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arXiv:1504.03070 · Journal of the Physical Society of Japan

Derivation of Markovian Master Equation by Renormalization Group Method

Yasusada Nambu, Shingo Kukita

Published Oct 28, 2016• 4 citations
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Abstract

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.

DOI
10.7566/jpsj.85.114002

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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 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.
For a model of a multilevel system interacting with several baths at zero temperature, it is shown that its dynamics becomes Markovian after the bath correlation time. We take into account not only the contribution of the bath spectral density, which leads to a continuous correlation function, but also the ohmic contribution to the spectral density, which leads to a renormalization of both equations and initial conditions. An explicit Gorini–Kossakowski–Sudarshan–Lindblad equation describing the dynamics of the system after the bath correlation time is derived, and the form of initial conditions for this equation is obtained. They do not coincide with the exact initial conditions due to the renormalization associated with the ohmic contribution and due to the short initial non-Markovian time interval preceding the bath correlation time.
Abstract For a model of a multilevel system interacting with several baths at zero temperature, it is shown that its dynamics becomes Markovian after the bath correlation time. We take into account not only the contribution of the bath spectral density, which leads to a continuous correlation function, but also the ohmic contribution to the spectral density, which leads to a renormalization of both equations and initial conditions. An explicit Gorini–Kossakowski–Sudarshan–Lindblad equation describing the dynamics of the system after the bath correlation time is derived, and the form of initial conditions for this equation is obtained. They do not coincide with the exact initial conditions due to the renormalization associated with the ohmic contribution and due to the short initial non-Markovian time interval preceding the bath correlation time.
The theory of open quantum systems is one of the most essential tools for the development of quantum technologies. Furthermore, the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) master equation plays a key role as it is the most general generator of Markovian dynamics in quantum systems. In this paper, we present this equation together with its derivation and methods of resolution. The presentation tries to be as self-contained and straightforward as possible to be useful to readers with no previous knowledge of this field.
A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0 . We show that for fixed, small values of the coupling constant λ , the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.
This preprint presents a self-contained and mathematically explicit reconstruction of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation starting from exact microscopic unitary dynamics through the Nakajima-Zwanzig projection operator formalism. Beginning with the exact non-local generalized master equation, we systematically analyze the hierarchy of approximations required for the emergence of a Markovian quantum dynamical semigroup. The Born (weak-coupling), Markov (time-localization), and secular approximations are derived and interpreted as successive information-reduction steps, each associated with clearly identifiable mathematical assumptions and physical consequences. Particular attention is devoted to the role of secularization in transforming the Redfield generator into a completely positive GKSL structure. To provide a concrete worked example, the formalism is applied to a damped two-level system coupled to a bosonic thermal reservoir, where the Redfield and GKSL equations are derived explicitly and compared at the microscopic level. Beyond the derivation itself, the work develops a systematic map of validity limits and failure regimes. The breakdown of the Born, Markov, secular, factorization, and stationarity assumptions is analyzed in terms of environmental correlation times, spectral-density structure, non-Markovian memory effects, information backflow, and loss of semigroup behavior. The resulting framework clarifies the precise operational domain in which GKSL dynamics can be regarded as a physically reliable effective description and provides a rigorous foundation for applications involving entropy-production theorems, quantum thermodynamics, and open-system information theory.