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arXiv:nlin/0012028

Quantum manifestation of Levy-type flights in a chaotic system

Alexander Iomin, G. M. Zaslavsky

Published Dec 13, 2000• 0 citations
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Abstract

Semi--classical dynamics of quantum wave packets spreading is studied for a kicked rotor. Quantum flights are established for a specific, "magic" value of a chaos control parameter when the classical stickiness of trajectories is most effective. By studying of survival probability and distribution of the accelerations we identify the presence of quantum Levy-type flights.

DOI
10.48550/arxiv.nlin/0012028

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Semi--classical dynamics of quantum wave packets spreading is studied for a kicked rotor. Quantum flights are established for a specific, "magic" value of a chaos control parameter when the classical stickiness of trajectories is most effective. By studying of survival probability and distribution of the accelerations we identify the presence of quantum Levy-type flights.
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].
We consider the quantum dynamics of the kicked rotor model in the regime where the last stable resonance islands in the classical phase space disappear. It is found that phase space regions with lower local diffusion rates are able to preserve quantum population. The low-diffusion regions can be moved through phase space in a quasi-adiabatic way by smoothly varying the parameters of the kicking field. With appropriately chosen time-scales for the variation of the field parameters, wave packets initially localized around the low-diffusion areas remain localized around these regions as the regions are moved. The wave packets can then be carried to desired regions of space despite the predominantly chaotic character of the classical phase space. We also comment on the Gong–Brumer scenario of coherent control of quantum chaotic diffusion [Phys. Rev. Lett. 86 1741 (2001)]. While our phase-space inspired control has a clear semiclassical interpretation, it appears that the Gong–Brumer control has a predominantly quantum origin. Furthermore, it is found that our control over wave packet localization disappears in the deeply chaotic regime, while the Gong–Brumer control over the chaotic diffusion persists in this regime but becomes increasingly unstable with increasing chaoticity.
Focusing on the standard map of the kicked rotor, we investigate the effects of classical stickiness (orbits temporarily confined to a region of a chaotic phase space) on an open quantum system. The regions of stickiness that we identify emerge in the strongly chaotic map. By scanning the system's phase space with a leak, we show that stickiness reduces the degree of delocalization of the quantum states and the decay rate of generic initial states. We find an excellent correspondence between the classical dwell time and finite-time Lyapunov exponents with the quantum dwell time and Wehrl entropies of the quantum states. Knowledge of the structure of the classically chaotic trajectories can thus be used to determine the best placement of leaks to enhance or decrease the degree of delocalization of quantum states and to modify the quantum dynamics of open systems.
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.
One of the principal goals of controlling classical chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. It is shown that a generalization to chaotic quantum systems is possible in the semiclassical regime, but requires tailored perturbations whose effects must undo the dynamical spreading of the evolving quantum state. The procedure described here is applied to initially minimum uncertainty wave packets in the quantum kicked rotor, a preeminent quantum chaotic paradigm, to illustrate the method, and investigate its accuracy. The method's error can be made to vanish as ℏ→0.
We investigate both the quantum and classical dynamics of a non-Hermitian system via a kicked rotor model with P T symmetry. For the quantum dynamics, both the mean momentum and mean square of momentum exhibit the staircase growth with time when the system parameter is in the neighborhood of the P T symmetry breaking point. If the system parameter is much larger than the P T symmetry breaking point, the accelerator mode results in the directed spreading of the wavepackets as well as the ballistic diffusion in momentum space. For the classical dynamics, the non-Hermitian kicking potential leads to the exponentially-fast increase of classical complex trajectories. As a consequence, the imaginary part of the trajectories exponentially diffuses with time, while the real part exhibits the normal diffusion. Our analytical prediction of the exponential diffusion of imaginary momentum and its breakdown time is in good agreement with numerical results. The quantum signature of the chaotic diffusion of the complex trajectories is reflected by the dynamics of the out-of-time-order correlators (OTOC). In the semiclassical regime, the rate of the exponential increase of the OTOC is equal to that of the exponential diffusion of the complex trajectories.