Stochastic quantization: Stabilizing quantum models

Kunio Yasue

Ana Marı́a Cetto, Luis de la Peña

Chapter 17 begins by approaching the general problem of a quantum system subject to a time-dependent perturbation. We then apply perturbation theory to the interaction of atomic matter with the radiation field and derive the most important radiative corrections. Atomic transitions – both induced and spontaneous – are discussed in detail, with some important applications. The formal introduction of field quantization allows us to make contact with quantum electrodynamics and quantum optics and discuss the Jaynes–Cummings model. Stochastic electrodynamics, in which the matter–field interaction is not a perturbation but is present ab initio, is used to explain atomic stability and its breakdown leading to radiative transitions. In the same spirit, the quantization of the field is physically justified by deriving the conventionally postulated field operators and their basic commutators. The chapter ends with a brief introduction to second quantization, an elegant and powerful formalism for describing and analyzing quantum many-body systems.

Swarnav Banik, Elliot Greenwald, Xing Pan

Laser frequency stabilization is conventionally analyzed using continuous-time control theory, which accurately models analog feedback but is insufficient for digital implementations where quantization, sampling, and stochastic noise shape the dynamics. In modern digital laser systems, such as Photonic Integrated Circuit (PIC)-based lasers, finite discriminator and actuator resolution, sampling delays, and measurement noise introduce stochastic behavior that deterministic models do not capture. We present a discrete-time Markov-state framework that models the evolution of the quantized actuator in a digital laser frequency lock, with state-transition probabilities determined by the frequency discriminator response, noise statistics, and implemented digital control logic. The steady-state actuator and locked-laser frequency distributions are obtained directly from the unit-eigenvalue solution of the transition matrix, providing immediate access to key stability metrics without long time-domain simulations. For white frequency noise, we show that the Markov formulation is exact under decorrelated sampling and update schemes, while correlated discriminator sampling introduces a predictable inflation of actuator variance without shifting the operating point. In the presence of colored noise, long-range temporal correlations induce sampling-dependent deviations in both actuator mean and variance, defining the regime of validity of the memoryless Markov description. This framework provides a compact and physically transparent tool for analyzing and optimizing digitally stabilized lasers in integrated photonic systems.

Stanley Gudder

Arash Kh. Sichani, Igor G. Vladimirov, Ian R. Petersen

This paper is concerned with open quantum systems whose dynamic variables satisfy canonical commutation relations and are governed by quantum stochastic differential equations. The latter are driven by quantum Wiener processes which represent external boson fields. The system-field coupling operators are linear functions of the system variables. The Hamiltonian consists of a nominal quadratic function of the system variables and an uncertain perturbation which is represented in a Weyl quantization form. Assuming that the nominal linear quantum system is stable, we develop sufficient conditions on the perturbation of the Hamiltonian which guarantee robust mean square stability of the perturbed system. Examples are given to illustrate these results for a class of Hamiltonian perturbations in the form of trigonometric polynomials of the system variables.