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The Journal of Physical Chemistry B

Markovian Approximation in the Relaxation of Open Quantum Systems

Yuan‐Chung Cheng, R. Silbey

Published Jul 1, 2005• 60 citations
Publisher

Abstract

In this paper, we examine the validity of the Markovian approximation and the slippage scheme used to incorporate short time transient memory effects in the Markovian master equations (Redfield equations). We argue that for a bath described by a spectral function, J(omega), that is dense and smoothly spread out over the range omega(d), a time scale of tau(b) approximately 1/omega(d) exists; for times of t > tau(b), the Markovian approximation is applicable. In addition, if J(omega) decays to zero reasonably fast in both the omega --> 0 and omega --> infinity limits, then the bath relaxation time, tau(b), is determined by the width of the spectral function and is weakly dependent on the temperature of the bath. On the basis of this criterion of tau(b), a scheme to incorporate transient memory effects in the Markovian master equation is suggested. Instead of using slipped initial conditions, we propose a concatenation scheme that uses the second-order perturbation theory for short time dynamics and the Markovian master equation at long times. Application of this concatenation scheme to the spin-boson model shows that it reproduces the reduced dynamics obtained from the non-Markovian master equation for all parameters studied, while the simple slippage scheme breaks down at high temperatures.

DOI
10.1021/jp051303o

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In this paper, we examine the validity of the Markovian approximation and the slippage scheme used to incorporate short time transient memory effects in the Markovian master equations (Redfield equations). We argue that for a bath described by a spectral function, J(omega), that is dense and smoothly spread out over the range omega(d), a time scale of tau(b) approximately 1/omega(d) exists; for times of t > tau(b), the Markovian approximation is applicable. In addition, if J(omega) decays to zero reasonably fast in both the omega --> 0 and omega --> infinity limits, then the bath relaxation time, tau(b), is determined by the width of the spectral function and is weakly dependent on the temperature of the bath. On the basis of this criterion of tau(b), a scheme to incorporate transient memory effects in the Markovian master equation is suggested. Instead of using slipped initial conditions, we propose a concatenation scheme that uses the second-order perturbation theory for short time dynamics and the Markovian master equation at long times. Application of this concatenation scheme to the spin-boson model shows that it reproduces the reduced dynamics obtained from the non-Markovian master equation for all parameters studied, while the simple slippage scheme breaks down at high temperatures.
A close examination of the validity of the Markovian approximation in the context of relaxation theory is presented. In particular, we examine the question of positivity of various approximations to the reduced dynamics of an open system in interaction with a heat reservoir. It is shown that the Markovian equations of motion obtained in the weak coupling limit (Redfield equations) are a consistent approximation to the actual reduced dynamics only if supplemented by a slippage in the initial conditions. This slippage captures the effects of the non-Markovian evolution that takes place in a short transient time, of the order of the relaxation time of the isolated bath.
For a slow open quantum subsystem weakly coupled to a fast thermal bath, we derive the general form of the slippage to be applied to the initial conditions of the Redfield master equation. This slippage is given by a superoperator which describes the non-Markovian dynamics of the subsystem during the short-time relaxation of the thermal bath. We verify in an example that the Redfield equation preserves positivity after the slippage superoperator has been applied to the initial density matrix of the subsystem. For δ-correlated baths, the Redfield master equation reduces to the Lindblad master equation and the slippage of initial conditions vanishes consistently.
Employing a recently proposed measure for quantum non-Markovianity, we carry out a systematic study of the size of memory effects in the spin-boson model for a large region of temperature and frequency cutoff parameters. The dynamics of the open system is described utilizing a second-order time-convolutionless master equation without the Markov or rotating wave approximations. While the dynamics is found to be strongly non-Markovian for low temperatures and cutoffs, in general, we observe a special regime favoring Markovian behavior. This effect is explained as resulting from a resonance between the system's transition frequency and the frequencies of the dominant environmental modes. We further demonstrate that the corresponding Redfield equation is capable of reproducing the characteristic features of the non-Markovian quantum behavior of the model.
Quantum master equations are commonly used to model the dynamics of open quantum systems, but their accuracy is rarely compared with the analytical solution of exactly solvable models. In this work, we perform such a comparison for the damped Jaynes-Cummings model of a qubit in a leaky cavity, for which an analytical solution is available in the one-excitation subspace. We consider the non-Markovian time-convolutionless master equation up to the second (Redfield) and fourth orders as well as three types of Markovian master equations: the coarse-grained, cumulant, and standard rotating-wave approximation (RWA) Lindblad equations. We compare the exact solution to these master equations for three different spectral densities: impulse, Ohmic, and triangular. We demonstrate that the coarse-grained master equation outperforms the standard RWA-based Lindblad master equation for weak coupling or high qubit frequency (relative to the spectral density high-frequency cutoff ${\ensuremath{\omega}}_{c}$), where the Markovian approximation is valid. In the presence of non-Markovian effects characterized by oscillatory, nondecaying behavior, the time-convolutionless approximation closely matches the exact solution for short evolution times (in units of ${\ensuremath{\omega}}_{c}^{\ensuremath{-}1}$) even outside the regime of validity of the Markovian approximations. For long evolution times, all master equations perform poorly, as quantified in terms of the trace-norm distance from the exact solution. The fourth-order time-convolutionless master equation achieves the top performance in all cases. Our results highlight the need for reliable approximation methods to describe open-system quantum dynamics beyond the short-time limit.
Interaction of a small system S with a large reservoir R amounts to thermal relaxation of the reduced system density operator ρS(t). The presence of the reservoir enters the equation of motion for ρS(t) through the reservoir correlation functions fkl(τ) (defined in the text), which decay to zero for τ→∞ on a time scale τc. Commonly, this τc is much smaller than the inverse relaxation constants for the time evolution of ρS(t). Then a series of approximations can be made, which lead to a Markovian equation of motion for ρS(t). In this paper the assumption of a small reservoir correlation time is removed. The equation of motion for ρS(t) is solved, and it appears that the memory effect, due to τc≂/0, can be incorporated in a frequency dependence of the relaxation operator Γ̃(ω). Subsequently, (unequal-time) quantum correlation functions of two system operators are considered, where explicit expressions for (the Laplace transform of) the correlation functions are obtained. They involve again the relaxation operator Γ̃(ω), which accounts for the time regression. Additionally it is found that an initial-correlation operator Υ̃(ω) arises, as a consequence of the fact that the equal-time correlation functions do not factorize as ρS(t) times the reservoir density operator. It is pointed out that the frequency dependence of Γ̃(ω) and the occurrence of a nonzero T̃(ω) both arise as a result of τc≂/0, and should therefore be treated on an equal footing. Explicit evaluation of Γ̃(ω) and Υ̃(ω) shows that their matrix elements can be expressed entirely in f̃kl(ω), just as in the Markov approximation. Hence no essential complications appear if one should go beyond the limits of a small reservoir correlation time τc.
We report a study of a stochastic Schrödinger equation corresponding to the Redfield master equation with slipped initial conditions, which describes the dynamics of a slow subsystem weakly coupled to a fast thermal bath. Using the projection-operator method of Feshbach, we derive a non-Markovian stochastic Schrödinger equation of the generalized Langevin type, which simulates the time evolution of the quantum wave functions of the subsystem driven by the fluctuating bath. For δ-correlated baths, the non-Markovian stochastic Schrödinger equation reduces to the previously derived Markovian one. Numerical methods are proposed to simulate the time evolution under our non-Markovian stochastic Schrödinger equation. These methods are illustrated with the spin-boson model.