We revisit the issue of thermally assisted quantum annealing by a detailed study of the dissipative Landau-Zener problem in presence of a Caldeira-Leggett bath of harmonic oscillators,using both a weak-coupling quantum master equation and a quasi-adiabatic path-integral approach. Building on the known zero-temperature exact results (Wubs et al., PRL 97, 200404 (2006)), we show that a finite temperature bath can have a beneficial effect on the ground-state probability only if it couples also to a spin-direction that is transverse with respect to the driving field, while no improvement is obtained for the more commonly studied purely longitudinal coupling. In particular, we also highlight that, for a transverse coupling, raising the bath temperature further improves the ground-state probability in the fast-driving regime. We discuss the relevance of these findings for the current quantum-annealing flux qubit chips.

We study the dynamics of Josephson Parametric Amplifier (JPA) coupled to a mechanical oscillator, as realised with a dc Superconducting Quantum Interference Device (SQUID) with an embeddedmovable arm. We analyse this system in the regime when the frequency of the mechanical oscillator is comparable in magnitude with the plasma oscillation of the SQUID. When the nano-mechanical resonator is driven, it strongly affects the dynamics of the JPA. We show that this coupling can considerably modify the dynamics of JPA and induce its multistability rather than common bistability. This analysis is relevant if one considers a JPA for detection of mechanical motion.

We study the properties of an array of QED-cavities coupled by nonlinear elements in the presence of photon leakage and driven by a coherent source. The main effect of the nonlinearcouplings is to provide an effective cross-Kerr interaction between nearest-neighbor cavities. Additionally correlated photon hopping between neighboring cavities arises. We provide a detailed mean-field analysis of the steady-state phase diagram as a function of the system parameters, the leakage and the external driving, and show the emergence of a number of different quantum phases. A photon crystal associated to a spatial modulation of the photon blockade appears. The steady state can also display oscillating behavior and bi-stability. In some regions the crystalline ordering may coexist with the oscillating behavior. Furthermore we study the effect of short-range quantum fluctuations by employing a cluster mean-field analysis. Focusing on the corrections to the photon crystal boundaries, we show that, apart for some quantitative differences, the cluster mean field supports the findings of the simple single-site analysis. In the last part of the paper we concentrate on the possibility to build up the class of arrays introduced here, by means of superconducting circuits of existing technology. We consider a realistic choice of the parameters for this specific implementation and discuss some properties of the steady-state phase diagram.

We introduce and study the properties of an array of QED cavities coupled by
non-linear elements, in the presence of photon leakage and driven by a coherent
source. The non-linearcouplings lead to photon hopping and to nearest-neighbor
Kerr terms. By tuning the system parameters, the steady state of the array can
exhibit a photon crystal associated to a periodic modulation of the photon
blockade. In some cases the crystalline ordering may coexist with phase
synchronisation. The class of cavity arrays we consider can be built with
superconducting circuits of existing technology.