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 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 quantumclassical transition and, accordingly, a lower bound for the reservoir temperature is derived for our model.

Low-resolution measurements induced classicality

“…Secondly, the existence of entanglement in classical regime seems to have been well justified: It prevents the cat state formation and guarantees the classical notion of statistics (Liouvillian classical limit) during the entire evolution of the system. Thirdly, in spite of the simplicity of our model we believe it is able to capture the essential features of nondissipative decoherence that is expected to take place in the dynamics of most open conservative systems (see [8,11] for more detailed discussions). Finally, as the Newtonian classical limit is concerned the conditions for the quasi-determinism emergence [8] turn out to be those given by equation (12).…”

“…These results -rigorously obtained in an analytical problemshow that classical maths cannot be expected to emerge from the quantum structure in some formal limit. In addition, in the light of the results reported in [8], one may assert that decoherence cannot make the situation better for conservative systems: Although it can destruct the entanglement between the subsystems it is not capable of restraining the wave function spreading, thus not recovering Newtonian trajectories and quasideterminism.…”

“…While it is a well known fact that classical physics dramatically fails in explaining the microscopic world, we cannot say that there exists a decisive proof attesting the universality of quantum mechanics as a theory capable of accounting for all aspects of the classical world. The limit → 0 and the Ehrenfest theorem [1] have recurrently been proved not to be sufficient to guarantee the classical limit both mathematically and conceptually [2,3,4,5,6,7,8]. More modern approaches such as the environment induced decoherence (EID) program also have been claimed to present some conceptual difficulties (see [8,9] and references therein for more detailed discussions), as for instance: i-) The incapability of diffusive EID in restraining the wave function spreading and hence recovering the classical determinism and ii-) the apparently paradoxical presence of entanglement -an exclusively quantum resourcein semiclassical regimes.…”

“…Recently, approaches based on observational aspects have been considered in order to address the quantum-to-classical transition in closed spin systems [10] as well as in bosonic systems under diffusive decoherence [8]. As has been suggested by these works, the apparent Newtonian motion of a falling ball in the vicinity of the Earth is nothing more than an effect of both the low resolution power of our experimental apparatus and the short time scales usually involved in the phenomenon.…”

“…In this case, another scale is required which will inevitably ignore the fine quantum details of the macroscopic body dynamics. In this case, a kind of deterministic behavior ("quasi-deterministic" [8], to be precise) is observed to exist. According to this view, classical behavior only exists as an approximated notion derived from low-resolution measurements.…”