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

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 ${\alpha }_{\mathrm{cr}}=(d+2)/2$. The exciton always diffuses anomalously: a superdiffusive motion is associated to a Lévy stable distribution with long-range algebraic tails for $\alpha \le {\alpha }_{\mathrm{cr}}$, while for $\alpha >{\alpha }_{\mathrm{cr}}$ 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 Lévy-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.

5 pages, 2 figures

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• Numerically efficient unitary evolution for Hamiltonians beyond nearest-neighbors