A quantum effect in the classical limit: nonequilibrium tunneling in the Duffing oscillator

TL;DR

This paper investigates nonequilibrium tunneling in the classical limit of the quantum Duffing oscillator, revealing non-unique crossover points and the importance of the ratio of quantum to thermal energy, with numerical and semi-analytical evidence.

Contribution

It demonstrates that the crossover point in the quantum Duffing oscillator is non-unique and depends on the order of limits, highlighting a conceptual difference from equilibrium tunneling.

Findings

Crossover point is non-unique and order-dependent.

The ratio of quantum to thermal energy remains key in the classical limit.

Numerical and semi-analytical methods confirm the results.

Abstract

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.