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arXiv:quant-ph/0512192 · Quantum and Semiclassical Optics Journal of the European Optical Society Part B

A stochastic Hamiltonian approach for quantum jumps, spontaneous localizations, and continuous trajectories

V P Belavkin, O Melsheimer

Published Feb 1, 1996• 18 citations
arXivHTMLPDF

Abstract

We give an explicit stochastic Hamiltonian model of discontinuous unitary evolution for quantum spontaneous jumps like in a system of atoms in quantum optics, or in a system of quantum particles that interacts singularly with `bubbles' which admit a continual counting observation. This model allows one to watch a quantum trajectory in a photodetector or in a cloud chamber by spontaneous localizations of the momenta of the scattered photons or bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained from a Hamiltonian singular interaction as a result of quantum filtering, i.e. a sequential time continuous conditioning of an input quantum process by the output measurement data. We show that in the case of indistinguishable particles or atoms, the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the non-stochastic Schrödinger equation only in the mean-field approximation, whereas the central limit yields Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.

DOI
10.1088/1355-5111/8/1/013

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Cambridge University Press eBooks•Nov 12, 2009•0 citations
We give an explicit stochastic Hamiltonian model of discontinuous unitary evolution for quantum spontaneous jumps like in a system of atoms in quantum optics, or in a system of quantum particles that interacts singularly with `bubbles' which admit a continual counting observation. This model allows one to watch a quantum trajectory in a photodetector or in a cloud chamber by spontaneous localizations of the momenta of the scattered photons or bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained from a Hamiltonian singular interaction as a result of quantum filtering, i.e. a sequential time continuous conditioning of an input quantum process by the output measurement data. We show that in the case of indistinguishable particles or atoms, the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the non-stochastic Schrödinger equation only in the mean-field approximation, whereas the central limit yields Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.
We give an explicit stochastic Hamiltonian model of discontinuous unitary evolution for quantum spontaneous jumps like in a system of atoms in quantum optics, or in a system of quantum particles that interacts singularly with "bubbles" which admit a continual counting observation. This model allows to watch a quantum trajectory in a photodetector or in a cloud chamber by spontaneous localisations of the momentums of the scattered photons or bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained from a Hamiltonian singular interaction as a result of quantum filtering, i.e., a sequential time continuous conditioning of an input quantum process by the output measurement data. We show that in the case of indistinguishable particles or atoms the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the nonstochastic Schroedinger equation only in the mean field approximation, whereas the central limit yelds Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.
We develop a rigorous treatment of discontinuous stochastic unitary evolution for a system of quantum particles that interacts singularly with quantum "bubbles" at random instants of time. This model of a "cloud chamber" allows to watch and follow with a quantum particle along the trajectory in the cloud chamber by sequential unsharp localization of spontaneous scatterings of the bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained as the result of quantum filtering theory, i.e., a theory describing the conditioning of the a priori quantum state by the measurement data. We show that in the case of indistinguishable particles the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the nonstochastic Schroedinger equation only in the mean field approximation, whereas the central limit yields Gaussian mixing fluctuations described by stochastic reduction equations of diffusive type.
Measurements continuous in time have been consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after a measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented for the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.
Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of ``typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of ``exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...
Introduction A very general concept of a quantum trajectory would be the path taken by the state of a quantum system over time. This state could be conditioned upon measurement results, as we considered in Chapter 1. This is the sort of quantum trajectory we are most interested in, and it is generally stochastic in nature. In ordinary use, the word trajectory usually implies a path that is also continuous in time. This idea is not always applicable to quantum systems, but we can maintain its essence by defining a quantum trajectory as the path taken by the conditional state of a quantum system for which the unconditioned system state evolves continuously. As explained in Chapter 1, the unconditioned state is that obtained by averaging over the random measurement results which condition the system. With this motivation, we begin in Section 4.2 by deriving the simplest sort of quantum trajectory, which involves jumps (that is, discontinuous conditioned evolution). In the process we will reproduce Lindblad's general form for continuous Markovian quantum evolution as presented in Section 3.6. In Section 4.3 we relate these quantum jumps to photon-counting measurements on the bath for the model introduced in Section 3.11, and also derive correlation functions for these measurement records. In Section 4.4 we consider the addition of a coherent field (the ‘local oscillator’) to the output before detection. In the limit of a strong local oscillator this is called homodyne detection, and is described by a continuous (diffusive) quantum trajectory.