The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath
ϱB(t) and the system
ϱS(t) are written down. A formal solution of the equation for
ϱB(t), having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in
ϱS(t), is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of
ϱB(t) and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.
We compare different quantum master equations for the time evolution of the reduced density matrix. The widely applied secular approximation (rotating wave approximation) applied in combination with the Born-Markov approximation generates a Lindblad-type master equation ensuring for completely positive and stable evolution and is typically well applicable for optical baths. For phonon baths however, the secular approximation is expected to be invalid. The usual Markovian master equation does not generally preserve positivity of the density matrix. As a solution we propose a coarse-graining approach with a dynamically adapted coarse-graining time scale. For some simple examples we demonstrate that this preserves the accuracy of the integro-differential Born equation. For large times we analytically show that the secular approximation master equation is recovered. The method can in principle be extended to systems with a dynamically changing system Hamiltonian, which is of special interest for adiabatic quantum computation. We give some numerical examples for the spin-boson model of cases where a spin system thermalizes rapidly, and other examples where thermalization is not reached.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of
N molecules interacting via a two-body potential and a quantum-optical system of
N two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's
F1(t) and
F2(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO
R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
Data of the numerical solution of the time-dependent Schrödinger equation of a system containing one spin-1/2 particle interacting with a bath of up to 32 spin-1/2 particles is used to construct a Markovian quantum master equation describing the dynamics of the system spin. The procedure of obtaining this quantum master equation, which takes the form of a Bloch equation with time-independent coefficients, accounts for all non-Markovian effects inasmuch the general structure of the quantum master equation allows. Our simulation results show that, with a few rather exotic exceptions, the Bloch-type equation with time-independent coefficients provides a simple and accurate description of the dynamics of a spin-1/2 particle in contact with a thermal bath. A calculation of the coefficients that appear in the Redfield master equation in the Markovian limit shows that this perturbatively derived equation quantitatively differs from the numerically estimated Markovian master equation, the results of which agree very well with the solution of the time-dependent Schrödinger equation.