Several small Josephson junctions in a resonant cavity: Deviation from the Dicke model

We have studied quantum mechanically a system of several small identical Josephson junctions in a lossless single-mode cavity for different initial states, under conditions such that the system is at resonance. This system is analogous to a collection of identical atoms in a cavity, which is described under appropriate conditions by the Dicke model. We find that our system can be well approximated by a reduced Hamiltonian consisting of two levels per junction. The reduced Hamiltonian is similar to the Dicke Hamiltonian, but contains an additional term resembling a dipole-dipole interaction between the junctions. This extra term can be understood as a natural consequence of degenerate second-order (Lowdin) perturbation theory. For typical, physically reasonable values of the junction-cavity coupling, we find that this perturbation treatment is an adequate way to include the junction energy levels beyond the lowest two. As in the Dicke model, we find that, when N junctions are present in the cavity, the junction-cavity interaction is enhanced by rootN, with a corresponding decrease in the Rabi oscillation period. We find that this enhancement survives even if the junctions differ slightly from one another, as expected in a realistic system. Since coherence effects thus reduce the Rabi period, it may become smaller than the decoherence time due to dissipation, making these oscillations observable.