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arXiv:quant-ph/9705008 · Physical Review Letters

Coupling Classical and Quantum Variables using Continuous Quantum Measurement Theory

Lajos Diósi, J. J. Halliwell

Published Oct 5, 1998• 84 citations
arXivHTMLPDF

Abstract

Continuous quantum measurement theory is used to construct a phenomenological description of the interaction of a quasiclassical variable $X$ with a quantum variable $x$, where the quasiclassical nature of $X$ is assumed to have come about as a result of decoherence. The state of the quantum subsystem evolves according to the stochastic nonlinear Schr\"odinger equation of a continuously measured system, and the classical system couples to a stochastic $c$ number $\overline{x}(t)$ representing the imprecisely measured value of $x$. The theory gives intuitively sensible results even when the quantum system starts out in a superposition of well-separated localized states.

DOI
10.1103/physrevlett.81.2846

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Continuous quantum measurement theory is used to construct a phenomenological description of the interaction of a quasiclassical variable XXX with a quantum variable xxx, where the quasiclassical nature of XXX is assumed to have come about as a result of decoherence. The state of the quantum subsystem evolves according to the stochastic nonlinear Schr"odinger equation of a continuously measured system, and the classical system couples to a stochastic ccc number x‾(t)\overline{x}(t)x(t) representing the imprecisely measured value of xxx. The theory gives intuitively sensible results even when the quantum system starts out in a superposition of well-separated localized states.
The measuring process is an external intervention in the dynamics of a quantum system. It involves a unitary interaction of that system with a measuring apparatus, a further interaction of both with an unknown environment causing decoherence, and then the deletion of a subsystem. This description of the measuring process is a substantial generalization of current models in quantum measurement theory. In particular, no ancilla is needed. The final result is represented by a completely positive map of the quantum state $\ensuremath{\rho}$ (possibly with a change of the dimensions of $\ensuremath{\rho}).$ A continuous limit of the above process leads to Lindblad's equation for the quantum-dynamical semigroup [Commun. Math. Phys. 48,48,48, 119 (1976)].
We study the dynamics of classical and quantum systems undergoing a continuous measurement of position by schematizing the measurement apparatus with an infinite set of harmonic oscillators at finite temperature linearly coupled to the measured system. Selective and non-selective measurement processes are then introduced according to a selection of or an average over all possible initial configurations of the measurement apparatus. At quantum level, the selective processes are described by a nonlinear stochastic Schrödinger equation whose solutions evolve into properly defined coherent states in the case of linear systems. For arbitrary measured systems, classical behaviour is always recovered in the macroscopic limit.
We study the dynamics of classical and quantum systems undergoing a continuous measurement of position by schematizing the measurement apparatus with an infinite set of harmonic oscillators at finite temperature linearly coupled to the measured system. Selective and nonselective measurement processes are then introduced according to a selection of or an average over all possible initial configurations of the measurement apparatus. At quantum level, the selective processes are described by a nonlinear stochastic Schrödinger equation whose solutions evolve into properly defined coherent states in the case of linear systems. For arbitrary measured systems, classical behavior is always recovered in the macroscopic limit.
We use the decoherent histories approach to quantum theory to derive the form of an effective theory describing the coupling of classical and quantum variables. The derivation is carried out for a system consisting of a large particle coupled to a small particle with the important additional feature that the large particle is also coupled to a thermal environment producing the decoherence necessary for classicality. The effective theory is obtained by tracing out both the environment and the small particle variables. It consists of a formula for the probabilities of a set of histories of the large particle, and depends on the dynamics and initial quantum state of the small particle. It has the form of an almost classical particle coupled to a stochastic variable whose probabilities are determined by a formula very similar to that given by quantum measurement theory for continuous measurements of the small particle's position. The effective theory gives intuitively sensible answers when the small particle is in a superposition of localized states (unlike the simple mean field approach of coupling to the expectation values of the small system). The derived effective theory suggests a form of the semiclassical theory even when the quantum theory of the large system is not known, as is the case, for example, when a classical gravitational field is coupled to a quantized matter field, thus offering a new approach to the back reaction problem.
Decoherence of a quantum system (which then starts to display classical features) results from the interaction of the system with the environment, and is well described in the framework of the theory of continuous quantum measurements (CQM). Reviewed are the various approaches to the CQM theory, and the approach based on the effective complex Hamiltonians is discussed in greater detail. The effective complex Hamiltonian is derived from the restricted path integral, which emphasizes the role of information in the dynamics of the system being measured. The complex Hamiltonian is used for analyzing the CQM of energy in a two-level system. Such measurement is demonstrated to be capable of monitoring the quantum transition, and the back effect of monitoring on the probability of transition is analyzed. The realization of this type of measurement by a long series
Quantum and classical stochastic evolution equations are derived for the statistical state of a system subject to continuous observation of a position variable. In the classical case this results in a stochastic Fokker - Planck equation for the conditional state of the system conditioned on observation records. In the quantum case a stochastic master equation results, or a stochastic Schrödinger equation if the system starts in a pure state. In both cases the resulting equations are nonlinear. A simple modification of the model results in a linear stochastic Schrödinger equation which can also be interpreted as a conditional evolution equation for observations which give no information about the measured system but indicate the size of the random perturbations induced by the environment coupling.