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Journal of Mathematical Physics

A posterior Schrödinger equation for continuous nondemolition measurement

V. P. Belavkin

Published Dec 1, 1990• 74 citations
Publisher

Abstract

A continuous model for a nondemolition observation of an atom is given. An equation for the corresponding instrument is found and a stochastic dissipative Schrödinger equation for the unnormalized posterior wave function of the atom is derived. It is shown that the continuously observed isolated atom relaxes to the ground state without mixing.

DOI
10.1063/1.528946

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A continuous model for a nondemolition observation of an atom is given. An equation for the corresponding instrument is found and a stochastic dissipative Schrödinger equation for the unnormalized posterior wave function of the atom is derived. It is shown that the continuously observed isolated atom relaxes to the ground state without mixing.
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{)}^{1/2})1/2, where \ensuremath{\lambda} is the accuracy coefficient of an indirect nondemolition measurement of the particle's position.
Abstract. We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous– spectrum and continuous-in-time observations. The “collapsed state–vector” after the “objectification ” is simply treated as a random vector of the a posteriori state given by the quantum filtering, i.e., the conditioning of the a priori induced state on the corresponding reduced algebra. The nonlinear phenomenological equation of “continuous spontaneous localization ” has been derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments. 1. The status of quantum measurement theory
A continuous quantum-nondemolition measurement scheme that is based on intensity dependent atomic scattering in three-state systems is analyzed. The proposed scheme measures the number of photons in an optical cavity mode by observing the scattering rate of the atomic medium. The time evolution of the coupled atom-field system is analyzed using the Monte Carlo wave-function approach. The wave-function collapse occurs continuously as a result of the presence or absence of scattering events at each time instant. It is shown that the measurement is logically reversible and the information contained in the premeasurement wave function is preserved during the measurement interaction.
We derive the stochastic Schr\"odinger equation for the limit of continuous weak measurement where the observables monitored are canonical position and momentum. To this end we extend an argument due to Smolianov and Truman from the von Neumann model of indirect measurement of position to the Arthurs and Kelly model for simultaneous measurement of position and momentum. We require only unbiasedness of the detector states and an integrability condition sufficient to ensure a central limit effect. Despite taking a weak interaction as opposed to a weak measurement limit, the resulting stochastic wave equation is of the same form as that derived in a recent paper by Scott and Milburn for the specific case of joint Gaussian states.