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arXiv:0705.2794 · Physical Review A

Relaxation phenomena in a system of two harmonic oscillators

Antonia Chimonidou, E. C. G. Sudarshan

Published Mar 28, 2008• 10 citations
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Abstract

We study the process by which quantum correlations are created when an interaction Hamiltonian is repeatedly applied to a system of two harmonic oscillators for some characteristic time interval. We show that, for the case where the oscillator frequencies are equal, the initial Maxwell-Boltzmann distributions of the uncoupled parts evolve to a new Maxwell-Boltzmann distribution through a series of transient Maxwell-Boltzmann distributions, or quasistationary, nonequilibrium states. Further, we discuss why the equilibrium reached when the two oscillator frequencies are unequal is not a thermal one. All the calculations are exact and the results are obtained through an iterative process, without using perturbation theory.

DOI
10.1103/physreva.77.032121

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We study the process by which quantum correlations are created when an interaction Hamiltonian is repeatedly applied to a system of two harmonic oscillators for some characteristic time interval. We show that, for the case where the oscillator frequencies are equal, the initial Maxwell-Boltzmann distributions of the uncoupled parts evolve to a new Maxwell-Boltzmann distribution through a series of transient Maxwell-Boltzmann distributions, or quasistationary, nonequilibrium states. Further, we discuss why the equilibrium reached when the two oscillator frequencies are unequal is not a thermal one. All the calculations are exact and the results are obtained through an iterative process, without using perturbation theory.
A method is developed for generating relaxation by introducing a fundamental interval $\ensuremath{\tau}$ and a stirring hypothesis. The application to spin and harmonic oscillator systems is discussed in some detail. All the results are obtained by exact calculations without applying perturbation theory as the systems considered are simple and completely soluble. Equations similar to phenomenological Bloch equations are derived in the case of spin systems. The relaxation times obtained by the application of the theory are not only proportional to the strength of interaction, but also to the fundamental interval $\ensuremath{\tau}$ which plays an important role in the theory. It is shown that in the case of a harmonic oscillator system, an initial Boltzmann distribution relaxes to a final equilibrium Boltzman distribution through a sequence of transient Boltzman distributions.
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We propose an exactly soluble W*-dynamical system generated by repeated harmonic perturbations of the one-mode quantum oscillator. In the present paper we deal with the case of isolated system. Although dynamics is Hamiltonian and quasi-free, it produces relaxation of initial state of the system to the steady state in the large-time limit. The relaxation is accompanied by the entropy production and we found explicitly the rate for it. Besides, we study evolution of subsystems to elucidate their eventual correlations and convergence to equilibrium state. Finally we prove a universality of the dynamics driven by repeated harmonic perturbations in a certain short-time interaction limit.
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