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Communications in Theoretical Physics

Quantum to Classical Transition in a System of Two Coupled Kicked Rotors

Wen-Lei Zhao, Jie Quan-Lin

Published Mar 1, 2009• 12 citations
Publisher

Abstract

We investigate the quantum-classical transition in a system of two coupled kicked rotors. We find that when the mass of one kicked rotor is much smaller than the other's, the influence of the light kicked rotor is still able to make decoherence of the heavy one. This leads to the quantum-classical transition of the heavy kicked rotor. We demonstrate this by two different coupling potentials.

DOI
10.1088/0253-6102/51/3/17

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We investigate the quantum-classical transition in a system of two coupled kicked rotors. We find that when the mass of one kicked rotor is much smaller than the other's, the influence of the light kicked rotor is still able to make decoherence of the heavy one. This leads to the quantum-classical transition of the heavy kicked rotor. We demonstrate this by two different coupling potentials.
We investigate the quantum-classical transition of a kicked rotor (KR) under perturbation by a second one. The influence of such a chaotic KR makes decoherence of the first one, resulting in the emergence of classical diffusion from its quantum dynamics. Such quantum-classical transition persists by decreasing the effective Planck's constant ℏ, and at the same time, decreasing the mass of the second KR and the interaction strength proportionally. In the limit of ħ → 0, due to vanishing small mass and interaction, the second KR has almost no effect on the classical dynamics of the first one. We demonstrate this via two different coupling potentials.
The understanding of how classical dynamics can emerge in closed quantum systems is a problem of fundamental importance. Remarkably, while classical behavior usually arises from coupling to thermal fluctuations or random spectral noise, it may also be an innate property of certain isolated, periodically driven quantum systems. Here, we experimentally realize the simplest such system, consisting of two coupled, kicked quantum rotors, by subjecting a coherent atomic matter wave to two periodically pulsed, incommensurate optical lattices. Momentum transport in this system is found to be radically different from that in a single kicked rotor, with a breakdown of dynamical localization and the emergence of classical diffusion. Our observation, which confirms a long-standing prediction for many-dimensional quantum-chaotic systems, sheds new light on the quantum-classical correspondence.
Via a system of two kicked particles that are coupled in an infinite square well, we numerically show that the interaction with a particle of very small mass is able to lead to a quantum-to-classical transition on condition that the corresponding classical dynamics is almost unaffected. With the decrease of the mass of one of the particles, its effect on the classical dynamics of the other one decreases. Such an effect is even negligible if the mass of the particle is small enough. The classically chaotic dynamics of this small particle is effective for promoting the decoherence of the heavy particle. Therefore its quantum behavior exhibits the transition from the dynamical localization to the classically chaotic diffusion with the decrease of the effective Planck's constant ħ. Under the perturbation from the small particle, the linear entropy is rapidly saturated as time passes by. With the decrease of ħ, the time-averaged linear entropy exponentially increases from zero to almost unity.
Lindblad dynamics is one of the most popular approaches to study open quantum systems. This paper studies a kicked rotor; in particular, a two rotor system where one of the rotors is treated as the environment and the other is studied with the Lindblad master equation. The energy levels are solved explicitly in terms of Bessel functions and the results show that the energy levels exhibit dissipative behavior with variance in momentum increasing as ⟨p^2⟩=2Dt\langle\hat{p}^2\rangle=2Dt⟨p^​2⟩=2Dt similar to the classical kicked rotor. This result shows us how interaction between systems can break down quantum phenomena and result in more classical results, bridging the gap between the quantum and the classical world.
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.