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Physica Scripta

Theory of relaxation and pumping of quantum oscillator non-resonantly coupled with the other oscillator

Andrew I. Trubilko, Ashat M. Basharov

Published Jan 22, 2020• 18 citations
Publisher

Abstract

Abstract The paper shows mechanisms of both the pumping and energy decay of an ‘isolated’ oscillator. The oscillator is only coupled non-resonantly with the adjacent oscillator which resonantly interacts with the thermal bath environment. Under these conditions the ‘isolated’ oscillator begins interacting with the thermal bath environment of the adjacent oscillator. The conclusion is based on the kinetic equation derived relative to anti-rotating terms of the initial Hamiltonian. The original model is reduced to two ‘independent’ open systems, with each containing original oscillators connected with their effective independent thermal baths, whose parameters are determined. The paper also gives a description of the obtained model in terms of quantum stochastic differential equations.

DOI
10.1088/1402-4896/ab6eb5

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Abstract The paper shows mechanisms of both the pumping and energy decay of an ‘isolated’ oscillator. The oscillator is only coupled non-resonantly with the adjacent oscillator which resonantly interacts with the thermal bath environment. Under these conditions the ‘isolated’ oscillator begins interacting with the thermal bath environment of the adjacent oscillator. The conclusion is based on the kinetic equation derived relative to anti-rotating terms of the initial Hamiltonian. The original model is reduced to two ‘independent’ open systems, with each containing original oscillators connected with their effective independent thermal baths, whose parameters are determined. The paper also gives a description of the obtained model in terms of quantum stochastic differential equations.
The paper shows mechanisms of both the pumping and energy decay of an "isolated" oscillator. The oscillator is only non-resonantly coupled with the adjacent oscillator which resonantly interacts with the thermal bath environment. Under these conditions the "isolated" oscillator begins interacting with the thermal bath environment of the adjacent oscillator. The conclusion is based on the kinetic equation derived relative to anti-rotating terms of the initial Hamiltonian, with the latter being the Hamiltonian of two oscillators and environment of one of them.
Abstract The mechanisms for the pumping and decay of an “isolated” oscillator that can interact only nonresonantly with a neighboring oscillator of different frequency are presented. We show that if this neighboring oscillator is coupled to a broadband heat-bath field, the isolated oscillator begins to interact with this heat-bath field. A new relaxation channel attributable to the quantum interference of the interacting systems that is difficult or impossible to justify in terms of traditional approaches emerges as a result.
Abstract We investigate the classical problem of two resonantly interacting oscillators each of which is coupled to “its own” heat bath based on the effective Hamiltonian method and the quantum stochastic differential equation (in contrast to the well-known “global” and “local” approaches). We show that in the second order of the algebraic perturbation theory, each of the oscillators turns out to be also coupled to the “foreign” heat bath. We calculate the steady-state heat flows and prove that there is no heat flow from the cold heat bath to the hot one, as evidenced by some results of the local approach.
We present an approximation of a non-Markovian solution for the time evolution of the mean number of quanta in an anharmonic oscillator linearly and stochastically coupled to a thermal bath. A colored Gaussian bath is explicitly considered, and a quantum treatment of the bath correlation functions, based on the use of a relation between their real and imaginary parts, is applied. The behavior of the oscillator is characterized by a reduced set of (five) parameters referring to the oscillator, the bath, and their mutual interaction. A comparison is made between the prediction of the approach for the mean stationary number of quanta and its exact value, which enables us to analyze its accuracy. When the Markovian limit holds, an explicit expression for a parameter measuring the time scale for the relaxation of the oscillator is obtained in terms of the characteristic parameters of the model. Except for very high temperatures, the anharmonicity has a small influence on this relaxation time. A numerical study of the time dependence of the mean number of quanta is presented on the basis of the reported analytical solution. We find that the anharmonicity is not a significant parameter in the appearance nor in the increase of non-Markovian effects.
We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.
We study the dynamics of relaxation and thermalization in an exactly solvable model of a quantum particle interacting with a thermal bath of oscillators with the goal of understanding the effects of off-shell processes. The focus is to compare the exact evolution of the distribution function with different approximations to the relaxational dynamics: Boltzmann, non-Markovian and Markovian quantum kinetics. The time evolution of the distribution function is evaluated exactly using different approaches where each method provides different insights. There are two different cases that are studied in detail: (i) no stable particle states below threshold of the bath and a quasiparticle resonance above it and (ii) a stable discrete exact ``particle'' state below threshold. The exact solution for the evolution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of particles and quasiparticles. For the case of quasiparticles in the continuum (resonances) the exact quasiparticle distribution asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes such as the strength of the quasiparticle poles, the width of the unstable particle and proximity to thresholds. In case (ii), the distribution of particles does not thermalize with the bath. We study the kinetics of thermalization and relaxation by deriving a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation are obtained from the quantum kinetic equation in the limit of a wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian and Boltzmann approximations are compared to the exact result of the model. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important. Implications for thermalization in field theory models are discussed.