Reduction of modal length using Josephson junction array confined cavity

We propose an architecture in which the coupling between a cavity and superconducting qubits is enhanced by reducing the cavity modal length. The cavity is connected to outside transmission lines at the both ends by two short Josephson junction arrays. The latter plays a role of distributed reflector, which can provide an arbitrary negative phase to the reflected wave below its photonic bandgap. With this phase contribution from the Josephson junction arrays, we show, in the context of Fabry-Perot resonance, that the modal length can be 2 orders shorter than the wavelength.