure leads to resolution of the frequency of ion oscillation in two mutually perpendicular directions in the radial plane which is otherwise unresolved in a symmetric structure in absence of dc potential. Such asymmetric linear trap, in principle, finds importance in the study of Coulomb crystal, mass spectrometry and relevant trapped ion dynamics. It also opens up the possibility to remove undesired ion species (often known as dark ion) within a chain of physical qubits.},
	journal = {Int. J. Mass Spectrom.},
	volume = {498},
	pages = {117217},
	numpages = {},
	year = {2024},
	month = {Apr},
	issn = {1387-3806},
	publisher = {Elsevier},
	doi = {10.1016/j.ijms.2024.117217},
	**url = {https://www.sciencedirect.com/science/article/pii/S1387380624000289}**
}

@incollection{Vogel24,
	author = {Vogel, Manuel}, 
	editor = {Babb, James and Bandrauk, Andre D. and Bartschat, Klaus and Joachain, Charles J. and Keidar, Michael and Lambropoulos, Peter and Leuchs, Gerd and Velikovich, Alexander},
	booktitle = {Particle Confinement in Penning Traps: An Introduction},
	series = {Springer Series on Atomic, Optical, and Plasma Physics},
	volume = {100} ,
	edition = {2nd},
	year = {2024},
	month = {Apr},
	day = {2},
	publisher = {Springer},
	address = {Cham},
	ISBN = {978-3-031-55419-3},
	e-ISBN = {978-3-031-55420-9}, 
	issn = {1615-5653},
	e-issn = {2197-6791},
	doi = {10.1007/978-3-031-55420-9},
  **url = {https://doi.org/10.1007/978-3-031-55420-9}**
}

@article{Mih24a,
	title = {Mathieu–Hill Equation Stability Analysis for Trapped Ions: Anharmonic Corrections for Nonlinear Electrodynamic Traps},
	author = {Mihalcea, Bogdan M.},
	keywords = {Mathieu-Hill equation, Floquet theory, Sturm-Liouville problem, perturbation method, Paul trap, stability diagram},
	abstract = {The stability properties of the Hill equation are discussed, especially those of the Mathieu equation that characterizes ion motion in electrodynamic traps. The solutions of the Mathieu-Hill equation for a trapped ion are characterized by employing the Floquet theory and Hill’s method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters a and q that are real. Characteristic curves are introduced naturally by the Sturm–Liouville problem for the well-known even and odd Mathieu equations $ce_m (z, q)$ and $se_m (z, q)$. In the case of a Paul trap, the stable solution corresponds to a superposition of harmonic motions. The maximum amplitude of stable oscillations for ideal 	conditions (taken into consideration) is derived. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for both axial and radial motion, where the former is described by the canonical Mathieu equation. Anharmonic corrections for nonlinear Paul traps are discussed within the frame of perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter and we demonstrate they are shifted towards negative values of the a parameter. The applications of the results include but are not restricted to 2D and 3D ion traps used for different applications such as mass spectrometry (including nanoparticles), high resolution atomic spectroscopy and quantum engineering applications, among which we mention optical atomic clocks and quantum frequency metrology.},
	journal = {Photonics},
	volume = {11},
	pages = {551},
	numpages = {22},
	year = {2024},
	month = {June},
	date = {11},
	issn = {},
	publisher = {MDPI Basel},
	doi = {10.3390/photonics11060551},
	**url = {https://doi.org/10.3390/photonics11060551}**
}

@article{Coel24,
	author = {Stanley S Coelho and Lucas Queiroz and Danilo T Alves},
	title = {Squeezing equivalence of quantum harmonic oscillators under different frequency modulations},
	journal = {Phys. Scr.},
	abstract = {The papers by Janszky and Adam [Phys. Rev. A 46, 6091 (1992)] and Chen et al [Phys. Rev. Lett. 104, 063 002 (2010)] are examples of works where one can find the following equivalences: quantum harmonic oscillators subjected to different time-dependent frequency modulations, during a certain time interval $\tau$, exhibit exactly the same final null squeezing parameter ($r_f  = 0$). In the present paper, we discuss a more general case of squeezing equivalence, where the final squeezing parameter can be non-null ($r_f \geq 0$). We show that when the interest is in controlling the forms of the frequency modulations, but keeping free the choice of the values of $r_f$  and $\tau$, this in general demands numerical calculations to find these values leading to squeezing equivalences (a particular case of this procedure recovers the equivalence found by Jansky and Adams). On the other hand, when the interest is not in previously controlling the form of these frequencies, but rather $r_f$  and $\tau$ (and also some constraints, such as minimization of energy), one can have analytical solutions for these frequencies leading to squeezing equivalences (particular cases of this procedure are usually applied in problems of shortcuts to adiabaticity, as done by Chen et al). In this way, this more general squeezing equivalence discussed here is connected to recent and important topics in the literature as, for instance, generation of squeezed states and the obtaining of shortcuts to adiabaticity.},
	volume = {99},
	number = {8},
	pages = {085104},
	numpages = {15},
	year = {2024},
	month = {Jul},
	publisher = {IOP Publishing},
	doi = {10.1088/1402-4896/ad56d6},
  **url = {https://doi.org/10.1088/1402-4896/ad56d6}**
}

@article{Leib24,
	title = {Prospects of a thousand-ion Sn$^{2+}$ Coulomb-crystal clock with sub-$10^{-19}$ inaccuracy},
	author = {Leibrandt, David R. Porsev, Sergey G. and Cheung, Charles and Safronova, Marianna S.},
	keywords = {},
	abstract = {Optical atomic clocks are the most accurate and precise measurement devices of any kind, enabling advances in international timekeeping, Earth science, fundamental physics, and more. However, there is a fundamental tradeoff between accuracy and precision, where higher precision is achieved by using more atoms, but this comes at the cost of larger interactions between the atoms that limit the accuracy. Here, we propose a many-ion optical atomic clock based on three-dimensional Coulomb crystals of order one thousand Sn$^{2+}$ ions confined in a linear RF Paul trap with the potential to overcome this limitation. Sn$^{2+}$ has a unique combination of features that is not available in previously considered ions: a $^1S_0 \leftrightarrow ^3P_0$ clock transition between two states with zero electronic and nuclear angular momentum (I = J = F = 0) making it immune to nonscalar perturbations, a negative differential polarizability making it possible to operate the trap in a manner such that the two dominant shifts for three-dimensional ion crystals cancel each other, and a laser-accessible transition suitable for direct laser cooling and state readout. We present calculations of the differential polarizability, other relevant atomic properties, and the motion of ions in large Coulomb crystals, in order to estimate the achievable accuracy and precision of Sn$^{2+}$ Coulomb-crystal clocks.},
	journal = {Nat. Comm.},
	volume = {15},
	number = {1},
	pages = {5663},
	numnpages = {12},
	year = {2024},
	month = {July},
	date = {05},
	issn = {2041-1723},
	doi = {10.1038/s41467-024-49241-w},
	**url = {https://doi.org/10.1038/s41467-024-49241-w}**
}

@article{Huo24,
	title = {A simple numerical simulation model can elucidate the key factors for designing a miniaturized ion trap mass spectrometer},
	author = {Xinming Huo and Lin Gan and Xinyue Ding and Quan Yu and Bin Zhou and Jianhua Zhou and Xiang Qian},
	keywords = {Numerical simulation, Ion trap, Quadrupole, Mass spectrometer, Resonance excitation},
	abstract = {Background
	Miniature ion trap mass spectrometer enables mass-to-charge ratio analysis of ions via quadrupole field in a low vacuum environment. It plays an important role in on-site detection due to its portability and specificity. In order to gain a deeper understanding of the analysis mechanism of miniature ion trap mass spectrometers, a quadrupole MS ion trajectory numerical simulation model (QITNS) is established in this paper for ions trajectory calculation under the action of quadrupole field, exciting field and neutral gas molecule collision. Compared with the existing methods, the model in this paper is simpler and more direct, which effectively explored the effects of dipole excitation and quadrupole excitation on ion manipulation under high background pressure.
	Results
	The simulation results demonstrate that high RF amplitude, low auxiliary AC amplitude and quadrupole excitation can effectively improve the isolation resolution. Besides, it clarified the difference between the analysis mechanism of ion trap mass spectrometers under high background pressure (above 13.332 Pa) and absolute vacuum conditions. The relevant results are consistent with the conclusions of previous experiments and other theories, proving the applicability and accuracy of the proposed calculation model and solution method.
	Significance
	This research bears the guiding significance for further understanding the mechanism of quadrupole mass spectrometry as well as designing and developing miniature mass spectrometers.},
	journal = {Anal. Chim. Acta},
	volume = {1318},
	pages = {342943},
	numpages = {},
	year = {2024},
	month = {Aug},
	date = {22},
	issn = {0003-2670},
	doi = {10.1016/j.aca.2024.342943},
	**url = {https://www.sciencedirect.com/science/article/pii/S000326702400744X}**
}

@article{Fu24,
	title = {Single-frequency ion parking in a digital 3D quadrupole ion trap},
	author = {Liangxuan Fu and Gregory S. Eakins and Mark S. Carlsen and Scott A. McLuckey},
	abstract = {Single-frequency ion parking, a useful technique in electrospray mass spectrometry (ESI-MS), involves gas-phase charge-reduction ion/ion reactions in an electrodynamic ion trap in conjunction with the application of a supplementary oscillatory voltage to selectively inhibit the reaction rate of an ion of interest. The ion parking process provides a means for limiting the extent of charge reduction in a controlled fashion and allows for ions distributed over a range of charge states to be concentrated into fewer charge states (a single charge state under optimal conditions). As charge reduction inherently leads to an increase in the mass-to-charge $(m/z)$ ratio of the ions, it is important that the means for storing and analyzing ions be able to accommodate ions of high $m/z$ ratios. The so-called ‘digital ion trap’ (DIT), which uses a digital waveform as the trapping RF, has been demonstrated to be well-suited for the analysis of high m/z ions by taking advantage of its ability to manipulate the waveform frequency. In this study, the feasibility of ion parking in a 3D quadrupole ion trap operated as a DIT using a slow-amplitude single-frequency sine-wave for selective inhibition of an ion/ion reaction is demonstrated. A recently described model that describes ion parking has been adjusted for the DIT case and is used to interpret experimental data for proteins ranging in mass from 8600 Da to 467,000 Da.},
	journal = {Int. J. Mass Spectrom.},
	volume = {503},
	pages = {117282},
	year = {2024},
	month = {Sep},
	issn = {1387-3806},
	doi = {https://doi.org/10.1016/j.ijms.2024.117282},
**url = {https://www.sciencedirect.com/science/article/pii/S1387380624000939}**
}