# Dynamics of Open Quantum Systems II, Markovian Approximation

Marco Merkli

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, A1C 5S7, Canada

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### Abstract

A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature $T \gt 0$. We show that for fixed, small values of the coupling constant $\lambda$, 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 $O(|\lambda|^{1/4})$ for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy $O(|\lambda|^{1/4})$ 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 $1/t^{4}$ or faster, which is a significant improvement relative to the previously required exponential decay.

We consider a finite-dimensional quantum system interacting with a thermal Bose field (a reservoir consisting of non-interacting oscillators). The total dynamics is given by a Hamiltonian containing a system, a reservoir plus an interaction term. We give a rigorous argument showing that the reduced dynamics of the system (after tracing out the reservoir degrees of freedom) is approximated by the solution of the markovian master equation. The difference between the approximate (markovian) and the true system density matrix is small in the interaction between the system and reservoir, equally so for all times $t\ge0$. We show this result for reservoirs with polynomially decaying correlation functions (previously, exponentially quick decay was needed).

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