Superconducting circuits and trapped ions are promising architectures for quantum information processing. However, the natural frequencies for controlling these systems -- radio frequency ion control and microwave domain superconducting qubit control
-- make direct Hamiltonian interactions between them weak. In this paper we describe a technique for coupling a trapped ions motion to the fundamental mode of a superconducting circuit, by applying to the circuit a carefully modulated external magnetic flux. In conjunction with a non-linear element (Josephson junction), this gives the circuit an effective time-dependent inductance. We then show how to tune the external flux to generate a resonant coupling between the circuit and ions motional mode, and discuss the limitations of this approach compared to using a time-dependent capacitance.
