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Bulletin of the Russian Academy of Sciences Physics

On the nonlinear dynamics of a quantum oscillator

С. В. Сазонов

Published Dec 1, 2014• 1 citation
Publisher

Abstract

A quantum version of the asymptotic approach in the theory of anharmonic oscillator motion is proposed. It is shown that quantum transitions from the most likely stationary state to the nearest quantum level correspond to the classical Eigen frequency. This frequency depends on the oscillator amplitude. Quantum corrections to the motion of a classical nonlinear oscillator are determined.

DOI
10.3103/s1062873814120272

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A quantum version of the asymptotic approach in the theory of anharmonic oscillator motion is proposed. It is shown that quantum transitions from the most likely stationary state to the nearest quantum level correspond to the classical Eigen frequency. This frequency depends on the oscillator amplitude. Quantum corrections to the motion of a classical nonlinear oscillator are determined.
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A theory allowing to describe of quantum chaos in the term of nonlinear classical dynamics is proposed. It is showed that a solution of a quantum problem in quasiclassical region is connected with a solution of a corresponding classical problem which is the classical limit of the quantum one. It is showed also that the wave function phase is satisfied an own equation of motion and it is a classical dynamic equation. Dynamical chaos of nonlinear classical dynamics results to irregularity of the wave function phase and a strange attractor in a classical dynamics of the wave function phase leads to irregularity in probabilities of transitions of anharmonic oscillator in an external time-dependent field. In the pure quantum and the pure classical region corresponding solutions are independent.
We develop a method for the determination of the dynamics of dissipative quantum systems in the limit of large number of quanta N , based on the 1/ N -expansion of Heidmann et al. (1985 Opt. Commun. 54 189) and the quantum-classical correspondence. Using this method, we find analytically the dynamics of nonclassical states generation in the higher-order anharmonic dissipative oscillators for an arbitrary temperature of a reservoir. We show that the quantum correction to the classical motion increases with time quadratically up to some maximal value, which is dependent on the degree of nonlinearity and a damping constant, and then it decreases. Similarities and differences with the corresponding behavior of the quantum corrections to the classical motion in the Hamiltonian chaotic systems are discussed. We also compare our results obtained for some limiting cases with the results obtained by using other semiclassical tools and discuss the conditions for validity of our approach.
The paper introduces a simple quantum model to calculate in a general way allowed frequencies and energy levels of the anharmonic oscillator. The theoretical basis of the approach has been introduced in two early papers aimed to infer the properties of quantum systems exploiting the uncertainty principle only. For clarity the anharmonic oscillator is described having in mind the lattice oscillations of atoms/ions, yet quantum formalism of the model and approach have general character and can be extended to any oscillating system. The results show that the harmonic energy levels split into a complex system of energy levels dependent upon the number of anharmonic terms that characterize the oscillator.
The nonlinear-oscillating system in action-angle variables is characterized by the dependence of frequency of oscillation $\ensuremath{\omega}(I)$ on action III. Periodic perturbation is capable of realizing in the system a stable nonlinear resonance at which the action III adapts to the resonance condition $\ensuremath{\omega}({I}_{0})\ensuremath{\simeq}\ensuremath{\omega}$, that is, ``sticking'' in the resonance frequency. For a particular physical problem there may be a case when $I\ensuremath{\gg}\ensuremath{\hbar}$ is the classical quantity, whereas its correction $\ensuremath{\Delta}I\ensuremath{\simeq}\ensuremath{\hbar}$ is the quantum quantity. Naturally, dynamics of $\ensuremath{\Delta}I$ is described by the quantum equation of motion. In particular, in the moderate nonlinearity approximation $\ensuremath{\varepsilon}\ensuremath{\ll}(d\ensuremath{\omega}/dI)(I/\ensuremath{\omega})\ensuremath{\ll}1/\ensuremath{\varepsilon}$, where $\ensuremath{\varepsilon}$ is the small parameter, the description of quantum state is reduced to the solution of the Mathieu-Schr\"odinger equation. The state formed as a result of sticking in resonance is an eigenstate of the operator $\ensuremath{\Delta}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{I}$ that does not commute with the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$. Expanding the eigenstate wave functions in Hamiltonian eigenfunctions, one can obtain a probability distribution of energy level population. Thus, an inverse level population for times lower than the relaxation time can be obtained.