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Physical Review A

Non-Markovian effects on an anharmonic oscillator stochastically coupled to a thermal bath

Juan A. White, Santiago Velasco

Published Sep 1, 1989• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.40.3156

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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.
The influence of a finite bath correlation time and the bath temperature on the onset of a non-Markovian behaviour in the relaxation of a harmonic oscillator linearly coupled to a coloured thermal bath is analysed. A fully quantum mechanical treatment of the time autocorrelation function of the oscillator-bath interaction is made. Special attention is paid to the temperature dependence. In the low-temperature regime a non-exponential time decay of the mean number of quanta is found even though the correlation time associated with the oscillator-bath interaction is zero.
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.
Two different approaches (quantum and semiclassical) used to calculate time-dependent transition rates for a multilevel molecular system stochastically interacting with a thermal bath are presented. While the former is based on the use of a relationship between the real and the imaginary parts of the time autocorrelation function (TAF) associated with the interaction Hamiltonian, the latter relies on the replacement of a quantum TAF by the corresponding anticommutator. Two well-known examples for a stochastic interaction Hamiltonian, an Ornstein-Uhlenbeck process and a white-noise process, have been considered. In both cases, explicit expressions for the transition rates in terms of a reduced set of parameters characterizing the bath are obtained. In the Gaussian regime both quantum and semiclassical transition rates show a time-damped oscillatory behavior towards the same steady value. In the white-noise (\ensuremath{\delta}-correlated) regime, while the quantum transition rates are time dependent, the semiclassical rates are not. The high- and the low-temperature limits for the quantum and the semiclassical transition rates are also reported. The semiclassical results reduce to the quantum results only in the high-temperature limit.
The memory kernel appearing in a non-Markovian population equation for a multilevel quantum system stochastically coupled to a colored Gaussian bath is calculated on the basis of a fully quantum-mechanical treatment to construct the time autocorrelation function (TAF) associated to the system–bath interaction Hamiltonian. The limiting cases of high and low temperatures and white noise bath are also analyzed. These examples allow us to discuss in a very simple way the dependence of the memory kernel on the Bohr frequencies of the system, the intensity of the system–bath interaction, its correlation time, and the bath temperature. In particular, it is shown that a true Markovian population equation is only achieved at high temperatures with a bath correlation time much shorter than the relaxation process characteristic time.
The system of nonlinear Langevin equations was obtained by using Hamiltonian's operator of two coupling quantum oscillators which are interacting with heat bath. By using the analytical solution of these equations, the analytical expressions for transport coefficients was found. Generalized Langevin equations and fluctuation-dissipation relations are derived for the case of a nonlinear non-Markovian noise. The explicit expressions for the time-dependent friction and diffusion coefficients are presented for the case of linear couplings in the coordinate between the collective two coupled harmonic oscillators and heat bath.
The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.