  For suitable parameters, the classical Duffing oscillator has a known
bistability in its stationary states, with low- and high-amplitude branches. As
expected from the analogy with a particle in a double-well potential,
transitions between these states become possible either at finite temperature,
or in the quantum regime due to tunneling. In this analogy, besides local
stability, one can also discuss global stability by comparing the two potential
minima. For the Duffing oscillator, the stationary states emerge dynamically so
that a priori, a potential-minimum criterion for them does not exist. However,
global stability is still relevant, and definable as the state containing the
majority population for long times, low temperature, and close to the classical
limit. Further, the crossover point is the parameter value at which global
stability abruptly changes from one state to the other.
  For the double-well model, the crossover point is defined by
potential-minimum degeneracy. Given that this analogy is so effective in other
respects, it is thus striking that for the Duffing oscillator, the crossover
point turns out to be non-unique. Rather, none of the three aforementioned
limits commute with each other, and the limiting behaviour depends on the order
in which they are taken. More generally, as both $\hbar\To0$ and $T\To0$, the
ratio $\hbar\omega_0/k_\mathrm{B}T$ continues to be a key parameter and can
have any nonnegative value. This points to an apparent conceptual difference
between equilibrium and nonequilibrium tunneling. We present numerical evidence
by studying the pertinent quantum master equation in the photon-number basis.
Independent verification and some further understanding are obtained using a
semi-analytical approach in the coherent-state representation.
