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Journal of Physics Conference Series

A classical counting process and non-Wiener stochastic differential equations in quantum optics

Ashat M. Basharov

Published Jul 1, 2019• 0 citations
Publisher

Abstract

Abstract The von Neumann projection postulate under the registration of a spontaneously emitted photon is described through the stochastic counting process as a non-Wiener linear stochastic differential equation for a non-normalized atomic state vector.
DOI
10.1088/1742-6596/1283/1/012002

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Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.
An equation has been obtained for the non-normalized vector of an a posteriori state of a two-level particle at its resonant pumping by a coherent field at the detection of photons emitted by the particle. The equation is linear and belongs to the class of stochastic differential equations controlled by the classical Wiener and Poisson (reading) processes. The reading process describes photo detection and the Wiener process describes an additional dephasing of particles, which has not yet been taken into account.
Abstract Quantum stochastic differential equations are shown to naturally arise because of the mathematical background of the correctness of the Schrödinger equation, which describes the interaction of an open quantum system with broadband quantized fields in the Markov approximation. The main classical and quantum stochastic processes appear in the notation in the Markov approximation of the operator of the electric broadband quantized field strength and the effective Hamiltonian of the interaction of a quantum system with it obtained by the methods of algebraic resonance perturbation theory. All basic stochastic quantum processes, including the counting and two-quantum Lévy processes, are constructed based on quantum white noise, which differs significantly from the classical case. At the same time, the apparatus of stochastic differential equations demonstrates consistency and parallelism.