Phase-Drift Limits and Adaptive Quadrature Readout in Programmable Photonic Processors

The gist The study develops a theory for phase-drift limits and adaptive quadrature readout in programmable photonic processors by analyzing sequential versus simultaneous measurement strategies.

Experimental Basis and Optical Transformation

Programmable photonic processors implement linear transformations through interference in meshes of tunable Mach–Zehnder interferometers (MZIs) The output powers of a multi-input processor depend on the relative optical phases Measurements on an eight-mode programmable processor characterized the free-running relative-phase fluctuations and revealed both broad stochastic evolution and structured spectral components associated with correlated disturbances The processor contains tunable MZIs and external thermo-optic phase shifters

Relative-Phase Increment Model

The paper models the relative phase change during a reconfiguration interval as a zero-mean local increment, defined as δτ = ∆ϕ(t + τ) − ∆ϕ(t) This local transition model is sufficient for the receiver analysis and does not require a separate fit of a diffusion coefficient The relation between the two optical paths and the relative-phase increment is also useful

Estimation of Phase-Increment Variance from Synchronized Records

The synchronized four-port data set provides an empirical route for evaluating the relative-phase change accumulated over a selected measurement interval The pooled estimate of the phase-increment variance can then be formed from the complete set of recordings as Qbτm = X r R r=1 NrX−m−1 n=0 h δe(r) m [n] i squared X r R (Nr − m − 1) This quantity estimates Qτm = Var (δτm), which is the phase-increment variance used in the receiver analysis

Fixed-Order Drift Calculation

For the order C → S, to first order, the reconstructed phase is given by ϕbC→S = ϕ0 + δτ cos squared ϕ0 + O(δ squared τ) The desired end-time estimation error is then eC→S = δτ cos squared ϕ0 − 1 + O(δ squared τ) Similarly, for the reverse S → C ordering, the first-order estimate is ϕbS→C = ϕ0 + δτ sin squared ϕ0 + O(δ squared τ) The corresponding end-time error is eS→C = δτ sin squared ϕ0 − 1 + O(δ squared τ)

Phase-Predicted Adaptive Ordering

A phase-predicted ordering rule measures the locally less informative quadrature first and the more informative quadrature second Its uniform first-order penalty is (3/8−1/π)Qτ, which is 84.9% below the fixed-order value This adaptive ordering rule yields a relative MSE reduction of 1 − 3/8 − 1/π = 0.8488, or 84.9%

Increment-Aware End-Time Estimation

The increment-aware estimator reduces the Fisher information from I to I/(1 + IQτ) by marginalizing the unknown phase increment The resulting MSE is MSES→S,IA = 1/(IS + IC / (1 + IC Qτ)) This correction prevents an estimator from continuing to trust an old phase sample after the uncertainty of the intervening phase change has become dominant

Photon Resources and Architecture Criterion

The information-gain factor g = It / Isp, where It = I(t)C + I(t)S, is defined to compare total Fisher information available to the simultaneous spatial receiver during one complete phase estimation cycle The boundary for adaptive readout is defined by IspQτ < 1 − 1/g 3/8 − 1/π This boundary shows that temporal operation is favored when IspQτ 1

Monte Carlo Validation and Design Study

Exact nonlinear Monte Carlo simulations validate the perturbative laws, quantify robustness to prediction error, and compare simultaneous, fixed-order, increment-aware, and adaptive receivers under a common measurement-noise model The exact simulation confirms the fixed-order first-order expression 3Q/8 in the small-increment regime The adaptive result approaches (3/8 − 1/π)Q as Q → 0

Conclusion

The study developed a detailed theory of spatial and sequential quadrature readout for programmable photonic processors The resulting crossover conditions separate the possible optical-information advantage of temporal operation from the phase change accumulated during reconfiguration The information-gain factor g then produces explicit dimensionless boundaries in the product IspQτ for fixed-order, symmetric local increment-aware, and adaptive readout Exact Monte Carlo simulations validate the analytical laws, quantify robustness to prediction error, and show the additional limitations introduced by nonlinear quadrature observations The synchronized four-port records provide an empirical route for estimating Qτ at a selected delay

Acknowledgments

This work was supported by the Federal Ministry of Research, Technology and Space of Germany through the Q-TREX project under Grant 16KISR026 and the QDCamNetz project under Grant 16KISQ077, and by the Bavarian state government through the Munich Quantum Valley under the Hightech Agenda Bayern Plus

Disclosures

The authors declare no conflicts of interest Data availability No new experimental data were acquired for this study The processor recordings analyzed here, together with the simulation code and numerical data supporting the figures, are available from the authors upon reasonable request

References

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1. Adaptive Quadrature Readout: The AI system can implement an adaptive ordering law that measures the locally less informative quadrature first and the more informative quadrature second, achieving a uniform first-order penalty is (3/8−1/π)Qτ, which is 84.9% below the fixed-order value.

2. Increment-Aware Estimator: The system can use an increment-aware estimator from a local state-space model to reduce the variance of a stale phase observation by Qτ, leading to a Fisher information reduction from I to I/(1 + IQτ).

3. Resource Allocation Criterion: The AI can select between spatial and temporal readouts based on dimensionless boundaries in the product IspQτ, such as the condition that temporal readout is favored when IspQτ 1.

Phase fluctuations between optical inputs limit programmable photonic processors because their output powers depend on coherent interference. We study the phase-drift penalty that arises when sine and cosine quadratures are measured sequentially rather than simultaneously. The analysis is motivated by measurements from an eight-mode programmable photonic processor, including 35 free-running recordings of 300 s acquired at approximately 125 samples per second per channel. These recordings provide an empirical route for estimating the phase-increment variance at a selected reconfiguration interval. The estimate is defined at the time of the second measurement. For fixed quadrature order, perturbation of the atan2 reconstruction gives e C to S=-δ τ 2ϕ 0+O(δ τ 2) and e S to C=-δ τ 2ϕ 0+O(δ τ 2). Writing Q τ= Var(δ τ), uniform phase averaging gives the first-order drift mean-square error 3Q τ/8. A phase-predicted ordering rule measures the locally less informative quadrature first and the more informative quadrature second. Its uniform first-order penalty is (3/8-1/π)Q τ, which is 84.9 percent below the fixed-order value. We also derive an increment-aware estimator from a local state-space model. Marginalizing the unknown phase increment increases the variance of a stale phase observation by Q τ, reducing its Fisher information from I to I/(1+IQ τ). For ideal balanced Poisson detection, the Fisher information of each quadrature equals its detected signal-photon number. This yields dimensionless architecture boundaries in spatial information and phase-increment variance. Nonlinear Monte Carlo simulations validate the perturbative laws, quantify robustness to prediction error, and compare simultaneous, fixed-order, increment-aware, and adaptive receivers under a common noise model.