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-levels systems (qubits) strongly interacting with resonant cavity photons. We predict and analyze {a second(first)-order phase transition} between an incoherent (the high-temperature phase) and coherent (the low-temperatures phase) states of photons. In 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 imaginary-time dependent momentum of photon field $P(\tau)$. We show that the order parameter of this phase transition is the $P_{0}(\tau)$ minimizing the effective action of a 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 phase transition $T^\star$ is determined by the energy splitting of two-level systems $\Delta$, a number of SQUIDs in the array $N$, and the strength of the interaction $\eta$ between SQUIDs and photons in cavity.