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Physical Review A

Classical-quantum correspondence in the driven surface-state-electron model

Miroslaw Latka, Paolo Grigolini A, Bruce J. West

Published Jun 1, 1993• 18 citations
Publisher

Abstract

The quantum dynamics of minimum uncertainty wave packets in a system described by the surface-state-electron Hamiltonian are studied herein. The quantum evolution is found to be strongly dependent upon the orbit structure in the classical phase space. A Gaussian wave packet, initially centered on the part of the phase space incased by Kolmogorov-Arnold-Moser tori remains localized, while the packets located in the chaotic region spread rapidly, filling the available phase space. To quantify the observed spreading quantum uncertainty is introduced as a dynamical variable, and its time development is determined by the ballistic growth of the position uncertainty and the erratic oscillations of the momentum uncertainty. The quantum evolution is analyzed within the framework of the Floquet formalism. The Floquet spectrum of the stable packet is shown to be dominated by a few quasienergy states with the corresponding Husimi distribution embedded in the classical stability island. Further, the classical concept of diffusion previously used in this context is shown to be inappropriate.

DOI
10.1103/physreva.47.4649

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The quantum dynamics of minimum uncertainty wave packets in a system described by the surface-state-electron Hamiltonian are studied herein. The quantum evolution is found to be strongly dependent upon the orbit structure in the classical phase space. A Gaussian wave packet, initially centered on the part of the phase space incased by Kolmogorov-Arnold-Moser tori remains localized, while the packets located in the chaotic region spread rapidly, filling the available phase space. To quantify the observed spreading quantum uncertainty is introduced as a dynamical variable, and its time development is determined by the ballistic growth of the position uncertainty and the erratic oscillations of the momentum uncertainty. The quantum evolution is analyzed within the framework of the Floquet formalism. The Floquet spectrum of the stable packet is shown to be dominated by a few quasienergy states with the corresponding Husimi distribution embedded in the classical stability island. Further, the classical concept of diffusion previously used in this context is shown to be inappropriate.
Using the minimum uncertainty state of quantum integrable system as initial state, the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics. Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some pairs of neighboring components of the wave packet, the deterministic dynamical evolution becomes very complicated and appears to be chaotic. It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors. It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.
Using the minimum uncertainty state of quantum integrable system Ho as initial state, the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics. Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some pairs of neighboring components of the wave packet, the deterministic dynamical evolution becomes very complicated and appears to be chaotic. It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors. It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.
A methodology to calculate generalized coherent states for a periodically driven system is presented. We study wave packets constructed as a linear combination of suitable Floquet states of the three-dimensional Rydberg atom in a microwave field. The driven coherent states show classical space localization, spreading, and revivals and remain localized along the classical trajectory. The microwave strength and frequency have a great effect in the localization of Floquet states, since quasienergy avoided crossings produce delocalization of the Floquet states, showing that tuning of the parameters is very important. Using wavelet-based time-frequency analysis, the classical phase-space structure is determined, which allows us to show that the driven coherent state is located in a large regular region in which the zzz coordinate is in resonance with the external field. The expectation values of the wave packet show that the driven coherent state evolves along the classical trajectory.
The initial quantum dynamics of an initially well-localized, minimum-uncertainty Gaussian wave packet is studied analytically for the periodically kicked pendulum. The general results presented here include our earlier analytic result [Phys. Rev. A 41, 2952 (1990)] as a special case. We find that as long as the packet remains localized, it remains approximately Gaussian, while its center and widths evolve in time approximately according to, respectively, the classical map and a variance map which utilize the instantaneous values of the classical Jacobi matrix. When the classical map is very strongly chaotic, we predict that the average kinetic energy of the well-localized wave packet does not initially grow diffusively (linearly with time). We show that, instead, in the presence of chaos, the deviations initially grow, on the average, exponentially at a rate equal to the corresponding classical Liapunov exponent under the appropriate conditions. The time scale (Ehrenfest time) on which the Ehrenfest-Hamilton correspondence holds in the presence of chaos is estimated and is slightly different from a previous estimate made by Chirikov, Israilev, and Shepelyansky [Sov. Sci. Rev. C 2, 209 (1981)]. Numerical evidence in support of these analytical results is presented.
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.
We compare the dynamics of quantum wave packets with the dynamics of classical trajectory ensembles. The wave packets are Gaussian with expectation values of position and momenta which centers them in phase space. The classical trajectory ensembles are generated directly from the quantum wave packets via the Wigner transform. Quantum and classical dynamics are then compared using several quantum measures and the analogous classical ones derived from the Wigner equivalent formalism. Comparisons are made for several model potentials and it is found that there is generally excellent classical–quantum correspondence except for certain specific cases of tunneling and interference. In general, this correspondence is also very good in regions of phase space where there is classical chaos.