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Physical Review A

Time-dependent transition rates for a multilevel quantum system stochastically coupled to a thermal bath

Santiago Velasco, Juan A. White, Antonio Calvo Hernández

Published Apr 1, 1989• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.39.3653

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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.
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 fluctuation properties of a one-dimensional many-body quantum system composed of interacting bosons and investigate the regimes where quantum noise or, respectively, thermal excitations are dominant. For the latter, we develop a semiclassical description of the fluctuation properties based on the Ornstein-Uhlenbeck stochastic process. As an illustration, we analyze the phase correlation functions and the full statistical distributions of the interference between two one-dimensional systems, either independent or tunnel-coupled, and compare with the Luttinger-liquid theory.
We explore the noise-induced barrier crossing dynamics of a Brownian particle in the high temperature quantum regime under large damping. We assume the associated heat bath not to be in thermal equilibrium; it is rather driven by an externally applied random force which exposes the system particles to a nonequilibrium environment. We propose a system + reservoir model to study the stochastic Langevin dynamics. We also construct the corresponding Fokker–Planck equation in the said regime and solve it to explore the bistable kinetics. We investigate the role of different parameters in shaping the nature of such a bistable kinetics in detail and hence allowing one to get some insight into the very complicated dynamics of quantum dissipative system(s). Finally, we analyze the semiclassical rate vis-a-vis the classical analog.