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arXiv:quant-ph/0306043 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Coherent manipulation of quantum δ-kicked dynamics: Faster-than-classical anomalous diffusion

Jiangbin Gong, Hans Jakob Wörner, Paul W. Brumer

Published Aug 18, 2003• 10 citations
arXivHTMLPDF

Abstract

Large transporting regular islands are found in the classical phase space of a modified kicked rotor system in which the kicking potential is reversed after every two kicks. The corresponding quantum system, for a variety of system parameters and over long time scales, is shown to display energy absorption that is significantly faster than that associated with the underlying classical anomalous diffusion. The results are of interest to areas of both quantum chaos and quantum control.

DOI
10.1103/physreve.68.026209

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•May 3, 1998•0 citations
Large transporting regular islands are found in the classical phase space of a modified kicked rotor system in which the kicking potential is reversed after every two kicks. The corresponding quantum system, for a variety of system parameters and over long time scales, is shown to display energy absorption that is significantly faster than that associated with the underlying classical anomalous diffusion. The results are of interest to areas of both quantum chaos and quantum control.
Control over the quantum dynamics of chaotic kicked rotor systems is demonstrated. Specifically, control over a number of quantum coherent phenomena is achieved by a simple modification of the kicking field. These include the enhancement of the dynamical localization length, the introduction of classical anomalous diffusion assisted control for systems far from the semiclassical regime, and the observation of a variety of strongly nonexponential line shapes for dynamical localization. The results provide excellent examples of controlled quantum dynamics in a system that is classically chaotic and offer opportunities to explore quantum fluctuations and correlations in quantum chaos.
The de Broglie-Bohm (BB) approach to quantum mechanics gives trajectories similar to classical trajectories except that they are also determined by a quantum potential. The quantum potential is a "fictitious potential" in the sense that it is part of the quantum kinetic energy. We use quantum trajectories to treat quantum chaos in a manner similar to classical chaos. For the kicked rotor, which is a bounded system, we use the Benettin et al. method to calculate both classical and quantum Lyapunov exponents as a function of control parameter K and find chaos in both cases. Within the chaotic sea we find in both cases nonchaotic stability regions for K equal to multiples of π. For even multiples of π the stability regions are associated with classical accelerator mode islands and for odd multiples of π they are associated with new oscillator modes. We examine the structure of these regions. Momentum diffusion of the quantum kicked rotor is studied with both BB and standard quantum mechanics (SQM). A general analytical expression is given for the momentum diffusion at quantum resonance of both BB and SQM. We obtain agreement between the two approaches in numerical experiments. For the case of nonresonance the quantum potential is not zero and must be included as part of the quantum kinetic energy for agreement. The numerical data for momentum diffusion of classical kicked rotor is well fit by a power law DNβ in the number of kicks N. In the anomalous momentum diffusion regions due to accelerator modes the exponent β(K) is slightly less than quadratic, except for a slight dip, in agreement with an upper bound (K2/2)N2. The corresponding coefficient D(K) in these regions has three distinct sections, most likely due to accelerator modes with period greater than one. We also show that the local Lyapunov exponent of the classical kicked rotor has a plateau for a duration that depends on the initial separation and then decreases asymptotically as O(t-1lnt), where t is the time. This behavior is consistent with an upper bound that is determined analytically.
The study of quantum resonances in the chaotic atom-optics kicked rotor system is of interest from two different perspectives. In quantum chaos, it marks out the regime of resonant quantum dynamics in which the atomic cloud displays ballistic mean energy growth due to coherent momentum transfer. Secondly, the sharp quantum resonance peaks are useful in the context of measurement of Talbot time, one of the parameter that helps in precise measurement of fine structure constant. Most of the earlier works rely on fidelity based approach and have proposed Talbot time measurement through experimental determination of the momentum space probability density of the periodically kicked atomic cloud. Fidelity approach has the disadvantage that phase reversed kicks need to be imparted as well which potentially leads to dephasing. In contrast to this, in this work, it is theoretically shown that, without manipulating the kick sequences, the quantum resonances through position space density can be measured more accurately and is experimentally feasible as well.
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.
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].