Quantum magnetomechanics: Towards the ultrastrong coupling regime

In this paper we investigate a hybrid quantum system comprising a mechanical oscillator coupled via magnetic induced electromotive force to an $LC$ resonator. We derive the Lagrangian and Hamiltonian for this system and find that the interaction can be described by a charge-momentum coupling with a strength that has a strong geometry dependence. We focus our study on a mechanical resonator with a thin-film magnetic coating which interacts with a nanofabricated planar coil. We determine that the coupling rate between these two systems can enter the strong and ultrastrong coupling regimes with experimentally feasible parameters. This magnetomechanical configuration allows for a range of applications including electromechanical state transfer and weak-force sensing.