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IEEE Conference on Decision and Control (CDC)

Pathwise solution of a class of quantum filtering equations

I. Kurniawan, M. R. James

Published Jan 1, 2004• 2 citations
Publisher

Abstract

The purpose of this paper is to apply Clark's pathwise reformulation of the equations of nonlinear filtering to a class of filtering equations that arise in quantum physics. Specifically, we consider a class of stochastic master equations driven by white noise. We also derive Clark's robust approximation and illustrate the ideas applying them to an imperfectly observed two-level atom continuously monitored by homodyne photodetection.

DOI
10.1109/cdc.2004.1428947

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The purpose of this paper is to apply Clark's pathwise reformulation of the equations of nonlinear filtering to a class of filtering equations that arise in quantum physics. Specifically, we consider a class of stochastic master equations driven by white noise. We also derive Clark's robust approximation and illustrate the ideas applying them to an imperfectly observed two-level atom continuously monitored by homodyne photodetection.
In this paper we consider an alternative formulation of a class of stochastic wave and master equations with scalar noise that are used in quantum optics for modelling open systems and continuously monitored systems. The reformulation is obtained by applying J.M.C. Clark's pathwise reformulation technique from the theory of classical nonlinear filtering. The pathwise versions of the stochastic wave and master equations are defined for all driving paths and depend continuously on them. In the case of white noise equations, we derive analogs of Clark's robust approximations. The results in this paper may be useful for implementing filters for the continuous monitoring and measurement feedback control of quantum systems, and for developing new types of numerical methods for unravelling master equations. The main ideas are illustrated by an example.
In this paper we consider an alternative formulation of a class of stochastic wave and master equations with scalar noise that are used in quantum optics for modelling open systems and continuously monitored systems. The reformulation is obtained by applying J.M.C. Clark's pathwise reformulation technique from the theory of classical nonlinear filtering. The pathwise versions of the stochastic wave and master equations are defined for all driving paths and depend continuously on them. In the case of white noise equations, we derive analogs of Clark's robust approximations. The results in this paper may be useful for implementing filters for the continuous monitoring and measurement feedback control of quantum systems, and for developing new types of numerical methods for unravelling master equations. The main ideas are illustrated by an example.
We study quantum filters that are driven by basic quantum noises and construct classical versions. Our approach is based on exploiting the quantum markovian component of the observation and measurement processes of the filters. This approach leads in a natural way the classical versions for a class of quantum filters. We consider quantum white noises derived from Wiener and Poisson processes that drive the signal and measurement processes and derive the recursive filtering equations using classical machinery.
The aim of this paper is to determine quantum master and filter equations for systems coupled to continuous-mode single photon fields. The system and field are described using a quantum stochastic unitary model, where the continuous-mode single photon state for the field is determined by a wavepacket pulse shape. The master equation is derived from this model and is given in terms of a system of coupled equations. The output field carries information about the system from the scattered photon, and is continuously monitored. The quantum filter is determined with the aid of an embedding of the system into a larger system, and is given by a system of coupled stochastic differential equations. An example is provided to illustrate the main results.