Adiabatic quantum computing is an analog quantum computing scheme with various applications in solving optimization problems. In the parity picture of quantum optimization, the problemis encoded in local fields that act on qubits which are connected via local 4-body terms. We present an implementation of a parity annealer with Transmon qubits with a specifically tailored Ising interaction from Josephson ring modulators.

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 consider a superconducting coplanar waveguide resonator where the central conductor is interrupted by a series of uniformly spaced Josephson junctions. The device forms an extendedmedium that is optically nonlinear on the single photon level with normal modes that inherit the full nonlinearity of the junctions but are nonetheless accessible via the resonator ports. For specific plasma frequencies of the junctions a set of normal modes clusters in a narrow band and eventually become entirely degenerate. Upon increasing the intensity of a red detuned drive on these modes, we observe a sharp and synchronized switching from low occupation quantum states to high occupation classical fields, accompanied by a pronounced jump from low to high output intensity.

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.

We introduce a circuit quantum electrodynamical setup for a quantum single
photon transistor. In our approach single photons propagate in two open
transmission lines that are coupledvia two interacting transmon qubits. The
interaction is such that photons are not exchanged between the two transmission
lines but a photon in one line can completely block respectively enable the
propagation of photons in the other line. High on-off ratios can be achieved
for feasible experimental parameters. Our approach is inherently scalable as
all photon pulses can have the same pulse shape and carrier frequency such that
output signals of one transistor can be input signals for a consecutive
transistor.

We present the Josephson junction intersected superconducting transmission
line resonator. In contrast to the Josephson parametric amplifier, Josephson
bifurcation amplifier and Josephsonparametric converter we consider the regime
of few microwave photons. We review the derivation of eigenmode frequencies and
zero point fluctuations of the nonlinear transmission line resonator and the
derivation of the eigenmode Kerr nonlinearities. Remarkably these
nonlinearities can reach values comparable to Transmon qubits rendering the
device ideal for accessing the strongly correlated regime. This is particularly
interesting for investigation of quantum many-body dynamics of interacting
particles under the influence of drive and dissipation. We provide current
profiles for the device modes and investigate the coupling between resonators
in a network of nonlinear transmission line resonators.

We explore photon coincidence counting statistics in the ultrastrong-coupling
regime where the atom-cavity coupling rate becomes comparable to the cavity
resonance frequency. In thisregime usual normal order correlation functions
fail to describe the output photon statistics. By expressing the electric-field
operator in the cavity-emitter dressed basis we are able to propose correlation
functions that are valid for arbitrary degrees of light-matter interaction. Our
results show that the standard photon blockade scenario is significantly
modified for ultrastrong coupling. We observe parametric processes even for
two-level emitters and temporal oscillations of intensity correlation functions
at a frequency given by the ultrastrong photon emitter coupling. These effects
can be traced back to the presence of two-photon cascade decays induced by
counter-rotating interaction terms.

We investigate a network of coupled superconducting transmission line
resonators, each of them made nonlinear with a capacitively shunted Josephson
junction coupling to the odd fluxmodes of the resonator. The resulting
eigenmode spectrum shows anticrossings between the plasma mode of the shunted
junction and the odd resonator modes. Notably, we find that the combined device
can inherit the complete nonlinearity of the junction, allowing for a
description as a harmonic oscillator with a Kerr nonlinearity. Using a dc SQUID
instead of a single junction, the nonlinearity can be tuned between 10 kHz and
4 MHz while maintaining resonance frequencies of a few gigahertz for realistic
device parameters. An array of such nonlinear resonators can be considered a
scalable superconducting quantum simulator for a Bose-Hubbard Hamiltonian. The
device would be capable of accessing the strongly correlated regime and be
particularly well suited for investigating quantum many-body dynamics of
interacting particles under the influence of drive and dissipation.

We investigate a chain of superconducting stripline resonators, each
interacting with a transmon qubit, that are capacitively coupled in a row. We
show that the dynamics of this systemcan be described by a Bose-Hubbard
Hamiltonian with attractive interactions for polaritons, superpositions of
photons and qubit excitations. This setup we envisage constitutes one of the
first platforms where all technological components that are needed to
experimentally study chains of strongly interacting polaritons have already
been realized. By driving the first stripline resonator with a microwave source
and detecting the output field of the last stripline resonator one can
spectroscopically probe properties of the system in the driven dissipative
regime. We calculate the stationary polariton density and density-density
correlations $g^{(2)}$ for the last cavity which can be measured via the output
field. Our results display a transition from a coherent to a quantum field as
the ratio of on site interactions to driving strength is increased.