Abstract
We study exact solutions of the steady state behaviour of several non-linear open quantum systems which can be applied to the eld of circuit quantum electrodynamics. Using Fokker-Planck 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 elds, along with the Husimi Q-function for the transmon. Second, we revisit exact solutions of quantum Du ng 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 Schrodinger cat states and the generation of squeezed states in parametric ampli ers, in addition to studying the Q-functions of the di erent phases of the quantum system. The eld of superconducting circuits, with its strong nonlinearities and couplings, has provided access to new parameter regimes in which returning to these exact quantum optics methods can provide valuable insights.