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A general form of a many-body Hamiltonian is considered, which includes aninteracting fermionic sub-system coupled to non-interacting extended fermionicand bosonic systems. We show that the exact dynamics of the extended bosonicsystem can be derived from the reduced density matrix of the sub-system alone,despite the fact that the latter contains information about the sub-systemonly. The advantage of the formalism is immediately clear: While the reduceddensity matrix of the sub-system is readily available, the formalism offersaccess to observables contained in the full density matrix, which is oftendifficult to obtain. As an example, we consider an extended Holstein model andstudy the nonequilibrium dynamics of the, so called, "reaction mode" fordifferent model parameters. The effects of the phonon frequency, the strengthof the electron-phonon couplings, and the source-drain bias voltage on thephonon dynamics across the bistability are discussed.

Phonon dynamics in correlated quantum systems driven away from equilibrium