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arXiv:cond-mat/0406135 · Physical Review Letters

Quantum-to-Classical Crossover of Quasibound States in Open Quantum Systems

Henning Schomerus, J. Tworzydło

Published Oct 7, 2004• 121 citations
arXivHTMLPDF

Abstract

In the semiclassical limit of open ballistic quantum systems, we demonstrate the emergence of instantaneous decay modes guided by classical escape faster than the Ehrenfest time. The decay time of the associated quasibound states is smaller than the classical time of flight. The remaining long-lived quasibound states obey random-matrix statistics, renormalized in compliance with the recently proposed fractal Weyl law for open systems [W.T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)]. We validate our theory numerically for a model system, the open kicked rotator.

DOI
10.1103/physrevlett.93.154102

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In the semiclassical limit of open ballistic quantum systems, we demonstrate the emergence of instantaneous decay modes guided by classical escape faster than the Ehrenfest time. The decay time of the associated quasibound states is smaller than the classical time of flight. The remaining long-lived quasibound states obey random-matrix statistics, renormalized in compliance with the recently proposed fractal Weyl law for open systems [W.T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)]. We validate our theory numerically for a model system, the open kicked rotator.
States supported by chaotic open quantum systems fall into two categories: a majority showing instantaneous ballistic decay, and a set of quantum resonances of classically vanishing support in phase space. We present a theory describing these structures within a unified semiclassical framework. Emphasis is put on the quantum diffraction mechanism which introduces an element of probability and is crucial for the formation of resonances. Our main result is boundary conditions on the semiclassical propagation along system trajectories. Depending on whether the trajectory propagation time is shorter or longer than the Ehrenfest time, these conditions describe deterministic escape, or probabilistic quantum decay.
Using the supersymmetry technique, we analytically derive the recent result of Casati, Maspero, and Shepelyansky [Phys. Rev. E 56, R6233 (1997)] according to which the quantum dynamics of open chaotic systems follows the classical decay up to a new quantum relaxation time scale ${t}_{q}\ensuremath{\sim}\sqrt{{t}_{c}{t}_{H}}.$ This scale is larger than the classical escape time tc{t}_{c}tc​ but still much smaller than the Heisenberg time tH.{t}_{H}.tH​. For systems with orthogonal or unitary symmetry the quantum decay is slower than the classical one while for the symplectic case there is an intermediate regime in which the quantum decay is slightly faster.
We address the decay in open chaotic quantum systems and calculate semiclassical corrections to the classical exponential decay. We confirm random matrix predictions and, going beyond, calculate Ehrenfest time effects. To support our results we perform extensive numerical simulations. Within our approach we show that certain (previously unnoticed) pairs of interfering, correlated classical trajectories are of vital importance. They also provide the dynamical mechanism for related phenomena such as photoionization and photodissociation, for which we compute cross-section correlations. Moreover, these orbits allow us to establish a semiclassical version of the continuity equation.
We use the open kicked rotator to model the chaotic scattering in a ballistic quantum dot coupled by two point contacts to electron reservoirs. By calculating the system-size-over-wave-length dependence of the shot-noise power we study the crossover from wave to particle dynamics. Both a fully quantum-mechanical and a semiclassical calculation are presented. We find numerically in both approaches that the noise power is reduced exponentially with the ratio of Ehrenfest time and dwell time, in agreement with analytical predictions.
We show that particle transport, as characterized by the equilibrium mean square displacement, in a uniform, quantum multibaker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semiclassical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We expect that, for a large class of 1D quantum random walks similar to the quantum multibaker, a sufficient condition for diffusion in the semiclassical limit is classically chaotic dynamics in each cell. The systems described generalize known quantum random walks and may have applications for quantum computation.