Quantum manifestations of classical stochasticity. II. Dynamics of wave packets of bound states

Abstract

Articles you may be interested inTunneling dynamics with a mixed quantum-classical method: Quantum corrected propagator combined with frozen Gaussian wave packetsIn this paper we consider some of the consequences of classical stochasticity phenomena in the quantum mechanical Henon-Heiles and Barbanis systems. To explore classical--quantum analogies we have introduced a quantum mechanical phase space based on the coherent state representation. Quantum Poincare maps (QPMs) were constructed from contour plots of the stationary phase-space densities. We have observed a correlation between the topological features of a QPM at an eigenenergy E and the sensitivity IdE /d€ I of that energy to the strength E of the nonlinear coupling. States with extreme values of IdE/d€l, corresponding to both high and to low values of IdE/d€l, show regular regions in the quantum phase space, while states with intennediate values of IdE/dEl span large parts of the quantum phase-space regions. These general features of the QPMs apply for almost each energy multiplet, and no qualitative change in the character of the QPMs was observed above the classical critical energy E, for the stochastic transition. To investigate the quantum analog of classical trajectories, we have studied the dynamics of initially coherent Gaussian wave packets. We have discovered two limiting types of time evolution of intially coherent wave packets, which exhibit quasiperiodic time evolution or rapid dephasing. Quasiperiodic quantum dynamics is analogous to quasiperiodic classical trajectory dynamics, while rapid dephasing of the wave packet reflects the effects of spreading of the wave packet or/and stochastic classical dynamics. We were able to establish a correspondence between the topology of the quantum phase-space density and the wave packet dynamics.Quasiperiodic time evolution and rapid dephasing are exhibited by wave packets initially located in the regular and in the irregular regions of the quantum phase space, respectively.

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“…We note that there has been an earlier work wherein similar correlations were observed. 50 To start with, note that the matrix element ͗n͉͑a 1 † ͒ m ͑ a 2 ͒ n ͉nЈ͘ϭV nm ͑ n 1 Ј ,n 2 Ј͒␦ n 1 ,n 1 Јϩm ␦ n 2 ,n 2 ЈϪn ͑5͒ yields an expression for the level velocity in the following convenient form:…”

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