Sub-Ohmic to super-Ohmic crossover behavior in nonequilibrium quantum systems with electron-phonon interactions

The transition from weakly damped coherent motion to localization in the context of the spin-boson model has been the subject of numerous studies with distinct behavior depending on the form of the phonon-bath spectral density, $J\left(\omega\right)\propto\omega^{s}$. Sub-Ohmic ($s<1$) and Ohmic ($s=1$) spectral densities show a clear localization transition at zero temperature and zero bias, while for super-Ohmic ($s>1$) spectral densities this transition disappears. In this work, we consider the influence of the phonon-bath spectral density on the \emph{nonequilibrium} dynamics of a quantum dot with electron-phonon interactions described by the extended Holstein model. Using the reduced density matrix formalism combined with the multi-layer multiconfiguration time-dependent Hartree approach, we investigate the dynamic response, the time scales for relaxation, as well as the existence of multiple long-lived solutions as the system-bath coupling changes from the sub- to the super-Ohmic cases. Bistability is shown to diminish for increasing powers of $s$ similar to the spin-boson case. However, the physical mechanism and the dependence on the model parameters such as the typical bath frequency $\omega_{c}$ and the polaron shift $\lambda$ are rather distinct.