Diffusive behavior from a quantum master equation

Abstract

We study a general class of translation invariant quantum Markov evolutions for a particle on $\bbZ^d$. The evolution consists of free flow, interrupted by scattering events. We assume spatial locality of the scattering events and exponentially fast relaxation of the momentum distribution. It is shown that the particle position diffuses in the long time limit. This generalizes standard results about central limit theorems for classical (non-quantum) Markov processes.Comment: 21 pages. Minor corrections of previous versio

Diffusive Limit for a Quantum Linear Boltzmann Dynamics

“…With the results from previous sections, the proof of the convergence for the rescaled position distribution in Thm. 2.1 has the pattern found in [3]. The convergence of the moments requires more effort due to the unbounded operators in this case.…”

“…2.1 is a standard one, which is formulated using a decomposition arising from the translation invariance of the model. The articles [26,3,7] are similarly based on a translation symmetry. The translation invariance of the model implies a "decomposition" of the state space into fibers…”

“…A consequence of this symmetry is that the dynamics admits a fiber decomposition in which individual fibers obey autonomous dynamics. Similar fiber decompositions can, for instance, be found in the study of periodic Schrödinger potentials [36, Ch.XIII.16] and in the study of certain translation invariant noisy dynamics for a quantum particle living on Z d , [26,3,7]. Also, the autonomous Markovian evolution for the momentum distribution (2.7) is a special case of the fiber decomposition and provides a useful tool for proving a solution of an unbounded Lindblad equation is unique when the Lindbladian is translation-covariant [20] (as discussed in the previous section).…”

“…Let m, M, m * = mM m+M refer to the mass of a single reservoir particle, the mass of the test particle, and the relative mass respectively. Let η be the density of the gas and r( k) = ( β 2πm ) 3 2 e − β 2m k 2 be the momentum distribution for a single reservoir particle. Define f : R + × [0, π] → C to be the scattering amplitude determined by the interaction potential between the test particle and a gas particle, and denote f ( p 2 , p 1 ) := f(p, θ), where p = p 1 2 = p 2 2 and θ is the angle between p 1 and p 2 .…”

Scattering induced current in a tight-binding band

“…There has recently been intensified interest in obtaining rigorous results along those lines. 8,11,12 We present here a simple, explicitly solvable, fully quantum mechanical and fully Hamiltonian model of such a particle-environment system within the repeated interaction scheme, in which the environment is described by a chain of two-level atoms. We show that a dc current is created due to the interaction of the particle with its environment.…”