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arXiv:2304.08627 · Lobachevskii Journal of Mathematics

Time-Convolutionless Master Equations for Composite Open Quantum Systems

A. Yu. Karasev, Alexander Evgen’evich Teretenkov

Published Jun 1, 2023• 8 citations
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Abstract

Abstract In this work we consider the master equations for composite open quantum systems. We provide purely algebraic formulae for terms of perturbation series defining such equations. We also give conditions under which the Bogolubov–van Hove limit exists and discuss some corrections to this limit. We present an example to illustrate our results. In particular, this example shows, that inhomogeneous terms in time-convolutionless master equations can vanish after reservoir correlation time, but lead to renormalization of initial conditions at such a timescale.

DOI
10.1134/s1995080223060239

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Abstract In this work we consider the master equations for composite open quantum systems. We provide purely algebraic formulae for terms of perturbation series defining such equations. We also give conditions under which the Bogolubov–van Hove limit exists and discuss some corrections to this limit. We present an example to illustrate our results. In particular, this example shows, that inhomogeneous terms in time-convolutionless master equations can vanish after reservoir correlation time, but lead to renormalization of initial conditions at such a timescale.
The time-convolutionless quantum master equation is an exact description of the nonequilibrium dynamics of open quantum systems, with the advantage of being local in time. We derive a perturbative expansion to arbitrary order in the system-reservoir coupling for its generator, which contains significantly fewer terms than the ordered-cumulant expansion. We show that the derived expansion also admits a simple recursive formulation. The derived series is then used to describe the nonequilibrium dynamics of a quantum dot, including coherences. We find a relation between the generator and a generalization of the TTT-matrix. Even though the TTT-matrix rate equations are plagued by divergences, we show that these cancel order by order in the generator of the time-convolutionless master equation. This generalizes previous work on the time-convolutionless Pauli master equation, which does not include coherences, and lays the ground for possible numerical evaluations and analytical resummations.
Abstract We study the perturbative corrections to the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation which arises in the weak coupling limit. The spin-boson model in the rotating wave approximation at zero temperature is considered. We show that the perturbative part of the density matrix satisfies the time-independent GKSL equation for arbitrary order of the perturbation theory (if all the moments of the reservoir correlation function are finite). But to reproduce the right asymptotic precision at long times, one should use an initial condition different from the one for exact dynamics. Moreover, we show that the initial condition for this master equation even fails to be a density matrix under certain resonance conditions.
Abstract The model of multi-level open quantum system interacting with a non-vacuum reservoir in the rotating wave approximation is considered. We provide an exact integral representation for the reduced density matrix of the system. For identical uncorrelated reservoirs in diagonal states we have obtained the first perturbative correction for such dynamics in the Bogolubov–van Hove limit. We have shown that after initial state renormalization it can be completely described in terms of finite-dimensional semigroup. The method we provide can also be applied to the further orders of perturbation theory with Bogolubov–van Hove scaling.
The time-convolutionless master equation provides a general framework to model the non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to perturbative time-convolutionless quantum master equations. We further introduce a general iterative approach to construct nonperturbative approximations for the time-local generator as nested time-ordered exponential operators.
The time-convolutionless master equation provides a general framework to model non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to the perturbative time-convolutionless quantum master equations. We further introduce a nonperturbative approach that approximates the time-convolutionless generator as a nested time-ordered exponential function.
We consider a general open system dynamics and we provide a recursive method to derive the associated non-Markovian master equation in a perturbative series. The approach relies on a momenta expansion of the open system evolution. Unlike previous perturbative approaches of this kind, the method presented in this paper provides a recursive definition of each perturbative term. Furthermore, we give an intuitive diagrammatic description of each term of the series, which provides a useful analytical tool to build them and to derive their structure in terms of commutators and anticommutators. We eventually apply our formalism to the evolution of the observables of the reduced system, by showing how the method can be applied to the adjoint master equation, and by developing a diagrammatic description of the associated series.