Impurity coupled to a lattice with disorder

TL;DR

This paper investigates how an impurity coupled to a disordered lattice affects the impurity's long-term occupation, revealing that bound states and localization influence the return probability, with implications for ultracold atom experiments.

Contribution

It provides exact formulas for return probability in lattice-impurity systems and links localization length to return probability in disordered regimes.

Findings

Return probability is zero in extended states unless bound states exist.

Bound state existence depends on lattice dimension.

Return probability correlates with localization length in disordered systems.

Abstract

We study the time-dependent occupation of an impurity state hybridized with a continuum of extended or localized states. Of particular interest is the return probability, which gives the long-time limit of the average impurity occupation. In the extended case, the return probability is zero unless there are bound states of the impurity and continuum. We present exact expressions for the return probability of an impurity state coupled to a lattice, and show that the existence of bound states depends on the dimension of the lattice. In a disordered lattice with localized eigenstates, the finite extent of the eigenstates results in a non-zero return probability. We investigate different parameter regimes numerically by exact diagonalization, and show that the return probability can serve as a measure of the localization length in the regime of weak hybridization and disorder. Possible experimental realizations with ultracold atoms are discussed.