Anomalous diffusion in the Long-Range Haken-Strobl-Reineker model

TL;DR

This paper studies how excitons propagate in a lattice with long-range hopping and dephasing, revealing a transition from superdiffusive to mixed Gaussian-Lévy behavior at a critical decay exponent, with implications for experimental systems.

Contribution

It analytically characterizes anomalous exciton diffusion in a long-range Haken-Strobl-Reineker model, identifying a critical decay exponent and describing the resulting spatial distributions.

Findings

Superdiffusive Lévy stable distribution for lpha  _{ m cr}

Mixed Gaussian profile with algebraic tails for lpha > _{ m cr}

Faster thermalization with longer hopping range

Abstract

We analyze the propagation of excitons in a $d$-dimensional lattice with power-law hopping $\propto 1/r^\alpha$ in the presence of dephasing, described by a generalized Haken-Strobl-Reineker model. We show that in the strong dephasing (quantum Zeno) regime the dynamics is described by a classical master equation for an exclusion process with long jumps. In this limit, we analytically compute the spatial distribution, whose shape changes at a critical value of the decay exponent $\alpha_{\rm cr} = (d+2)/2$. The exciton always diffuses anomalously: a superdiffusive motion is associated to a L\'evy stable distribution with long-range algebraic tails for $\alpha\leq\alpha_{\rm cr}$, while for $\alpha > \alpha_{\rm cr}$ the distribution corresponds to a surprising mixed Gaussian profile with long-range algebraic tails, leading to the coexistence of short-range diffusion and long-range L\'evy-flights. In the many-exciton case, we demonstrate that, starting from a domain-wall exciton profile, algebraic tails appear in the distributions for any $\alpha$, which affects thermalization: the longer the hopping range, the faster equilibrium is reached. Our results are directly relevant to experiments with cold trapped ions, Rydberg atoms and supramolecular dye aggregates. They provide a way to realize an exclusion process with long jumps experimentally.