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arXiv:1006.0345 · Chaos An Interdisciplinary Journal of Nonlinear Science

Chaotic dephasing in a double-slit scattering experiment

Zoran Levnajić, Tomaž Prosen

Published Nov 17, 2010• 2 citations
arXivHTMLPDF

Abstract

We design a computational experiment in which a quantum particle tunnels into a billiard of variable shape and scatters out of it through a double-slit opening on the billiard's base. The interference patterns produced by the scattered probability currents for a range of energies are investigated in relation to the billiard's geometry which is connected to its classical integrability. Four billiards with hierarchical integrability levels are considered: integrable, pseudointegrable, weak-mixing, and strongly chaotic. In agreement with the earlier result by Casati and Prosen [Phys. Rev. A 72, 032111 (2005)], we find the billiard's integrability to have a crucial influence on the properties of the interference patterns. In the integrable case, most experiment outcomes are found to be consistent with the constructive interference occurring in the usual double-slit experiment. In contrast to this, nonintegrable billiards typically display asymmetric interference patterns of smaller visibility characterized by weakly correlated wave function values at the two slits. Our findings indicate an intrinsic connection between the classical integrability and the quantum dephasing, which is responsible for the destruction of interference.

DOI
10.1063/1.3509768

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We design a computational experiment in which a quantum particle tunnels into a billiard of variable shape and scatters out of it through a double-slit opening on the billiard's base. The interference patterns produced by the scattered probability currents for a range of energies are investigated in relation to the billiard's geometry which is connected to its classical integrability. Four billiards with hierarchical integrability levels are considered: integrable, pseudointegrable, weak-mixing, and strongly chaotic. In agreement with the earlier result by Casati and Prosen [Phys. Rev. A 72, 032111 (2005)], we find the billiard's integrability to have a crucial influence on the properties of the interference patterns. In the integrable case, most experiment outcomes are found to be consistent with the constructive interference occurring in the usual double-slit experiment. In contrast to this, nonintegrable billiards typically display asymmetric interference patterns of smaller visibility characterized by weakly correlated wave function values at the two slits. Our findings indicate an intrinsic connection between the classical integrability and the quantum dephasing, which is responsible for the destruction of interference.
We investigate the influence that classical dynamics has on interference patterns in coherence experiments. We calculate the time-integrated probability current through an absorbing screen and the conductance through a doubly connected ballistic cavity, both in an Aharonov-Bohm geometry with forward scattering only. We show how interference fringes in the probability current generically disappear in the case of a chaotic system with small openings, and how they may persist in the case of an integrable cavity. Simultaneously, the typical, sample dependent amplitude of the flux-sensitive g(phi) part of the conductance survives in all cases, and becomes universal in the case of a chaotic cavity. In the presence of dephasing by fluctuations of the electric potential in one arm of the Aharonov-Bohm loop, we find an exponential damping of the flux-dependent part of the conductance, g(phi) proportional exp[-tauL/tauphi, in term of the traversal time tauL through the arm and the dephasing time tauphi. This extends previous works on dephasing in ballistic systems to the case of many conducting channels.
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.
The present doctoral thesis is concerned, on the one hand, with the investigation of the decay probability in a quantum billiard, and, on the other, the realization of the double-slit experiment in microwave billiards. For both purposes transmission elements of the scattering matrix are experimentally measured in microwave cavities with regular and chaotic dynamics. One or more pieces of the billiards' boundary are removed and the leaked outward eld is investigated. The general aim is to distinguish the eect of the billiard dynamics on the temporal decay behaviour and on the interference patterns. The decay probability in an open quantum billiard can be explored by means of the scattering matrix formalismus derived in the framework of nuclear physics. In a circular billiard, archetype of regular systems, the opening breaks the spatial symmetry causing a splitting of the degenerate states observable at low frequencies. One state of the doublet remains unaected contributing to the long-time behavior inside the cavity (survival probability), while the short-lived one is strongly influenced by the opening and decays much faster. The case of chaotic dynamics is studied by means of a tilted stadium billiard. For sufficiently long times the decay probability shows an essentially nonexponential behaviour. Agreement with theoretical results derived by Harney, Dittes and Muller is obtained for a few of the open channels and intermediate times. For both, the circle and the stadium,the short time decay resembles the classical behaviour. Furthermore, a double-slit experiment is performed with a wave packet starting inside a microwave billiard domain, both for regular and chaotic dynamics. Stationary and temporal patterns formed by the leaking of the wave packet are experimentally determined. The evolution of a directed initial state is constructed by means of the electromagnetic propagator. Also single-slit experiments are carried out and compared with the corresponding double-slit experiments. The present work sheds light on the fundamental question of the decay of open quantum systems and provides plentiful information about the structure of the diracted eld from microwave billards with a double-slit on the boundary connecting classical properties with the corresponding quantum features in both cases.
Single and double-slit experiments are performed with two microwave billiards with the shapes of a rectangle and a quarter stadium, respectively. The classical dynamics of the former is regular, whereas that of the latter is chaotic. Microwaves can leave the billiards via slits in the boundary, forming interference patterns on a screen. The aim is to determine the effect of the billiard dynamics on their structure. For this the development of a method for the construction of a directed wave packet by means of an array of multiple antennas was crucial. The interference patterns show a sensitive dependence not only on the billiard dynamics but also on the initial position and direction of the wave packet.
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.
We investigate numerically the occurrence of the interference fringes in experiments where an initial Gaussian wave packet evolves inside billiard domains of several shapes with two narrow slits on a side. Contrarily to what claimed in the literature, our results seem to show that the occurrence of interference fringes does not depend on the billiard integrability but on a spatial reflection symmetry concerning both the billiard domain and the initial Gaussian wave packet. Indeed, whether the billiard is regular or chaotic, we find clear interference fringes when this symmetry is verified whereas when it is not verified, we find that the interference patterns are perturbed in various measure according to the effects of the symmetry violation.