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Classical and Quantum Chaotic States of A Nonautonomous System, pp. 237-262 $100.00
Authors:  (Wenhua Hai, Dept. of Physics, Hunan Normal Univ., Key Laboratory of Quantum Structure and Manipulation of Ministry of Education, Changsha, China)
Abstract:
We investigate a general nonautonomous and nonlinear system with damping and periodic perturbation, and study classical and quantum chaotic properties of the system. The asymptotic equilibrium solution (AES) of the system is defined, which contains the homoclinic and heteroclinic ones. Bounded perturbed corrections to the AES are constructed as a series of discrete solutions by the direct perturbation method, which is determined by some discrete parameters of the external field. The Lyapunov analysis to the linear stability
shows that the perturbed orbits may be unstable, because of their unboundedness. Conditions for controlling the unboundedness are suggested as fitting some relationships among the system parameters and initial conditions, which lead to sensitive dependence of the system on initial conditions and physical parameters, and result in the zero Perron-Lyapunov exponent. The relationships contain the Melnikov criterion for the onset of chaos, indicating the existence of deterministic chaos and revealing the conditional stability of chaos. The numerical incomputability and theoretical unpredictability of the classical chaos are evinced through the chaotic solutions. Given the above-mentioned discrete external field parameters and boundedness conditions, we can generate or suppress chaos by setting and adjusting the system parameters to fit or avoid them. The linearized systems governing the n′ discrete chaotic states are equivalent to some pure quantum systems mathematically, and the n n′
exact solutions of the both kind of systems are found as n′ sets of orthonormalized chaotic coherent states for the integers n, n′ . It is demonstrated that under such quantum chaotic states the expectation coordinate and momentum are equal to the correspondingly classical chaotic ones and the expectation energy contains the classical energies of n′ harmonic oscillators and n quantum chaotic bands. Thus we show a direct correspondence between the classical and quantum chaos and supply a method for controlling the classical and quantum chaotic motions through the manipulation to the external fields. As an example, we evidence the classical and quantum chaotic motions, through a single atom confined in an anharmonic potential well with a sech-squared-shaped time-manipulation and perturbed by a periodic force and a damping. Some analytical results are confirmed numerically and good agreement between them is found. 


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Classical and Quantum Chaotic States of A Nonautonomous System, pp. 237-262