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Europhysics Letters (EPL)

Continuous Measurements on the Quantum Kicked Rotor

Thomas Dittrich, R. Graham

Published Apr 1, 1990• 19 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/11/7/002

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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.
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.
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.
We consider the behavior of a classically chaotic quantum system, the periodically driven pendulum, under the influence of a continuous measurement of its angular momentum. Without measurement the system shows dynamical localization, a quantum interference effect which suppresses the classical chaotic diffusion in momentum space. The coupling of the system to a measuring device destroys its coherence and thus leads to delocalization. This is studied on the level of individual systems including the recording of the measurement results. For that purpose we analyze the appropriate stochastic Schr\"odinger equation, from which a stochastic quantum map is derived as an effective tool for numerical simulation of measurement trajectories. We show that a continuous momentum measurement restores the diffusive behavior of the system in momentum space, but that for sufficiently low accuracy of measurement the corresponding diffusion constant is smaller than the classical one. This is reflected by an equal diffusive growth of the recorded measurement results. Thus we find signatures of the classical chaos and of the dynamical localization both in the behavior of the measured quantum system and in the corresponding signal of the measuring device.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.
The kicked rotor provides a simple yet powerful model for introducing many of the central concepts of classical and quantum chaos. Despite its apparent simplicity, it exhibits rich dynamical behavior and has found applications across a wide range of fields, including atomic and optical physics, condensed matter physics, and emerging quantum technologies. This chapter begins by exploring foundational ideas using the kicked rotor as a unifying framework. We first discuss the transition from regular to chaotic motion in the classical system, and then introduce key quantum phenomena such as dynamical localization and quantum resonances. Special attention is devoted to the emergence of characteristic time scales and their role in the quantum-classical correspondence. To make these ideas more concrete, we also provide a brief overview of experimental realizations of the kicked rotor and its variants, illustrating how theoretical concepts are implemented in practice. In the second part of the chapter, we guide the reader toward more recent and advanced developments. Topics include near-resonant dynamics, topological features of kicked systems, the emergence of quantum dynamical phases inferred from classical transport properties, and extensions to non-Hermitian physics. We conclude with a discussion of open problems and future perspectives, outlining directions in which the kicked rotor continues to offer valuable insights.