  We analyze the quantum evolution of a weakly nonlinear resonator due to a
classical near-resonant drive and damping. The resonator nonlinearity leads to
squeezing and heating of the resonator state. Using a hybrid
phase-space--Fock-space representation for the resonator state within the
Gaussian approximation, we derive evolution equations for the four parameters
characterizing the Gaussian state. Numerical solution of these four ordinary
differential equations is much simpler and faster than simulation of the full
density matrix evolution, while providing good accuracy for the system analysis
during transients and in the steady state. We show that steady-state squeezing
of the resonator state is limited by 3 dB; however, this limit can be exceeded
during transients.
