Quantum diffusion with drift and the Einstein relation. II

Abstract

Diffusive behavior from a quantum master equationThis paper is a companion to Paper I [W. De Roeck, J. Fröhlich, and K. Schnelli, "Quantum diffusion with drift and the Einstein relation. I," J. Math. Phys. 55, 075206 (2014)]. The purpose of this paper is to describe and prove a certain number of technical results used in Paper I, but not proven there. Both papers concern long-time properties (diffusion, drift) of the motion of a driven quantum particle coupled to an array of thermal reservoirs. The main technical results derived in the present paper are: (1) an asymptotic perturbation theory applicable for small driving force, and (2) the construction of time-dependent correlation functions of particle observables. C 2014 AIP Publishing LLC. [http://dx.

Quantum diffusion with drift and the Einstein relation. I

“…and we claim that this limit exists for any exponentially localized ρ S and any O 1 , O 2 in A or X. We refer to [6] for a proof of this claim. Apart from an initial state (density matrix) of the product form ρ S ⊗ ρ R,β , we also consider the Gibbs state of the coupled system when the external force field vanishes, χ = 0.…”

“…We only describe the main ideas and present formal arguments. Mathematically precise arguments are given in [6], Sections 4 and 5, where elaborate expansion techniques are developed that can also be used to analyze correlation functions.…”

“…Before we discuss the Einstein relation, let us mention that our model relaxes exponentially fast to equilibrium at vanishing external field; cf., Theorem 3.3. in [6]:…”

“…The strategy we follow to control the long-time behavior when λ is small but non-zero, is basically the same as in [20]: We represent the Laplace transform of the fibered dynamics (Z [0,λ −2 τ ]) λ 2 κ as a small (in λ) perturbation of the resolvent of the Boltzmann generator M κ,χ . This is accomplished by appropriately resumming diagrams, and this is the tedious part of our analysis, which is described in [6]. The ideas underlying this analysis are elementary, and our technique is actually a time-dependent counterpart of the use of 'translation-analyticity in the spectral form factor' first applied to the study of 'confined' open quantum systems by [14].…”

“…Representation (6.7) is the starting point for the proof of Lemma 3.1. The details of our proof can be found in [6]; Section 5.1. Here we just indicate some of the main ideas underlying it: First, we recall the bound on U I from (3.2):…”