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arXiv:0903.3045 · Physical Review A

Renormalized coordinate approach to the thermalization process

G. Flores-Hidalgo, Adolfo P. C. Malbouisson, J. M. C. Malbouisson, Y. W. Milla, Ademir Eugenio de Santana

Published Mar 9, 2009• 7 citations
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Abstract

We consider a particle in the harmonic approximation coupled linearly to an environment modeled by an infinite set of harmonic oscillators. The system (particle environment) is considered in a cavity at thermal equilibrium. We employ the recently introduced notion of renormalized coordinates to investigate the time evolution of the particle occupation number. For comparison, we first present this study in bare coordinates. For a long elapsed time, in both approaches, the occupation number of the particle becomes independent of its initial value. The value of the occupation number of the particle is the physically expected one at the given temperature. So we have a Markovian process, describing the particle thermalization with the environment. With renormalized coordinates, no renormalization procedure is required, leading directly to a finite result.

DOI
10.1103/physreva.79.032105

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We consider a particle in the harmonic approximation coupled linearly to an environment modeled by an infinite set of harmonic oscillators. The system (particle environment) is considered in a cavity at thermal equilibrium. We employ the recently introduced notion of renormalized coordinates to investigate the time evolution of the particle occupation number. For comparison, we first present this study in bare coordinates. For a long elapsed time, in both approaches, the occupation number of the particle becomes independent of its initial value. The value of the occupation number of the particle is the physically expected one at the given temperature. So we have a Markovian process, describing the particle thermalization with the environment. With renormalized coordinates, no renormalization procedure is required, leading directly to a finite result.
We consider a particle in the approximation of a harmonic oscillator, coupled linearly to a field modeling an environment. The field is described by an infinite set of harmonic oscillators, and the system (particle--field) is considered in a cavity at thermal equilibrium. We employ the notions of bare and dressed coordinates to study the time evolution of the occupation number. With dressed coordinates no renormalization procedure is required, leading directly to a finite result. In particular, for a large time, the occupation number of the particle becomes independent of its initial value. So we have a Markovian process, describing the particle thermalization with the environment.
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An atom, coupled linearly to an environment, is considered in a harmonic approximation in thermal equilibrium inside a cavity. The environment is modeled by an infinite set of harmonic oscillators. We employ the notion of dressed states to investigate the time evolution of the atom initially in the first excited level. In a very large cavity (free space) for a long elapsed time, the atom decays and the value of its occupation number is the physically expected one at a given temperature. For a small cavity the excited atom never completely decays and the stability rate depends on temperature.
We study the time evolution of an atom suddenly coupled to a thermal radiation field. As a simplified model of the atom-electromagnetic field system we use a system composed by a harmonic oscillator linearly coupled to a scalar field in the framework of the recently introduced dressed coordinates and dressed states. We show that the time evolution of the thermal expectation values for the occupation number operators depend exclusively on the probabilities associated with the emission and absorption of field quanta. In particular, the time evolution of the number operator associated with the atom is given in terms of the probability of remaining in the first excited state and the decay probabilities from this state by emission of field quanta of frequencies ωkω_kωk​. Also, it is showed that independent of the initial state of the atom, it thermalizes with the thermal radiation field in a time scale of the order of the inverse coupling constant.
We analyze in full detail the time evolution of an open Gaussian quantum system rapidly bombarded by Gaussian ancillae. As a particular case this analysis covers the thermalization (or not) of a harmonic oscillator coupled to a thermal reservoir made of harmonic oscillators. We derive general results for this scenario and apply them to the problem of thermalization. We show that only a particular family of system-environment couplings will cause the system to thermalize to the temperature of its environment. We discuss that if we want to understand thermalization as ensuing from the Markovian interaction of a system with the individual microconstituents of its (thermal) environment then the process of thermalization is not as robust as one might expect.
There is presently considerable interest in accurately simulating the evolution of open systems for which Markovian master equations fail. Examples are systems that are time dependent and/or strongly damped. A number of elegant methods have now been devised to do this, but all use a bath consisting of a continuum of harmonic oscillators. While this bath is clearly appropriate for, e.g., systems coupled to the electromagnetic field, it is not so clear that it is a good model for generic many-body systems. Here we explore a different approach to exactly simulating open systems: using a finite bath chosen to have certain key properties of thermalizing many-body systems. To explore the numerical resources required by this method to approximate an open system coupled to an infinite bath, we simulate a weakly damped system and compare to the evolution given by the relevant Markovian master equation. We obtain the Markovian evolution with reasonable accuracy by using an additional averaging procedure, and elucidate how the typicality of the bath generates the correct thermal steady state via the process of "eigenstate thermalization."