Derivation of Some Translation-Invariant Lindblad Equations for a Quantum Brownian Particle

Abstract

We study the dynamics of a Brownian quantum particle hopping on an infinite lattice with a spin degree of freedom. This particle is coupled to free boson gases via a translation-invariant Hamiltonian which is linear in the creation and annihilation operators of the bosons. We derive the time evolution of the reduced density matrix of the particle in the van Hove limit in which we also rescale the hopping rate. This corresponds to a situation in which both the system-bath interactions and the hopping between neighboring sites are small and they are effective on the same time scale. The reduced evolution is given by a translation-invariant Lindblad master equation which is derived explicitly.KEY WORDS: out-of-equilibrium quantum statistical physics, open quantum systems, weak coupling limit, singular coupling limit, quantum Brownian motion.

Diffusion for a Quantum Particle Coupled to Phonons in d ≥ 3

“…Furthermore, we note that by the invariance property j(e, ′ dk ′ ; e, k) = j(e, ′ d(k ′ − k); e, 0), we can deduce that µ Q (k, e) = (1/2π) d µ Q (e) (independent of k). For more details on how the NESS might look like in a concrete example, we refer the reader to [11] where we construct a model that describes a ratchet. Starting from this example, it is straightforward to prove that the NESS is not an equilibrium state.…”

“…For more details on how the NESS might look like in a concrete example, we refer the reader to [11] where we construct a model that describes a ratchet. Starting from this example, it is straightforward to prove that the NESS is not an equilibrium state.…”

General Galilei Covariant Gaussian Maps

“…The Holevo characterizations play a major role in the description of several important physical phenomena such as environmental decoherence and relaxation phenomena [7][8][9][10][11][12][13][14]. Furthermore, they are also relevant for the foundation of quantum mechanics, where an intrinsic non-unitary dynamics is postulated to solve the measurement problem [22][23][24], the black hole information paradox [25], or to combine principles of general relativity with quantum mechanics [26].…”

“…The exact quantum dynamics of a system interacting with the surrounding environment can be very complicated: in general, heavy approximations and heuristical arguments are needed in order to arrive at an explicit useful expression for the system's effective dynamics. In this case, symmetries can be a guiding principle in constructing the effective dynamics, bypassing at least partially the complexity (or impossibility) of a direct calculation by imposing constraints, which are expected to hold not only at the fundamental level, but also at the effective level [7][8][9][10][11][12][13][14][15].…”

“…[64] shows that the predictions of collapse models depend on whether dissipative effects are taken into account (Fig. (8) of Ref. [64] shows that the bounds on the collapse model drastically change with the thermalization temperature T , which quantifies the dissipation in the model).…”