  We study exact solutions of the steady state behaviour of several non-linear
open quantum systems which can be applied to the field 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 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.
