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arXiv:quant-ph/0401136 · Physical Review E

Quantum-“classical” correspondence in a nonadiabatic transition system

Hiroshi Fujisaki

Published Mar 31, 2004• 2 citations
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Abstract

A nonadiabatic transition system which exhibits "quantum chaotic" behavior [H. Fujisaki and K. Takatsuka, Phys. Rev. E 63, 066221 (2001)] is investigated from quasiclassical aspects. Since such a system does not have a naive classical limit, we take the mapping approach [Stock and Thoss, Phys. Rev. Lett. 78, 578 (1997)] to represent the quasiclassical dynamics of the system. We numerically show that there is a sound correspondence between the quantum chaos and classical chaos for the system.

DOI
10.1103/physreve.69.037201

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A nonadiabatic transition system which exhibits "quantum chaotic" behavior [H. Fujisaki and K. Takatsuka, Phys. Rev. E 63, 066221 (2001)] is investigated from quasiclassical aspects. Since such a system does not have a naive classical limit, we take the mapping approach [Stock and Thoss, Phys. Rev. Lett. 78, 578 (1997)] to represent the quasiclassical dynamics of the system. We numerically show that there is a sound correspondence between the quantum chaos and classical chaos for the system.
The quantum-classical correspondence for dynamics of the nonlinear classically chaotic systems is analysed. The problem of quantum chaos consists of two parts: the quasiclassical quantisation of the chaotic systems and attempts to understand the classical chaos in terms of quantum mechanics. The first question has been partially solved by the Gutzwiller semiclassical trace formula for the eigenvalues of chaotic systems, while the classical chaos may be derived from quantum equations only introducing the decoherence process due to interaction with system's environment or intermediate frequent measurement. We may conclude that continuously observable quasiclassical system evolves essentially classically-like.
The behavior of the periodically kicked pendulum is determined classically and quantum mechanically. Our earlier analytic study [Phys. Rev. A 41, 2952 (1990)] is confirmed and extended by the numerical results presented here. We are able to construct minimum uncertainty, Gaussian wave packets that maintain their form, and shadow the corresponding classical motion for long times when the classical motion is nonchaotic or only weakly chaotic. However, when the classical motion is robustly chaotic, the quantum-mechanical probability distribution grows macroscopically broad exponentially fast and the quantum-classical correspondence breaks down.
We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos.
The simulation of the dynamics of quantum systems is very difficult, due to the fact that, in general, it cannot be calculated exactly for interacting many-body systems. Brute force simulations of quantum dynamics are simply not feasible, and approximations need to be made. In many instances a quantum system can be approximated as a quantum-classical system, where only a subsystem of interest is treated quantum mechanically, and the rest is considered as a classical bath. When energy is free to be exchanged between the subsystem and its environment, the dynamics that occur is said to be nonadiabatic. This type of dynamics is challenging to calculate on a computer, as it can lead to large statistical errors at long times. Hence, there is a need for improved algorithms for nonadiabatic dynamics. In this thesis, a recently introduced nonadiabatic sampling scheme [A. Sergi and F. Petruccione, Phys. Rev. E 81, 032101 (2010)] is used to calculate the long-time dynamics of a model system comprising a quantum spin coupled to a bath of harmonic oscillators. Also, various technical aspects of the algorithm are investigated.