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arXiv:cond-mat/0109368 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Aging effects in the quantum dynamics of a dissipative free particle: Non-Ohmic case

A. Mauger, No¨elle Pottier

Published May 3, 2002• 16 citations
arXivHTMLPDF

Abstract

We report the results related to the two-time dynamics of the coordinate of a quantum free particle, damped through its interaction with a fractal thermal bath (non-Ohmic coupling approximately omega(delta) with 0<delta <1 or 1<delta <2). When the particle is localized, its position does not age. When it undergoes anomalous diffusion, only its displacement may be defined. It is shown to be an aging variable. The finite temperature aging regime is self-similar. It is described by a scaling function of the ratio t(w)/tau of the waiting time to the observation time, as characterized by an exponent directly linked to delta.

DOI
10.1103/physreve.65.056107

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We report the results related to the two-time dynamics of the coordinate of a quantum free particle, damped through its interaction with a fractal thermal bath (non-Ohmic coupling approximately omega(delta) with 0<delta <1 or 1<delta <2). When the particle is localized, its position does not age. When it undergoes anomalous diffusion, only its displacement may be defined. It is shown to be an aging variable. The finite temperature aging regime is self-similar. It is described by a scaling function of the ratio t(w)/tau of the waiting time to the observation time, as characterized by an exponent directly linked to delta.
We investigate the exact quantum dynamics of a free particle damped through its interaction with an harmonic bath, when the effective coupling strength behaves as ωδ at low frequency. We find that various regimes can occur depending on the value of δ and of the temperature. At large times, the mean square displacement is shown either to diverge as tv, the exponent v being never greater than 2 or to tend towards a finite value, indicating a confinement in the broad sense. In addition, for δ < 1, an oscillation (absent in the uncoupled system) is found : due to the friction on the dominant low-frequency modes and to the existence of retardation effects, the particle is forced to relax in the mean towards its initial location, the net dynamical effect of the bath being similar to that of an applied potential. In the limit δ → 0, the particle is frozen.
Using the functional-integral method, we investigate the effect of a two-level systems thermal reservoir on the single particle dynamics. We find that at low temperatures, within the sub-ohmic regime, the particle becomes {}``dynamically'' localized at long times due to an effective potential generated by the particle-reservoir interaction. This behavior is different from the one obtained for the usual bath of harmonic oscillators and is fundamentally related with the non-Markovian character of the dissipative process.
The real-time dynamics of a particle in a slightly asymmetric double-well potential in the presence of ohmic dissipation is analysed by using standard quantum-statistical methods. At zero temperature, the particle eventually goes towards the lower potential well; the relaxation has an overall exponential character with possibly small oscillations; at finite temperature, the particle thermalizes over the two wells; two successive exponential regimes are observed in this case. The results about the final average position of the particle agree with the detailed balance principle and are in fact independent of the ohmic assumption. For an ohmic dissipation law, the localization phenomenon which occurs in the perfectly symmetric double-well potential at zero temperature disappears when an asymmetry is present, although a precursor behaviour is displayed for very low values of the asymmetry.
The time evolution of an initially localized state of a quantum particle coupled to a dissipative environment with spectral density I(\ensuremath{\omega})\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{\alpha}}}$ for low frequencies is discussed. At finite temperatures, the width of the state can grow subdiffusively or superdiffusively, depending on \ensuremath{\alpha}. For \ensuremath{\alpha}>2 damping becomes ineffective for long times and the state spreads kinematically. At zero temperature the spreading is slower and for \ensuremath{\alpha}<1 an initially localized state remains localized for all times.
We study the real-time dynamics of quantum models with long-range interactions coupled to a heat bath within the closed-time path-integral formalism. We show that quantum fluctuations depress the transition temperature. In the subcritical region there are two asymptotic time regimes with (i) stationary and (ii) slow aging dynamics. We extend the quantum fluctuation-dissipation theorem to the nonequilibrium case in a consistent way with the notion of an effective temperature that drives the system in the aging regime. The classical results are recovered for $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$.
The dissipative dynamics of a quantum Brownian particle is studied for different types of environment. We derive analytic results for the time evolution of the mean energy of the system for Ohmic, sub-Ohmic, and super-Ohmic environments, without performing the Markovian approximation. Our results allow one to establish a direct link between the form of the environmental spectrum and the thermalization dynamics. This in turn leads to a natural explanation of the microscopic physical processes ruling the system time evolution both in the short-time non-Markovian region and in the long-time Markovian one. Our comparative study of thermalization for different environments sheds light on the physical contexts in which non-Markovian dissipation effects are dominant.