  Bistability of nonlinear resonantly driven oscillator in the presence of
external noise is analyzed by means of classical Fokker-Planck equation in
quasienergy space with account for tunneling effects and by quantum master
equation in quasienergy states representation. Two time scales responsible for
different stages of bistable system relaxation have been obtained. We found out
that the slow relaxation rate caused by fluctuation--induced transitions
between different stable states can be enhanced by several orders due to
tunneling effects. It was also revealed that tunneling between nearly
degenerate quasienergy states and resonant multiphoton transitions between the
genuine eigenstates of the nonlinear oscillator are just the similar effects.
It was demonstrated that the quasienergy states in the bistability region
corresponding to higher amplitude are squeezed. The degree of squeezing is
determined by the ratio between nonlinearity and detuning, so the uncertainty
of one quadrature can be considerably smaller than the quantum limit. It was
found out that tunneling effects can enhance the generation of output
oscillator squeezed states. It is shown that 1D Fokker--Planck equation is a
quasiclassical limit of a quantum master equation.
