equations in the generalised P-representation we investigate the analytical solutions of two fundamental models. First, we solve for the steady-state response of a linear cavity that is coupled to an approximate transmon qubit and use this solution to study both the weak and strong driving regimes, using analytical expressions for the moments of both cavity and transmon fields, along with the Husimi Q-function for the transmon. Second, we revisit exact solutions of quantum Duffing oscillator which is driven both coherently and parametrically while also experiencing decoherence by the loss of single and pairs of photons. We use this solution to discuss both stabilisation of Schroedinger cat states and the generation of squeezed states in parametric amplifiers, in addition to studying the Q-functions of the different phases of the quantum system. The field of superconducting circuits, with its strong nonlinearities and couplings, has provided access to a new parameter regime in which returning to these exact quantum optics methods can provide valuable insights.

decoupling yields a good approximation to the dynamics of the system. We find that the stationary states are squeezed vacuum states of the non-interacting system which are enhanced by the Q-factor of the cavity. We show that this effective interaction could be realised with a cascaded superconducting cavity-qubit system in the strong dispersive regime in a setup similar to recent experiments. The qubit non-linearity, therefore, does not significantly influence the highly squeezed intracavity microwave field but, for a range of parameters, enables quantum state tomography of the cavity.