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Communications in Theoretical Physics

Spatio-Temporal Evolution of Initial Coherent State Under the Action of Quantum Integrable and Nonintegrable Systems

Xing Yong-Zhong, Gong-ou Xu, Yang YaTian

Published Oct 30, 2000• 0 citations
Publisher

Abstract

According to the exactly formulated concepts of quantum integrability and nonintegrability,an initial coherent state evolving under the action of an integrable Hamiltonian having the same dynamical symmetry is expressed as an analytical function of and . The expectation values of various dynamical variables should be expressed as correlated analytical functions of and . Thus, the general characters of initial coherent state would be maintained during the spatio-temporal evolution under the action of such a system even if the wavepacket is slightly distorted. In contrast, when the integrability condition is destroyed, these features will be lost during the evolution. This theoretical prediction is illustrated with numerical results computed for the two-level Lipkin model with a dou ble-well potential and dynamical explanations are given. And the Eherenfest time is verified with this model.

DOI
10.1088/0253-6102/34/3/531

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According to the exactly formulated concepts of quantum integrability and nonintegrability,an initial coherent state evolving under the action of an integrable Hamiltonian having the same dynamical symmetry is expressed as an analytical function of and . The expectation values of various dynamical variables should be expressed as correlated analytical functions of and . Thus, the general characters of initial coherent state would be maintained during the spatio-temporal evolution under the action of such a system even if the wavepacket is slightly distorted. In contrast, when the integrability condition is destroyed, these features will be lost during the evolution. This theoretical prediction is illustrated with numerical results computed for the two-level Lipkin model with a dou ble-well potential and dynamical explanations are given. And the Eherenfest time is verified with this model.
In order to have a complete description of the spatio-temporal evolution of a quantum state, the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry. Then we have qualitatively different situations where the initial coherent state evolves under the action of a system having or not having the same dynamical symmetry. In the former case, we have regular motion with the general character which tends to that of the corresponding classical case in the limit ℏ → 0. While in the latter case, we may have energy regions of global stochasticity with characters closely related to the initial dynamical symmetry strongly violated.
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.
The notion of quantum-classical correspondence is carefully investigated in order to prepare firm grounds for studying the spatiotemporal evolution of quantum states in the same spirit as for corresponding classical cases. Three relevant problems, (1) the integrability of dynamical equations of quantum systems, (2) the initial minimum uncertainty states one-to-one correspondent to classical phase points, and (3) the effective Planck constants for systems having analogous dynamical properties but exhibiting different quantum effects, have been successfully resolved. Then the solution ${\ensuremath{\rho}}_{\ensuremath{\gamma}}(t)$ of the dynamical equation of a quantum integrable system is shown to be expressed as an analytical functional of the initial minimum uncertainty state ${\ensuremath{\rho}}_{\ensuremath{\gamma}}^{0}$ varying smoothly with $\ensuremath{\gamma}$ and t. Such a general character of the quantum regular motion serves as a reference for the study of quantum irregular motion under the action of perturbed Hamiltonian.
The quantum dynamics of initial coherent states is studied in the Dicke model and correlated with the dynamics, regular or chaotic, of their classical limit. Analytical expressions for the survival probability, i.e. the probability of finding the system in its initial state at time t, are provided in the regular regions of the model. The results for regular regimes are compared with those of the chaotic ones. It is found that initial coherent states in regular regions have a much longer equilibration time than those located in chaotic regions. The properties of the distributions for the initial coherent states in the Hamiltonian eigenbasis are also studied. It is found that for regular states the components with no negligible contribution are organized in sequences of energy levels distributed according to Gaussian functions. In the case of chaotic coherent states, the energy components do not have a simple structure and the number of participating energy levels is larger than in the regular cases.
Taking coherent states as initial quantum states, the spatio-temporal evolution of the wave packet under the action of perturbed Hamiltonian showed one-to-one correspondence to the behavior of classical phase points. The character of the coherent state propagation in classically soft chaotic system is found to be associated with the statistical property of the spectrum contained in the coherent state wave packet. The spreading width of wave packet increases with time exponentially if the statistics of the spectrum contained in the wave packet show GOE distribution due to avoided level crossings.