photons. Using an approximate quadratic Hamiltonian model, we study the Gaussian entanglement that develops between the modes as the Josephson energy of the system is increased. We find that the modes in the system fall into a series of blocks, with bipartite entanglement generated between modes within a given block. Tripartite entanglement between modes within a given block is also widespread, though it is limited to certain ranges of the Josephson energy. The system could provide an alternative route to generating multimode microwave entanglement, an important resource in quantum technologies, without the need for ac excitation.

QED systems combining Josephson junctions with a microwave cavity and a drive achieved either through flux or voltage bias. For resonances involving 3 or more photons the system undergoes a sequence of two closely spaced dynamical transitions (the first one discontinuous and the second continuous) as the driving is increased leading to steady states that form complex periodic structures in phase space. In the vicinity of the transitions the system displays interesting bistable behaviour: we find that coherent effects can lead to surprising oscillations in the weight of the different dynamical states in the steady state of the system with increasing drive. We show that the dynamics is well-described by a simple effective rate model with transitions between states localised at different points in the phase space crystal. The oscillations in the weights of the dynamical states is reflected in corresponding oscillations in a time-scale that describes transitions between the states.