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On derivation of Markovian master equation with time coarse graining

Yasusada Nambu, Shingo Kukita

Published Apr 13, 2015• 0 citations
OpenAlex

Abstract

We present a derivation of the Markovian master equation using the time coarse graining method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then applying the renormalization group method, it is shown that dependence of the specific initial time in the solution can be removed and the Markovian semi-group master equation in Lindblad form is attained in the weak coupling limit.

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We present a derivation of the Markovian master equation using the time coarse graining method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then applying the renormalization group method, it is shown that dependence of the specific initial time in the solution can be removed and the Markovian semi-group master equation in Lindblad form is attained in the weak coupling limit.
We present a derivation of the Markovian master equation by the renormalization group method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then by applying the renormalization group method, we show that the dependence of the specific initial time on the perturbative solution can be removed and the Markovian semigroup master equation in the Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) form is obtained in the weak coupling limit.
We present a first-principles derivation of the Markovian semigroup master equation without invoking the rotating-wave approximation (RWA). Instead we use a time coarse-graining approach that leaves us with a free time-scale parameter, which we can optimize. Comparing this approach to the standard RWA-based Markovian master equation, we find that significantly better agreement is possible using the coarse-graining approach, for a three-level model coupled to a bath of oscillators, whose exact dynamics we can solve for at zero temperature. The model has the important feature that the RWA has a nontrivial effect on the dynamics of the populations. We show that the two different master equations can exhibit strong qualitative differences for the population of the energy eigenstates even for such a simple model. The RWA-based master equation misses an important feature which the coarse-graining-based scheme does not. By optimizing the coarse-graining time scale the latter scheme can be made to approach the exact solution much more closely than the RWA-based master equation.
This is a derivation of the quantum-optical master equation using coarse-graining of time, which aims at exploring Markovian evolution from a microscopic perspective. Here, coarse-graining refers to finding an approximate evolution operator by integrating over a time step Δ t , and considering the field states to be approximated by a similarly coarse-grained basis. By using these ideas and keeping the field states coarse-grained throughout the derivation, I explore the past-future independence of the quantum white noise formalism explicitly. Namely, that a quantum-optical system interacting with a vacuum field may be viewed as sequentially interacting with a series of initially uncorrelated and temporally coarse-grained vacuum states—this process leads to correlations only between the system and the time bins which have interacted with the system in the past, along with a Markovian evolution of the reduced system state. The derivation here may also be viewed in terms of a type of quantum-mechanical path integral.
The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to a further timescale with respect to which the coarse-graining time interval can be chosen. Different choices lead to different master equations that correspond to the local and global forms. These can be then understood as having different physical interpretations based on the role of the coarse-graining, as well as different limitations in application.
Inspired by holographic Wilsonian renormalization, we consider coarse graining a quantum system divided between short-distance and long-distance degrees of freedom (d.o.f.), coupled via the Hamiltonian. Observations using purely long-distance observables are described by the reduced density matrix that arises from tracing out the short-distance d.o.f. The dynamics of this density matrix is non-Hamiltonian and nonlocal in time, on the order of some short time scale. We describe this dynamics in a model system with a simple hierarchy of energy gaps $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{UV}}>\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{IR}}$, in which the coupling between high- and low-energy d.o.f. is treated to second order in perturbation theory. We then describe the equations of motion under suitable time averaging, reflecting the limited time resolution of actual experiments, and find an expansion of the master equation in powers of $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{IR}}/\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{UV}}$, after the fashion of effective field theory. The failure of the system to be Hamiltonian or even Markovian appears at higher orders in this ratio. We compute the evolution of the density matrix in three specific examples: coupled spins, linearly coupled simple harmonic oscillators, and an interacting scalar quantum field theory. Finally, we argue that the logarithm of the Feynman-Vernon influence functional is the correct analog of the Wilsonian effective action for this problem.