The motion of a nonlinearly oscillating particle under the influence of a periodic sequence of short impulses is investigated. We analyze the Schrödinger equation for the universal Hamiltonian. It is shown that the quantum criterion of overlapping of resonances is of the form lambdaK>or=1, where K is the classical coefficient of stochasticity and lambda is the functional defined with the use of Mathieu functions. The area of the maximal values of lambda is determined. The idea about the emerging of quantum chaos due to the adiabatic motion along the curves of Mathieu characteristics at multiple passages through the points of branching is advanced.
The pendulum is the simplest system having all the basic properties inherent in dynamic stochastic systems. In the present paper we investigate the pendulum with the aim to reveal the properties of a quantum analogue of dynamic stochasticity or, in other words, to obtain the basic properties of quantum chaos. It is shown that a periodic perturbation of the quantum pendulum (similarly to the classical one) in the neighborhood of the separatrix can bring about irreversible phenomena. As a result of recurrent passages between degenerate states, the system gets self-chaotized and passes from the pure state to the mixed one. Chaotization involves the states, the branch points of whose levels participate in a slow "drift" of the system along the Mathieu characteristics this "drift" being caused by a slowly changing variable field. Recurrent relations are obtained for populations of levels participating in the irreversible evolution process. It is shown that the entropy of the system first grows and, after reaching the equilibrium state, acquires a constant value.
The quantum-mechanical investigation of nonlinear resonance in terms of approximation to moderate nonlinearity is reduced to the investigation of eigenfunctions and eigenvalues of the Mathieu-Schrodinger equation. The eigenstates of the Mathieu-Schrodinger equation are nondegenerate in a certain area of pumping amplitude values in the neighborhood of the classical separatrix. Outside this area, the system finds itself in a degenerate state for both small and large pumping amplitude values. Degenerate energy terms arise as a result of merging and branching of pairs of nondegenerate energy terms. Equations are obtained for finding the merging points of energy terms. These equations are solved by numerical methods. The main objective of this paper is to establish a quantum analog of the classical stochastic layer formed in the separatrix area. With this end in view, we consider a nonstationary quantum-mechanical problem of perturbation of the state of the Mathieu-Schrodinger equation. It is shown that in passing through the branching point the system may pass from the pure state to the mixed one. At multiple passages through branching points there develops the irreversible process of "creeping" of the system to quantum states. In that case, the observed population of a certain number of levels can be considered, in our opinion, to be a quantum analog of the stochastic layer. The number of populated levels is defined by a perturbation amplitude.
Usually reason of irreversibility in open quantum-mechanical system is interaction with a thermal bath, consisting form infinite number of degrees of freedom. Irreversibility in the system appears due to the averaging over all possible realizations of the environment states. But, in case of open quantum-mechanical system with few degrees of freedom situation is much more complicated.Should one still expect irreversibility, if external perturbation is just an adiabatic force without any random features? Problem is not clear yet. This is main question we address in this review paper. We prove that key point in the formation of irreversibility in chaotic quantum-mechanical systems with few degrees of freedom, is the complicated structure of energy spectrum. We shall consider quantum mechanical-system with parametrically dependent energy spectrum. In particular, we study energy spectrum of the Mathieu-Schrodinger equation. Structure of the spectrum is quite non-trivial, consists from the domains of non-degenerated and degenerated stats, separated from each other by branch points. Due to the modulation of the parameter, system will perform transitions from one domain to other one. For determination of eigenstates for each domain and transition probabilities between them, we utilize methods of abstract algebra. We shall show that peculiarity of parametrical dependence of energy terms, leads to the formation of mixed state and to the irreversibility, even for small number of levels involved into the process. This last statement is important. Meaning is that, we are going to investigate quantum chaos in essentially quantum domain.In the second part of the paper, we will introduce concept of random quantum phase approximation. Then along with the methods of random matrix theory, we will use this assumption, for derivation of muster equation in the formal and mathematically strict way.
As a model of coupled nano-electromechanical resonantors we study two nonlinear driven oscillators with an arbitrary coupling strength between them. Analytical expressions are derived for the oscillation amplitudes as a function of the driving frequency and for the energy transfer rate between the two oscillators. The nonlinear restoring forces induce the expected nonlinear resonance structures in the amplitude-frequency characteristics with asymmetric resonance peaks. The corresponding multistable behavior is shown to be an efficient tool to control the energy transfer arising from the sensitive response to small changes in the driving frequency. Our results imply that the nonlinear response can be exploited to design precise sensors for mass or force detection experiments based on nano-electromechanical resonators.
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