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Numerically "exact" methods addressing the dynamics of coupledelectron--phonon systems have been intensively developed. Nevertheless, thecorresponding results for the electron mobility $\mu_\mathrm{dc}$ are scarce,even for the one-dimensional (1d) Holstein model. Building on our recentprogress on single-particle properties, here, we develop the momentum-spacehierarchical equations of motion (HEOM) method to evaluate real-timetwo-particle correlation functions of the 1d Holstein model at finitetemperature. We compute numerically "exact" dynamics of the current--currentcorrelation function up to real times sufficiently long to capture theelectron's diffusive motion and provide reliable results for $\mu_\mathrm{dc}$in a wide range of model parameters. In contrast to the smoothballistic-to-diffusive crossover in the weak-coupling regime, we observe atemporally limited slow-down of the electron on intermediate time scalesalready in the intermediate-coupling regime, which translates to afinite-frequency peak in the optical response. Our momentum-space formulationlowers the numerical effort with respect to existing HEOM-methodimplementations, while we remove the numerical instabilities inherent to theundamped-mode HEOM by devising an appropriate hierarchy closing scheme. Still,our HEOM remains unstable at too low temperatures, for too strongelectron--phonon coupling, and for too fast phonons.

Holstein polaron transport from numerically "exact" real-time quantum dynamics simulations