  We study switching between period-two states of an underdamped quantum
oscillator modulated at nearly twice its natural frequency. For all
temperatures and parameter values switching occurs via quantum activation: it
is determined by diffusion over oscillator quasienergy, provided the relaxation
rate exceeds the rate of interstate tunneling. The diffusion has quantum origin
and accompanies relaxation to the stable state. We find the semiclassical
distribution over quasienergy. For T=0, where the system has detailed balance,
this distribution differs from the distribution for $T\to 0$; the T=0
distribution is also destroyed by small dephasing of the oscillator. The
characteristic quantum activation energy of switching displays a typical
dependence on temperature and scaling behavior near the bifurcation point where
period doubling occurs.
