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arXiv:quant-ph/0004075 · Journal of Optics B Quantum and Semiclassical Optics

Decoherence and thermalization dynamics of a quantum oscillator

V. V. Dodonov, Salomon Sylvain Mizrahi, A L de Souza Silva, Salomon Sylvain Mizrahi

Published May 24, 2000• 37 citations
arXivHTMLPDF

Abstract

We introduce the quantitative measures characterizing the rates of decoherence and thermalization of quantum systems. We study the time evolution of these measures in the case of a quantum harmonic oscillator whose relaxation is described in the framework of the standard master equation, for various initial states (coherent, `cat', squeezed and number). We establish the conditions under which the true decoherence measure can be approximated by the linear entropy 1-Tr 2 . We show that at low temperatures and for highly excited initial states the decoherence process consists of three distinct stages with quite different time scales. In particular, the `cat' states preserve 50% of the initial coherence for a long time interval which increases logarithmically with the increase of the initial energy.

DOI
10.1088/1464-4266/2/3/309

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We introduce the quantitative measures characterizing the rates of decoherence and thermalization of quantum systems. We study the time evolution of these measures in the case of a quantum harmonic oscillator whose relaxation is described in the framework of the standard master equation, for various initial states (coherent, cat', squeezed and number). We establish the conditions under which the true decoherence measure can be approximated by the linear entropy 1-Tr 2 . We show that at low temperatures and for highly excited initial states the decoherence process consists of three distinct stages with quite different time scales. In particular, the cat' states preserve 50% of the initial coherence for a long time interval which increases logarithmically with the increase of the initial energy.
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In this paper we develop a method of solving the dynamics of minimal open systems of coupled oscillators and qubits. Extracting the time-dependence of the system from its underlying Lie algebra of operators we find the time evolution operator, allowing us to explore the evolution of various initial states of the system and environment. For these initial states we find the linear entropy, a measure of decoherence of the system, and visualize the dynamics using Husimi functions and Bloch spheres for oscillator and qubit systems, respectively. We find that increasing the interaction of the system with its environment results in more rapid decoherence. For coupled oscillators, under the rotating wave approximation we find periodic linear entropies whereas for the anti-rotating wave approximation we find that the linear entropy can lose periodicity. For a qubit system coupled to an oscillator environment we find oscillatory linear entropies that are pseudo-periodic.
The rate at which pure initial states deteriorate into mixtures is computed for a harmonic oscillator interacting with an environment in thermal equilibrium. The decoherence process resulting from this interaction selects a set of states characterized by maximal stability (or minimal loss of predictive power) which can be quantified by the rate of increase in either linear or statistical entropy. In the weak coupling limit, coherent states are shown to produce the least entropy, thus becoming the natural counterparts of classical points in phase space.