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Physical Science International Journal

Cumulative Effects of the Temperature and Damping on the Time Dependent Entropy and Decoherence in the Caldirola-Kanai Harmonic Oscillator

M. P. Tabue Djemmo, A. J. Fotué, François Béceau Pelap

Published Jan 10, 2015• 0 citations
Publisher

Abstract

The time Dependence of probability and Shannon entropy of the specific damped harmonic oscillator systems is studied by using a prototypical Schrodinger cat state through the Feynman path method. By averaging the probability distribution over the thermal distribution of velocities, we show that, the temperature and the damped coefficient or dissipation as well as the distance separating two consecutive wave functions influence the coherence of the system.

DOI
10.9734/psij/2015/15391

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The time Dependence of probability and Shannon entropy of the specific damped harmonic oscillator systems is studied by using a prototypical Schrodinger cat state through the Feynman path method. By averaging the probability distribution over the thermal distribution of velocities, we show that, the temperature and the damped coefficient or dissipation as well as the distance separating two consecutive wave functions influence the coherence of the system.
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We calculate the time-dependent Fisher information in position and momentum for the lowest lying state of two classes of quantum damped (Lane–Emden (LE) and Caldirola–Kanai (CK)) harmonic oscillators. The expressions of and are written in terms of a c-number quantity satisfying a nonlinear differential equation. Analytical solutions of were obtained. For the LE and CK oscillators, we observe that increases while decreases with increasing time. The product increases and tends to a constant value in the limit for the LE oscillator, while it is time-independent for the CK oscillator. Moreover, for the CK oscillator the product decreases as the damping increases. Relations among the Fisher information, Leipnik and Shannon entropies, and the Stam and Cramer–Rao inequalities are given. A discussion on the squeezing phenomenon in position for the oscillators is presented.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.
The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. We find that the dynamics must include a nonvanishing noise term to yield physical results for the purity and that the oscillator decoheres in time such that the late-time density operator is thermal. We show that the frequency spectrum or unequal-time correlations can, however, distinguish between the damped oscillator and an isolated oscillator in thermal equilibrium and obtain a generalized fluctuation-dissipation relation for the damped oscillator. We briefly consider time-nonlocal dissipation as well, to show that the fluctuation-dissipation is satisfied for a specific choice of dissipation kernels. Lastly, we develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results for time-local dissipation are in fact nonperturbative in the initial state.