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QUANTUM QUARTIC OSCILLATORS WITH TWO COUPLED DEGREES OF FREEDOM

A. Sanin, E. A. Semyonov

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

The numerical integration of the non-stationary two-dimensional Schrodinger equation is carried out. The dynamical properties (dynamical averages, frequency spectra, uncertainty relations, autocorrelators) for two quantum oscillators coupled by the Pallen- Edmonds potential are studied. As a result of the numerical simulations, it was established that the quantum system is sensitive to small changes in Hamiltonian that are caused by the Pallen-Edmonds coupling potential. In the regime of weak coupling high-frequency oscillations are generated, the spectral component number is increased at amplification of coupling.

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The numerical integration of the non-stationary two-dimensional Schrodinger equation is carried out. The dynamical properties (dynamical averages, frequency spectra, uncertainty relations, autocorrelators) for two quantum oscillators coupled by the Pallen- Edmonds potential are studied. As a result of the numerical simulations, it was established that the quantum system is sensitive to small changes in Hamiltonian that are caused by the Pallen-Edmonds coupling potential. In the regime of weak coupling high-frequency oscillations are generated, the spectral component number is increased at amplification of coupling.
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We have extended the applicability of the two-step approach to higher-dimensional problem, and have further developed it to evaluate the quantum dynamics and stationary states of a general class of coupled anharmonic oscillator system. This new approach has enabled us to investigate the topic of continuous variable entanglement within these coupled oscillator systems with good numerical efficiency. In particular, we have applied it to study the relation between entanglement, squeezing, and quantum uncertainties, of the systems’ ground states. Interestingly, while the relation between quadrature squeezing and entanglement depends sensitively on the nonlinear potential, the influence of nonlinear perturbation on the entropy-uncertainty relation is minimal, especially when the entanglement between the two oscillators is large. We have also explored into the dynamics of entanglement where we have found that entanglement dynamics can depend completely on the global classical dynamical regime, without being influenced by the local classical behavior.
In this work, we investigate the Schrödinger dynamics of photon excitation numbers and entanglement in a system composed by two non-resonant time-dependent coupled oscillators. By considering π periodically pumped parameters (oscillator frequencies and coupling) and using suitable transformations, we show that the quantum dynamics can be determined by two classical Meissner oscillators. We then study analytically the stability of these differential equations and the dynamics of photon excitations and entanglement in the quantum system numerically. Our analysis shows two interesting results, which can be summarized as follows: (i) Classical instability of classical analog of quantum oscillators and photon excitation numbers (expectations Nj) are strongly correlated, and (ii) photon excitations and entanglement are connected to each other. These results can be used to shed light on the link between quantum systems and their classical counterparts and provide a nice complement to the existing works studying the dynamics of coupled quantum oscillators.
The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.