Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom

Abstract

In this review we summarize and amplify recent investigations of coupled quantum dynamical systems with few degrees of freedom in the short-wavelength, semiclassical limit. Focusing on the correspondence between quantum and classical physics, we mathematically formulate and attempt to answer three fundamental questions. (i) How can one drive a small dynamical quantum system to behave classically? (ii) What determines the rate at which two single-particle quantum-mechanical subsystems become entangled when they interact? (iii) How does irreversibility occur in quantum systems with few degrees of freedom? These three questions are posed in the context of the quantum-classical correspondence for dynamical systems with few degrees of freedom, and we accordingly rely on two short-wavelength approximations to quantum mechanics to answer them: the trajectory-based semiclassical approach on the one hand, and random matrix theory on the other hand. We construct novel investigative procedures towards decoherence and the emergence of classicality out of quantumness in dynamical systems coupled to external degrees of freedom. In particular, we show how dynamical properties of chaotic classical systems, such as local exponential instability in phase space, also affects their quantum counterparts. For instance, it is often the case that the fidelity with which a quantum state is reconstructed after an imperfect time-reversal operation decays with the Lyapunov exponent of the corresponding classical dynamics. For related reasons, but perhaps more surprisingly, the rate at which two interacting quantum subsystems become entangled can also be governed by the subsystem's Lyapunov exponents. Our method allows us to differentiate quantum coherent effects (those related to phase interferences) from classical ones (those related to the necessarily extended envelope of quantal wavefunctions) at each stage in our investigations. This makes it clear that all occurrences of Lyapunov exponents we witness have a classical origin, although they require rather strong decoherence effects to be observed. We extensively rely on numerical experiments to illustrate our findings and briefly comment on possible extensions to more complex problems involving environments with many interacting dynamical systems, going beyond the uncoupled harmonic oscillators model of Caldeira and Leggett.

Keywords:

• decoherence
• entanglement
• quantum reversibility
• fidelity
• dynamical quantum chaos
• semiclassical methods

Acknowledgements

While working on the topics surveyed in this review, we had the pleasure and privilege to collaborate with İnanc Adagideli, Carlo Beenakker, Diego Bevilaqua, Rick Heller and Peter Silvestrov. We would like to express our gratitude to each of them. We also greatly benefited from and enjoyed sometimes lively and controversial but always interesting and fruitful discussions on these and related topics with S. Åberg, G. Casati, N. Cerrutti, D. Cohen, F. Cucchietti, D. Dalvit, J. Emerson, S. Fishman, A. Goussev, M. Gutiérrez, F. Haake, R. Jalabert, C. Jarzynski, P. Levstein, C. Lewenkopf, E. Mucciolo, H. Pastawski, J. P. Paz, K. Richter, H.-J. Stöckmann, A. Tanaka, S. Tomsovic, J. Vanicek, D. Waltner, R. Whitney, D. Wojcik, S. Wu and W. Zurek, among others. Our work on these projects was funded by the Alexander von Humboldt foundation and the National Science Foundation under grant No. DMR–0706319.