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.

Emergence of chaos in quantum systems far from the classical limit.

The dynamical status of isolated quantum systems is unclear as conventional measures fail to detect chaos in such systems. However, when quantum systems are subjected to observation--as all experimental systems must be--their dynamics is no longer linear and, in the appropriate limit(s), the evolution of expectation values, conditioned on the observations, closely approaches the behavior of classical trajectories. Here we show, by analyzing a specific example, that microscopic continuously observed quantum systems, even far from any classical limit, can have a positive Lyapunov exponent, and thus be truly chaotic.

The semiclassical regime of the chaotic quantum-classical transition.

An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.

Chaos and quantum mechanics.

The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.

Quantum-classical transition in nonlinear dynamical systems.

Viewed as approximations to quantum mechanics, classical evolutions can violate the positive semidefiniteness of the density matrix. The nature of the violation suggests a classification of dynamical systems based on classical-quantum correspondence; we show that this can be used to identify when environmental interaction (decoherence) will be unsuccessful in inducing the quantum-classical transition. In particular, the late-time Wigner function can become positive without any corresponding approach to classical dynamics. In the light of these results, we emphasize key issues relevant for experiments studying the quantum-classical transition.