In a MAMZI with phase noise, the phases of the system
(see equation (
2)) are affected by the phase noise
. To stabilize it, an externally manipulable control phase (
) will be used, then the phases of the system are modeled as
with
. Note that if it is possible to observe if
(for all
k), then
converges to the identity matrix. Also, as
, then it can be seen that the intensity output from (
3) is given by
. Therefore, it is possible to stabilize a multi-parameter phase noise on an MAMZI by maximizing one of its outputs, which is obtained when a control phase is the negative magnitude of the phase noise applied to each arm. Then, the proposed phase noise stabilization method is obtained from the follow optimization:
4.1. MAMZI Simulation with Phase Noise
To evaluate the controllability described above, we have simulated MAMZIs for dimensions 2x2 up to 8x8 using the phased array model, MBSs obtained from the equations Eq. (
1), and the phase matrix from Eq. (
2). Our simulation is a discrete system, where the phase
is provided by a matrix of
, where
represents the time length of the simulation, and each noise sample is a cumulative random sequence with a variability of
, to mimic the behaviour of fiber MAMZI [
22]. We performed
experiments for each simulated control process. The generated phase noise is introduced into the plant "sample by sample," perturbing the intensity equation (
5).
Figure 3 shown the open loop intensity response of applied phase noise over diverse MAMZI.
Figure 3.a shows one of the 100 matrices of
for the accumulated phase noise simulation, while
Figure 3.b-
3.g shows the open-loop response of applying N vectors of the noise matrix to the
system. Note, for example, that in
Figure 3.g, the 8 vectors are used, resulting in a random intensity response for all MAMZI outputs. Each noise matrix was generated, ensuring output intensity with symmetrical temporal distribution.
4.3. Analysis of the Results in Phase Noise Stabilization
Figure 4 presents the results of applying the
algorithm to the system with multidimensional noise (for N=2, 4, 6, and 8), according to the scheme shown in
Figure 2. Due to the precision in the simulation and to illustrate the system’s operation, we show the algorithm
operating with a value of
in each configuration on the complete set of noise matrices described above.
In addition,
Figure 4 shows the calculated control phases and the respective intensity responses. For example, the results for dimension
are presented in the vertical pair
Figure 4.a and
Figure 4.b, while for
they are shown in
Figure 4.c and
Figure 4.d, and so on for higher dimensions. The system control operates on each output sequentially, remaining controlled for
, generating ten temporary operation cycles (
), effectively keeping the outputs maximized and operating the changes at high speed. However, the signals in steady-state show some variability.
Figure 5 shows the system’s response of a MAMZI for
under feedback control, using the abovementioned parameters. In this figure, the steady-state variability and the operating speed differences throughout each work cycle can be seen more clearly. In our analysis of the control operation, we used the following four indicators to evaluate the performance of the proposed control method:
Control velocity: evaluates the speed at which the algorithm reaches its maximum intensity, equivalent to stabilizing the phase noise. Therefore, the time taken by the stabilized signal intensity to go from
(
) to
(
) is measured, as we show in the
Figure 5. It is important to note that the sampling period is an arbitrary variable, so the control speed (
) is given by:
Root Mean Square Error: quantifies the steady-state control error on the stabilized signal. Since the stabilization cycle is
, the signal is assumed to reach the steady state from time
. The upper limit is set at time
to obtain a robust statistic (see
Figure 5). The
calculation is performed considering this time interval, being estimated from:
where
is the total number of points sampled betweeen
and
.
Visibility: quantifies the optical interference characteristics of a communications channel. It is especially crucial in quantum systems, where the visibility of the system directly influences the fidelity of quantum state preparation and measurement. Visibility is defined by:
where
and
correspond to maximum and minimum value of
respectively.
Crosstalk: measures the ratio of unwanted information (
) from an external channel to the information that this channel can transmit (
) with
, mathematically it is expressed as:
Figure 6 shows the results obtained for the
algorithm using a fixed step considering
. It is observed that the RMSE remains below
for systems smaller than N=6, while for larger N the error does not exceed
with
. Values for
lower than three were discarded since they caused the system to lose controllability. Furthermore, an increase in
accelerates the control system’s response, although it increases the error, indicating a less precise operation. This same trend is reflected in the visibility and crosstalk graphs, where it can be seen that, for minimum values for
the system shows less contamination by crosstalk and greater visibility of interference. However, by increasing
, the control system works faster when seeking system stabilization, at the cost of more significant crosstalk contamination and a loss in visibility quality.
Based on the above, improving the system’s performance is possible by adapting the step () to increase when more speed is required and reduce when greater precision is needed. To implement this adaptive control, it is proposed that the step be proportional to the error, i.e., , where A is a system parameter that regulates the impact of the signal on the value , according to previous results, .
Figure 7 compares the mean and standard deviation values of RMSE and
as a function of
N and
A, using the proposed adaptive control. It is observed that both the RMSE and its standard deviation decrease as
A increases for systems with
. On the other hand, for systems with
, the improvements are minimal. Furthermore, the speed and its standard deviation increase with
A, and, as expected, less complex systems (fewer arms) perform better in terms of RMSE and speed.
Regarding visibility, it is observed that adaptive control maintains practically constant values, regardless of
A, for systems with
. For systems with a more significant number of arms (
), an increase in visibility is observed, accompanied by a decrease in its standard deviation as
A increases (
Figure 7). It is relevant to note that almost all visibility values exceed 0.99.
The application of adaptive control maintains practically constant values of visibility, regardless of the increase in speed for all the systems studied, and a decrease in its standard deviation is observed as
A increases (
Figure 7), which implies that the system is more stable despite being accelerated. It is relevant to highlight that almost all visibility values exceed 0.99. Similarly to visibility, crosstalk shows a mean value that tends to remain constant for all values of
N as
A increases, following a similar trend in its standard deviation. In addition, it presents lower values concerning
with fixed steps, reflecting a clear improvement in the control process.
In summary, the results obtained by the adaptive steps show that the steady-state error remains below for systems with and below for higher-order systems, even with higher A. This behavior directly influences the optical parameters, with visibility remaining above 0.99 and crosstalk below in all cases analyzed, regardless of the system order.