Quantum critical temperature of a modulated oscillator
  We show that the rate of switching between the vibrational states of a modulated nonlinear oscillator is characterized by a quantum critical temperature $T_c\propto\hbar^2$. The rate is independent of $T$ for $T<T_c$. Above $T_c$ there emerges a quantum crossover region where the slope of the logarithm of the distribution over the oscillator states displays a kink and the switching rate has the Arrhenius form with the activation energy independent of the modulation. The results demonstrate the limitations of the real-time instanton theory of switching in systems lacking detailed balance. 
Lingzhen Guo, Vittorio Peano, M. Marthaler, and M. I. Dykman
cond-mat.mes-hall cond-mat.stat-mech quant-ph