  A Superconducting Quantum Interference Device (SQUID) inserted in a
superconducting waveguide resonator imposes current and voltage boundary
conditions that makes it suitable as a tuning element for the resonator modes.
If such a SQUID element is subject to a periodically varying magnetic flux, the
resonator modes acquire frequency side bands. In this work we calculate the
multi-frequency eigenmodes of resonators coupled to periodically driven SQUIDs
and we use the Lagrange formalism to propose a theory for their quantization.
The elementary excitations of a multi-frequency mode can couple resonantly to
physical systems with different transition frequencies and this makes the
resonator an efficient quantum bus for state transfer and coherent quantum
operations in hybrid quantum systems. As an example of the application of our
multi-frequency modes, we determine their coupling to transmon qubits with
different frequencies and we present a bi-chromatic scheme for entanglement and
gate operations.
