A.L. Sanin, E.A. Semyonov

The wave packet dynamics of quantum coupled oscillators is a subject of intensive research. This article presents an analysis of a quantum system with two degrees of freedom in which the force acts on an electron due to the polynomial potential depending on two coordinates. The potential consists of three terms. The first of them describes the double-well potential, which depends on one of the two coordinates, the second is a quadratic potential, dependent on the other coordinate, and the third one is proportional to the product of two coordinates and defines the coupling between the degrees of freedom. If the coupling is missing, the quantum system consists of two independent oscillators: a double-well and with a quadratic potential. Oscillator with a double-well potential is a bistable system which was studied thoroughly in one dimension case. An oscillator with a fourth degree polynomial potential is described by the Duffing equation in the classical limit. Oscillator with a finite quadratic potential for the given below parameters and initial conditions is harmonic. Numerical integration of the Schrodinger equation was performed to investigate independent and coupled oscillators and to make a comparative analysis of oscillatory regimes. For all investigated regimes, the time realizations of mean positions, their Fourier transform, Lissajous figures, and the autocorrelation function were obtained. In both cases the oscillations of mean positions induced by the double-well potential are the dynamic processes with different time scales. One of them is associated with large-scale-time tunneling. Fourier component corresponding to the tunneling frequency has the largest amplitude in all regimes of oscillations. Small-scale oscillations causing the modulation are induced by the transitions from highly excited states into the low-energy states, their frequencies are much greater than the tunneling frequency and the Fourier amplitudes are small. In case of coupling absence mean positions oscillations induced by the quadratic potential are harmonic. Fourier spectrum contains a single frequency. If the coupling is introduced, the properties of the oscillator with a quadratic potential become more complicated. The frequency spectrum contains the frequencies of the double-well oscillator, including the component at the tunneling frequency. Thus, there is a transfer of the frequency spectrum of one oscillator to another. In addition, the variation of the coupling coefficient influences on the tunneling frequency or tunneling rate. Increasing the coupling coefficient leads to decreasing of the frequency of tunneling. Numerical calculations of the Lissajous figures and autocorrelation functions, the analysis of the frequency spectra allow to state that these processes are quasiperiodic. The performed investigations are motivated by theory development of quantum electron systems interacting with an electromagnetic field.