Phonon dynamics in correlated quantum systems driven away from equilibrium

A general form of a many-body Hamiltonian is considered, which includes an interacting fermionic subsystem coupled to noninteracting extended fermionic and bosonic systems. We show that the exact dynamics of the extended bosonic system can be derived from the reduced density matrix of the subsystem alone, despite the fact that the latter contains information about the subsystem only. The advantage of the formalism is immediately clear: While the reduced density matrix of the subsystem is readily available, the formalism offers access to observables contained in the full density matrix, which is often difficult to obtain. As an example, we consider an extended Holstein model and study the nonequilibrium dynamics of the so-called "reaction mode" for different model parameters. The effects of the phonon frequency, the strength of the electron-phonon couplings, and the source-drain bias voltage on the phonon dynamics across the bistability are discussed.