Generation of non-classical photon states in superconducting quantum metamaterials

We report a theoretical study of diverse non-classical photon states that can be realized in superconducting quantum metamaterials. As a particular example of superconducting quantum metamaterials, an array of SQUIDs incorporated in a low-dissipative transmission line (resonant cavity) will be studied. This system will be modeled as a set of two-level systems (qubits) strongly interacting with resonant cavity photons. We predict and analyze a second-order phase transition between incoherent (the high-temperature phase) and coherent (the low-temperatures phase) states of photons. In the equilibrium state the partition function Z of the electromagnetic field (EF) in the cavity is determined by the effective action S-eff{P(tau)} that, in turn, depends on the imaginary time dependent momentum of the photon field P(tau). We show that the order parameter of this phase transition is the P-0(tau) minimizing the effective action of the whole system. In the incoherent state, the order parameter P-0(tau) = 0 but at low temperatures we obtain various coherent states characterized by non-zero values of P-0(tau). This phase transition in many aspects resembles the Peierls metal-insulator and the metal-superconductor phase transitions. The critical temperature of such a phase transition T* is determined by the energy splitting of two-level systems 1, the number of SQUIDs in the array N and the strength of the interaction eta between SQUIDs and photons in the cavity.