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arXiv:2008.02820 · Journal of Physics A Mathematical and Theoretical

Non-perturbative effects in corrections to quantum master equations arising in Bogolubov–van Hove limit

Alexander Evgen’evich Teretenkov

Published May 17, 2021• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ac0201

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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.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account non-secular effects, but, nevertheless, has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form (in particular, preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The non-secular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
For the model of a multi-level system in the rotating wave approximation we obtain the corrections for a usual weak coupling limit dynamics by means of perturbation theory with Bogolubov-van Hove scaling. It generalizes our previous results on a spin-boson model in the rotating wave approximation. Additionally, in this work we take into account some dependence of the system Hamiltonian on the small parameter. We show that the dynamics is long-time Markovian, i.e. after the bath correlation time all the non-Markovianity could be captured by the renormalization of initial condition and correlation functions.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The nonsecular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
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.