Correspondence principle for the diffusive dynamics of a quartic oscillator: deterministic aspects and the role of temperature

Abstract

The correspondence principle is investigated in the framework of deterministic predictions for individual systems. Exact analytical results are obtained for the quantum and the Liouvillian dynamics of a nonlinear oscillator coupled to a phase-damping reservoir at a finite temperature. In this context, the time of critical wave function spreading - the Ehrenfest time - emerges as the characteristic time scale within which the concept of deterministic behavior is admissible in physics. A scenario of "quasi-determinism" may be then defined within which the motion is experimentally indistinguishable from the truly deterministic motion of Newtonian mechanics. Beyond this time scale, predictions for individual systems can be given only statistically and, in this case, it is shown that diffusive decoherence is indeed a necessary ingredient to establish the quantum-classical correspondence. Moreover, the high-temperature regime is shown to be an additional condition for the quantum-classical transition and, accordingly, a lower bound for the reservoir temperature is derived for our model.