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Reviews in Mathematical Physics

Localization and Analytic Properties of the Solutions of the Simplest Quantum Filtering Equation

Vassili Nikitich Kolokoltsov

Published Aug 1, 1998• 18 citations
Publisher

Abstract

The paper deals with the quantum Langevin equation describing a quantum particle with continuously observed position. Special Banach spaces of entire analytic functions are introduced and studied (including a theorem of Paley–Wiener type for them), which comprise all solutions of this equation and in which the uniform convergence (as time tends to infinity) of the solutions to the Gaussian function with a fixed dispersion (selflocalisation or continuous collapse) is proved. The asymptotic behavior at infinity of the mean position and momentum of the limit Gaussian wave packet (which satisfy classical Langevin equations) is also investigated.

DOI
10.1142/s0129055x98000264

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The paper deals with the quantum Langevin equation describing a quantum particle with continuously observed position. Special Banach spaces of entire analytic functions are introduced and studied (including a theorem of Paley–Wiener type for them), which comprise all solutions of this equation and in which the uniform convergence (as time tends to infinity) of the solutions to the Gaussian function with a fixed dispersion (selflocalisation or continuous collapse) is proved. The asymptotic behavior at infinity of the mean position and momentum of the limit Gaussian wave packet (which satisfy classical Langevin equations) is also investigated.
In the article, the complete investigation is given of the large times behavior of the solutions of the Belavkin quantum filtering equation describing a quantum particle with continuously observed coordinate. It turns out that these solutions have an extraordinary property, namely: as t → ∞, any solution will tend to a Gaussian function (for almost all realizations of innovating Wiener process). As a consequence, it follows that the dispersion of coordinate will always tend to the same limit, as t → ∞, not depending on initial data.
We discuss the time evolution of the wave function which is solution of a stochastic Schroedinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. Observing that there exist three time regimes, namely the collapse regime, after which the wave function is localized in space; the classical regime, during which the collapsed wave function moves along a classical path and the diffusive regime, in which diffusion overlaps significantly the deterministic motion, we study the long time behavior of the wave function. We assert that the general solution converges a.s. to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this.
We discuss the time evolution of the wave function which is the solution of a stochastic Schrödinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. We observe that there exist three time regimes: the collapse regime, the classical regime and the diffusive regime. Concerning the latter, we assert that the general solution converges almost surely to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this issue.
We analyze the time-behaviour of the Gaussian wave packet of a quantum particle moving in the field of a linear force, undergoing a continuous nondemolition observation of its position. In particular, the time-dependence of position and momentum dispersions, as well as position-momentum uncertainty is investigated. The discussion includes the case of a free particle and initially coherent state of a harmonic oscillator. It is found that in the beginning of observation the momentum dispersion and position-momentum uncertainty always increase in contrast to the position dispersion. It is also proved that all considered functions asymptotically take the form of harmonic oscillations, decaying to finite values. For initially coherent states, the asymptotic bounds for all considered functions are given.
A stochastic Schrödinger equation is presented to describe simultaneous continuous measurement of the position and momentum of a non-relativistic particle. The equation is solved to yield a state localised in position and momentum contingent on the uncertainty principle. The state is understood as that to which the particle tends after a period of continuous measurement. The solution takes the form of a wave packet of fixed Gaussian shape whose centre undergoes random motion in phase space. It is shown to be locally stable.
Stochastic differential equations in Hilbert space as random nonlinear modified Schrödinger equations have achieved great attention in recent years; of particular interest is the long-time behavior of their solutions. In this note we discuss the long-time behavior of the solutions of the stochastic differential equation describing the time evolution of a free quantum particle subject to spontaneous collapses in space. We explain why the problem is subtle and report on a recent rigorous result, which asserts that any initial state converges almost surely to a Gaussian state having a fixed spread both in position and momentum.