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arXiv:cond-mat/0206143 · Physical review. B, Condensed matter

Time-resolved dynamics of electron wave packets in chaotic and regular quantum billiards with leads

Igor V. Zozoulenko, Torbjörn Blomquist

Published Feb 28, 2003• 13 citations
arXivHTMLPDF

Abstract

We perform numerical studies of wave packet propagation through open quantum billiards whose classical counterparts exhibit regular and chaotic dynamics. We show that for $t\ensuremath{\lesssim}{\ensuremath{\tau}}_{H}({\ensuremath{\tau}}_{H}$ being the Heisenberg time), the features in the transmitted and reflected currents are directly related to specific classical trajectories connecting the billiard leads. When $t\ensuremath{\gtrsim}{\ensuremath{\tau}}_{H},$ the calculated quantum-mechanical current starts to deviate from its classical counterpart, with the decay rate obeying a power law that depends on the number of decay channels. In a striking contrast to the classical escape from chaotic and regular systems (exponentially fast ${e}^{\ensuremath{-}\ensuremath{\gamma}t}$ for the former versus power-law ${t}^{\ensuremath{-}\ensuremath{\xi}}$ for the latter), the asymptotic decay of the corresponding quantum systems does not show a qualitative difference.

DOI
10.1103/physrevb.67.085320

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Using the minimum uncertainty state of quantum integrable system Ho as initial state, the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics. Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some pairs of neighboring components of the wave packet, the deterministic dynamical evolution becomes very complicated and appears to be chaotic. It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors. It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.
Using the minimum uncertainty state of quantum integrable system as initial state, the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics. Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some pairs of neighboring components of the wave packet, the deterministic dynamical evolution becomes very complicated and appears to be chaotic. It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors. It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.
We report on the numerical simulation of the double-slit experiment, where the initial wave packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.
We investigate the sensitivity of the time evolution of semiclassical wave packets in two-dimensional chaotic billiards with respect to local perturbations of their boundaries. For this purpose, we address, analytically and numerically, the time decay of the Loschmidt echo (LE). We find the LE to decay exponentially in time, with the rate equal to the classical escape rate from an open billiard obtained from the original one by removing the perturbation-affected region of its boundary. Finally, we propose a principal scheme for the experimental observation of the LE decay.
We discuss the stability of quantum motion under system's perturbations in the light of the corresponding classical behavior. In particular we focus our attention on the so called "fidelity" or Loschmidt echo, its relation with the decay of correlations, and discuss the quantum-classical correspondence. We then report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.
The dynamics of a particle in square and circular billiards is studied within the framework of Bohm's quantum mechanics. While conventional quantum mechanics predicts that the system shows no indication of chaotic behavior for these geometries from either the eigenvalue spectra distribution or the structure of the eigenfunctions, we find that in Bohm's quantum mechanics these systems exhibit both regular and chaotic behavior, depending on the form of the initial wave packet and on the particle's initial position.