  Gateable semiconductor quantum dots (QDs) provide a versatile platform for
analog quantum simulations of electronic many-body systems. In particular, QD
arrays offer a natural representation of the interacting $\pi$-electron system
of small hydrocarbons. Here we investigate the prospects for extending QD
simulators to encompass also the nuclear degrees of freedom. We represent the
molecular vibrational modes by single-mode microwave resonators coupled
capacitively to the QDs and study the gate-tunable energy transfer from a
voltage-biased triple quantum dot (TQD) system to a single damped resonator
mode. We determine the QD population inversions, the corresponding charge and
energy currents as well as the resonator photon number, using Lindblad master
equations and lowest-order perturbation theory within Keldysh Green function
formalism. Along the way, we discuss the merits and shortcomings of the two
methods.A central result is the interrelation of a pronounced minimum in the
charge current with a maximum in energy transfer, arising from a gate-tunable
interference effect in the molecular orbitals of the TQD electron system.
