Semiclassics of the chaotic quantum-classical transition

We elucidate the basic physical mechanisms responsible for the quantum-classical transition in one-dimensional, bounded chaotic systems subject to unconditioned environmental interactions. We show that such a transition occurs due to the dual role of noise in regularizing the semiclassical Wigner function and averaging over fine structures in classical phase space. The results are interpreted in the context of applying recent advances in the theory of measurement and open systems to the semiclassical quantum regime. We use these methods to show how a local semiclassical picture is stabilized and can then be approximated by a classical distribution at later times. The general results are demonstrated explicitly via high-resolution numerical simulations of the quantum master equation for a chaotic Duffing oscillator.

Figure 1

Phase space rendering of the Wigner function for the Duffing system at time $t=314$ periods of driving. The nonlocal interference is significant and cannot be associated with specific classical structures. This plot is taken at a relatively small $D$ value $\left({10}^{-4}\right)$ for resolution purposes. The value of $\hslash$ is set equal to 1 in order to clearly demonstrate this effect.

Figure 2

Sectional cuts of Wigner functions (dashed lines) and classical distributions (solid lines) for a driven Duffing oscillator, after 149 drive periods, taken at $p=0$ for (a) $D={10}^{-5}$, (b) $D={10}^{-3}$, and (c) $D={10}^{-2}$. Parameter values are as stated in the text; the height is specified in scaled units.

Figure 3

(Color online) Cross-sectional slices of the Wigner function (solid) and classical distribution function (dashed) taken in phase space for $p=0$ and $D={10}^{-3}$. The upper plot is taken at $t=10$ and the lower plot is taken at $t=30$.

Figure 4

(Color online) Cross-sectional slices of the Wigner function (solid) and classical distribution function (dashed) taken in phase space for $p=0$ and $D={10}^{-2}$. The upper plot is taken at $t=8$ and the lower plot is taken at $t=20$.

Figure 5

(Color online) Cross-sectional slices of the Wigner function (solid) and classical distribution function (dashed) following Fig. 4. The upper panel has $D=0.1$ and the lower panel has $D=1$. Both snapshots are taken at $t=20$.

Figure 6

(Color online) Phase space rendering of the Wigner function at time $t=149$ periods of driving. The early-time part of the unstable manifold associated with the noise-free dynamics is shown in white (yellow online) (see text for discussion). The value of $D={10}^{-3}$ is not sufficient to wipe out all the quantum interference which, as expected, is most prominent near sharp turns in the manifold.