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arXiv:quant-ph/0512138 · Physical Review A

Nondemolition observation of a free quantum particle

V. P. Belavkin, P. Staszewski

Published Feb 1, 1992• 121 citations
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Abstract

A stochastic model of a continuous nondemolition observation of a quantum system is presented. The nonlinear stochastic wave equation describing the posterior dynamics of the observed quantum system is solved for a free particle of mass m>0. It is shown that the dispersion of the Gaussian wave packet does not increase to infinity as for a free unobserved particle, but tends to the finite limit ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\infty}}}^{2}$=(\ensuremath{\Elzxh}/2\ensuremath{\lambda}m${)}^{1/2}$, where \ensuremath{\lambda} is the accuracy coefficient of an indirect nondemolition measurement of the particle's position.

DOI
10.1103/physreva.45.1347

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A stochastic model of a continuous nondemolition observation of a quantum system is presented. The nonlinear stochastic wave equation describing the posterior dynamics of the observed quantum system is solved for a free particle of mass m>0. It is shown that the dispersion of the Gaussian wave packet does not increase to infinity as for a free unobserved particle, but tends to the finite limit \ensuremathτ\ensuremath∞2{\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\infty}}}^{2}\ensuremathτ\ensuremath∞2​=(\ensuremath{\Elzxh}/2\ensuremath{\lambda}m)1/2{)}^{1/2})1/2, where \ensuremath{\lambda} is the accuracy coefficient of an indirect nondemolition measurement of the particle's position.
Recently, the stochastic nonlinear wave equation describing the dynamics of an observed quantum system has been solved for a free particle [Phys. Rev. A 45, 1347 (1992)]. In this paper, transient effects that occur during a continuous observation of the particle position are discussed. It is shown that the dispersion of the Gaussian wave packet of the particle always decreases in the beginning of the observation, then takes the form of oscillations which, after a short time, decay rapidly to the asymptotic value.
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A brief presentation is given of the basic concepts in quantum probability theory in analogy to the classical ones, as well as the necessary information about noncommutative stochastic integration and its explicit representation in the Fock space. Algebraic differential relations that define quantum diffusion motion with nondemolition observation are derived. In the Markov case we obtain the stochastic equation of quantum diffusion filtering which is analogous to the Zakai equation of classical nonlinear filtering. A quantum linear stochastic model with continuous observation is given for which this equation is reduced to a linear stochastic quantum filtering equation of Kalman-Busi type and to the operator Riccati equation. We also consider an example of observing the coordinate of a free quantum particle, for which the solution of a stationary quantum filtering problem is obtained.
A quantum stochastic model for an open dynamical system (quantum receiver) and output multi-channel of observation with an additive nonvacuum quantum noise is given. A quantum stochastic Master equation for the corresponding instrument is derived and quantum stochastic filtering equations both for the Heisenberg operators and the reduced density matrix of the system under the nondemolition observation are found. Thus the dynamical problem of quantum filtering is generalized for a noncommutative output process, and a quantum stochastic model and optimal filtering equation for the dynamical estimation of an input Markovian process is found. The results are illustrated on an example of optimal estimation of an input Gaussian diffusion signal, an unknown gravitational force say in a quantum optical or Weber's antenna for detection and filtering a gravitational waves.