Equivalence of dynamics of disordered quantum ensembles and semi-infinite lattices

Abstract We develop a formalism for mapping the exact dynamics of an ensemble of disordered quantum systems onto the dynamics of a single particle propagating along a semi-infinite lattice, with interactions determined by the probability distribution of disorder realisations of the original heterogeneous quantum ensemble. This provides a geometric interpretation of the loss of coherence when averaging over the ensemble and allows computation of the observables of the ensemble in a single simulation. Conversely, the reverse map implies lattice particle dynamics to be equivalent to the dynamics of an ensemble of disordered quantum systems. The potential of the equivalence is demonstrated by computing the exact disorder-averaged density matrix dynamics for a qubit and a dimer ensemble, and interpreting dynamics of a tight-binding chain with constant and staggered hoppings in terms of an ensemble of disorder realisations.