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arXiv:1207.1196 · Journal of the Optical Society of America B

Filtering equation for measurement of a coherent channel

Anita Magdalena Dąbrowska, P. Staszewski

Published Apr 25, 2011• 3 citations
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Abstract

We propose a stochastic model for continuous photon counting and heterodyne measurement of a coherent source. A nonlinear filtering equation for the posterior state of a single-mode field in a cavity is derived by using the methods of the quantum stochastic calculus. The posterior dynamics is found for the observation of a Bose field being initially in a coherent state. The filtering equations for the counting and diffusion processes are given.

DOI
10.1364/josab.28.001238

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We propose a stochastic model for continuous photon counting and heterodyne measurement of a coherent source. A nonlinear filtering equation for the posterior state of a single-mode field in a cavity is derived by using the methods of the quantum stochastic calculus. The posterior dynamics is found for the observation of a Bose field being initially in a coherent state. The filtering equations for the counting and diffusion processes are given.
We present the Belavkin filtering equation for the intense balanced heterodyne detection in a unitary model of an indirect observation. The measuring apparatus modelled by a Bose field is initially prepared in a coherent state and the observed process is a diffusion one. We prove that this filtering equation is relaxing: any initial square-integrable function tends asymptotically to a coherent state with an amplitude depending on the coupling constant and the initial state of the apparatus. The time development of a squeezed coherent state is studied and compared with the previous results obtained for the measuring apparatus prepared initially in the vacuum state.
In this thesis we study continuously observed open quantum systems from the point of view of quantum filtering theory. The conditioned state of the open system obeys a non-commutative or quantum analogue of the Kushner-Stratonovich equation in classical filtering theory. The thesis consists out of four chapters. The first chapter is a brief introduction to quantum probability theory and quantum stochastic calculus. The second chapter is a description of a photon counting experiment in continuous time within the framework of Davies and Srinivas. The third chapter focusses on the derivation of the quantum filtering or Belavkin equation from the quantum stochastic differential equation governing the interaction between the open system and its environment, the electromagnetic field. The fourth chapter shows how the quantum filtering equation can be used to control quantum systems.
We analyze a simultaneous continuous measurement of photon-counting and homodyne detection. The stochastic master equation or stochastic Schrödinger equation describing the measurement process includes both jump-type and diffusive-type stochastic increments. Analytic expressions of the wave function conditioned on homodyne and photon-counting records are obtained, yielding the probability density distributions and generating functions of the measurement records. Formula for the expectation values of the homodyne records conditioned on a photon-counting event is also derived which quantitatively describes the measurement backaction of photon-counting on the homodyne output. The obtained results are applied to typical initial states --- coherent, number, thermal, and squeezed states. Monte Carlo simulations of the measurement processes are also presented to demonstrate the dynamics of the combined measurement process.
This paper is concerned with a filtering problem for a\nclass of nonlinear quantum stochastic systems with multichannel nondemolition\nmeasurements. The system-observation dynamics are governed\nby a Markovian Hudson-Parthasarathy quantum stochastic differential\nequation driven by quantum Wiener processes of bosonic fields in vacuum\nstate. The Hamiltonian and system-field coupling operators, as functions\nof the system variables, are represented in a Weyl quantization form.\nUsing the Wigner-Moyal phase-space framework, we obtain a stochastic\nintegro-differential equation for the posterior quasi-characteristic function\n(QCF) of the system conditioned on the measurements. This equation\nis a spatial Fourier domain representation of the Belavkin-Kushner-\nStratonovich stochastic master equation driven by the innovation process\nassociated with the measurements. We also discuss a more specific form\nof the posterior QCF dynamics in the case of linear system-field coupling\nand outline a Gaussian approximation of the posterior quantum state.
The cavity is a fundamental ingredient of quantum optical systems. This paper concerns the behavior of a quantum cavity driven by non-classical field in single-photon state. To this end, the number operator has been opted to reveal the number of photons inside the cavity. Then, the quantum filtering equations have been employed to derive a stochastic master equation for the cavity which is driven by single-photon and is observed by either Homodyne or photon-counting detector. Finally, the state of the cavity has been estimated by the derived equations, and the results have been compared with the conventional master equation.