  Driven-dissipative systems have recently attracted great attention due to the
existence of novel physical phenomena with no analog in the equilibrium case.
The Keldysh path-integral theory is a powerful tool to investigate these
systems. However, it has still been challenge to study strong nonlinear effects
implemented by recent experiments, since in this case the photon number is few
and quantum fluctuations play a crucial role in dynamics of system. Here we
develop a new approach for deriving exact steady states of driven-dissipative
systems by introducing the Keldysh partition function in the Fock-state basis
and then mapping the standard saddle-point equations into KeldyshHeisenberg
equations. We take the strong Kerr nonlinear resonators with/without the
nonlinear driving as two examples to illustrate our method. It is found that in
the absence of the nonlinear driving, the exact steady state obtained does not
exhibit bistability and agree well with the complex P-representation solution.
While in the presence of the nonlinear driving, the multiphoton resonance
effects are revealed and are consistent with the qualitative analysis. Our
method provides an intuitive way to explore a variety of driven-dissipative
systems especially with strong correlations.
