Exactly solvable model of quantum diffusion

TL;DR

This paper presents an exactly solvable quantum diffusion model using a translationally invariant tight-binding system coupled to an environment, revealing how diffusion emerges and disappears based on coupling strength.

Contribution

It provides an exact analytical solution for quantum diffusion in a finite system, including the spectrum of relaxation modes and the conditions for diffusion to occur.

Findings

Diffusion appears above a critical environmental coupling strength.

The diffusion coefficient depends on the energy band width and environmental noise.

Diffusion ceases below the critical coupling, indicating a transition in transport behavior.

Abstract

We study the transport property of diffusion in a finite translationally invariant quantum subsystem described by a tight-binding Hamiltonian with a single energy band and interacting with its environment by a coupling in terms of correlation functions which are delta-correlated in space and time. For weak coupling, the time evolution of the subsystem density matrix is ruled by a quantum master equation of Lindblad type. Thanks to the invariance under spatial translations, we can apply the Bloch theorem to the subsystem density matrix and exactly diagonalize the time evolution superoperator to obtain the complete spectrum of its eigenvalues, which fully describe the relaxation to equilibrium. Above a critical coupling which is inversely proportional to the size of the subsystem, the spectrum at given wavenumber contains an isolated eigenvalue describing diffusion. The other eigenvalues rule the decay of the populations and quantum coherences with decay rates which are proportional to the intensity of the environmental noise. On the other hand, an analytical expression is obtained for the dispersion relation of diffusion. The diffusion coefficient is proportional to the square of the width of the energy band and inversely proportional to the intensity of the environmental noise because diffusion results from the perturbation of quantum tunneling by the environmental fluctuations in this model. Diffusion disappears below the critical coupling.