# Circuit Quantum Electrodynamics of Granular Aluminum Resonators

The introduction of crystalline defects or dopants can give rise to so-called „dirty superconductors“, characterized by reduced coherence length and quasiparticle mean free
path. In particular, granular superconductors such as Granular Aluminum (GrAl), consisting of remarkably uniform grains connected by Josephson contacts have attracted interest since the sixties thanks to their rich phase diagram and practical advantages, like increased critical temperature, critical field, and kinetic inductance. Here we report the measurement and modeling of circuit quantum electrodynamics properties of GrAl microwave resonators in a wide frequency range, up to the spectral superconducting gap. Interestingly, we observe self-Kerr coefficients ranging from 10−2 Hz to 105 Hz, within an order of magnitude from analytic calculations based on GrAl microstructure. This amenable nonlinearity, combined with the relatively high quality factors in the 105 range, open new avenues for applications in quantum information processing and kinetic inductance detectors.

# Double resonance response of a superconducting quantum metamaterial: manifestation of non-classical states of photons

We report a theoretical study of ac response of superconducting quantum metamaterials (SQMs), i.e. an array of qubits (two-levels system) embedded in the low-dissipative resonator.
By making use of a particular example of SQM, namely the array of charge qubits capacitively coupled to the resonator, we obtain a second-order phase transition between an incoherent (the high-temperature phase) and coherent (the low-temperatures phase) states of photons. This phase transition in many aspects resembles the paramagnetic-ferromagnetic phase transition. The critical temperature of the phase transition, T⋆, is determined by the energy splitting of two-level systems δ, number of qubits in the array N, and the strength of the interaction η between qubits and photons in the cavity. We obtain that the photon states manifest themselves by resonant drops in the frequency dependent transmission D(ω) of electromagnetic waves propagating through a transmission line weakly coupled to the SQM. At high temperatures the D(ω) displays a single resonant drop, and at low temperatures a peculiar \emph{double resonance response} has to be observed. The physical origin of such a resonant splitting is the quantum oscillations between two coherent states of photons of different polarizations.

# 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.