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We present a novel hierarchical quantum master equation (HQME) approach whichprovides a numerically exact description of nonequilibrium charge transport innanosystems with electronic-vibrational coupling. In contrast to previous work[Phys. Rev. B $\bf{94}$, 201407 (2016)], the active vibrational degrees offreedom are treated in the reservoir subspace and are integrated out. Thisfacilitates applications to systems with very high excitation levels, forexample due to current-induced heating, while properties of the vibrationaldegrees of freedom, such as the excitation level and other moments of thevibrational distribution function, are still accessible. The method is appliedto a generic model of a nanosystem, which comprises a single electronic levelthat is coupled to fermionic leads and a vibrational degree of freedom.Converged results are obtained in a broad spectrum of parameters, ranging fromthe nonadiabatic to the adiabatic transport regime. We specifically investigatethe phenomenon of vibrational instability, that is, the increase ofcurrent-induced vibrational excitation for decreasing electronic-vibrationalcoupling. The novel HQME approach allows us to analyze the influence of levelbroadening due to both molecule-lead coupling and thermal effects. Resultsobtained for the first two moments suggest that the vibrational excitation isalways described by a geometric distribution in the weak electronic-vibrationalcoupling limit.

physical review. b./physical review. b

Hierarchical quantum master equation approach to electronic-vibrational  coupling in nonequilibrium transport through nanosystems: Reservoir  formulation and application to vibrational instabilities