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The Journal of Chemical Physics

Slippage of initial conditions for the Redfield master equation

Pierre Gaspard, Masataka Nagaoka

Published Oct 1, 1999• 163 citations
Publisher

Abstract

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.

DOI
10.1063/1.479867

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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.
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.
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.
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.
It is very common in the literature to write a Markovian quantum master equation in Lindblad form to describe a system with multiple degrees of freedom and weakly connected to multiple thermal baths which can, in general, be at different temperatures and chemical potentials. However, the microscopically derived quantum master equation up to leading order in a system-bath coupling is of the so-called Redfield form, which is known to not preserve complete positivity in most cases. Additional approximations to the Redfield equation are required to obtain a Lindblad form. We lay down some fundamental requirements for any further approximations to the Redfield equation, which, if violated, leads to physical inconsistencies such as inaccuracies in the leading order populations and coherences in the energy eigenbasis, violation of thermalization, and violation of local conservation laws at the nonequilibrium steady state. We argue that one or more of these conditions will generically be violated in all the weak system-bath-coupling Lindblad descriptions existing in the literature to our knowledge. As an example, we study the recently derived universal Lindblad equation and use these conditions to show the violation of local conservation laws due to inaccurate coherences but accurate populations in the energy eigenbasis. Finally, we exemplify our analytical results numerically in an interacting open quantum spin system.
We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxation timescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfield equation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity of the density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolution of pure states (rather than density matrices). We discuss how our method can be applied to static or driven quantum many-body systems, and illustrate its power through numerical simulation of a spin chain that would be challenging to treat by existing methods.
The propagation of a fast particle in a low-density gas at thermal equilibrium is studied in the context of quantum mechanics. A quantum master equation in the Redfield form governing the reduced density matrix of the particle is derived explicitly from first principles. Under some approximations, this equation reduces to the linear Boltzmann equation. The issue of the positivity of the time evolution is also discussed by means of a Lindblad form. The Born and Markov assumptions underlying these equations, as well as other approximations regarding the bath correlation function, are discussed in detail. Furthermore, all these master equations are shown to be equivalent with each other if the density matrix of the particle is diagonal in the momentum basis, or if the collision rate is independent of the particle momentum.