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Physical Review A

Continuous quantum measurements and chaos

Thomas Dittrich, R. Graham

Published Oct 1, 1990• 40 citations
Publisher

Abstract

The quantized standard map (kicked rotor) is coupled to macroscopic systems acting as measuring devices for the probability distribution of the action variable or some of its moments. As a result of such measurements, which are continuous in time but have a limited time resolution, localization of the action variable is destroyed in sufficiently long time scales and replaced by diffusion. The diffusion constant, in general, differs from that of the classical chaotic diffusion and depends on the measurement performed.

DOI
10.1103/physreva.42.4647

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The quantized standard map (kicked rotor) is coupled to macroscopic systems acting as measuring devices for the probability distribution of the action variable or some of its moments. As a result of such measurements, which are continuous in time but have a limited time resolution, localization of the action variable is destroyed in sufficiently long time scales and replaced by diffusion. The diffusion constant, in general, differs from that of the classical chaotic diffusion and depends on the measurement performed.
A simple model for the coupling of a quantum system to a macroscopic measuring device is proposed, which allows to describe continuous measurements with finite time resolution. It is used to derive a quantum map, generating the dynamics of the quantum kicked rotor under the influence of continuous measurements. Numerical experiments with this quantum map show that the incoherent back-action of the measuring device always destroys dynamical localization on a sufficiently long time scale, and renders the measured dynamics diffusive.
We investigate the quantum-classical transition in the $\ensuremath{\delta}$-function-kicked rotor and the attainment of the classical limit in terms of measurement-induced state localization. It is possible to study the transition by fixing the environmentally induced disturbance at a sufficiently small value, and examining the dynamics as the system is made more macroscopic. When the system action is relatively small, the dynamics is quantum mechanical and when the system action is sufficiently large there is a transition to classical behavior. The dynamics of the rotor in the region of transition, characterized by the late-time momentum diffusion coefficient, can be strikingly different from both the purely quantum and classical results. Remarkably, the early-time diffusive behavior of the quantum system, even when different from its classical counterpart, is stabilized by the continuous measurement process. This shows that such measurements can succeed in extracting essentially quantum effects. The transition regime studied in this paper is accessible in ongoing experiments.
It is well established that classical chaotic motion is suppressed by quantum mechanics. The paradigm theoretical system has been the ?-kicked rotor, where the suppression manifests itself as dynamical localization. This effect is saturation of momentum growth after the quantum break time, with a resulting exponentially localized distribution. Classical analyses of this system have also identified regimes of anomalous diffusion, characterized by “Levy-type” flights in phase space [1]. This behavior is considerably more complex than simple diffusion and has attracted much interest in recent years. The quantum system was predicted to exhibit an increase in localization length near regions of anomalous diffusion [2], and recent theoretical work predicts very large growth of energy in this case, even for systems that are far from the semiclassical limit [3].
The question of how to quantize a system whose classical dynamics is chaotic has long been a puzzle. The hallmark of classical chaos is sensitive dependence on initial conditions: A small perturbation of the system changes its evolution in a way that grows exponentially with time. But in quantum mechanics a small perturbation of the initial state will in general be mapped into a small perturbation of the final state. In reeveral cent years the study of quantum systems whose underlying classical dynamics is chaotic has been an active area of research.