Abstract
We develop a framework to analyze the dynamics of a finite-dimensional quantum systemSin contact with a reservoirR. The full, interactingSRdynamics is unitary. The reservoir has a stationary state but otherwise dissipative dynamics. We identify a main part of the full dynamics, which approximates it for small values of theSRcoupling constant, uniformly for all timest≥0. The main part consists of explicit oscillating and decaying parts. We show that the reduced system evolution is Markovian for all times. The technical novelty is a detailed analysis of the link between the dynamics and the spectral properties of the generator of theSRdynamics, based on Mourre theory. We allow forSRinteractions with little regularity, meaning that the decay of the reservoir correlation function only needs to be polynomial in time, improving on the previously required exponential decay.In this work we distill the structural and technical ingredients causing the characteristic features of oscillation and decay of theSRdynamics. In the companion paper [27] we apply the formalism to the concrete case of anN-level system linearly coupled to a spatially infinitely extended thermal bath of non-interacting Bosons.
Highlights
The fundamental evolution equation of quantum theory is the Schrödinger equation, which governs the dynamics of closed quantum systems, isolated from their surroundings
– For a large class of correlated initial SR states, the correlations decay polynomially in time. After this decay has happened, the reservoir is in thermal equilibrium and the system evolves according to the Markovian dynamics generated by the Davies generator, up to errors O(|λ|1/4), uniformly in times t ≥ 0
It is not our aim to present a detailed discussion of the huge literature on the dynamics of open quantum systems, as the goal of the current manuscript is the construction of a general mathematical framework
Summary
The fundamental evolution equation of quantum theory is the Schrödinger equation, which governs the dynamics of closed quantum systems, isolated from their surroundings. Is not affected much by the inteaction with the system These assumptions, called the Markov- and Born approximations, respectively, are quantified by the decay of the reservoir correlation function (quick memory loss) and a smallness condition on λ (weak coupling). The generator L(λ) has to be constructed starting from the full SR dynamics (interacting Hamiltonian H as above) and reducing the evolution to the system component alone by tracing out the reservoir degrees of freedom. The literature on this topic is enourmous, and there have been many proposals for L(λ). The proof is based on the fact that the estimate holds for Lλ replaced by L0, together with a suitable perturbation theory in λ
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