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Geometric Phase in Quantum Billiards with a Pointlike Scatterer  [PDF]
Taksu Cheon,Takaomi Shigehara
Physics , 1995, DOI: 10.1103/PhysRevLett.76.1770
Abstract: We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various levels. The associated Berry phase is illustrated, and the existence of localized wave functions is pointed out. In one dimension, even the ground state is shown to display the sign reversal with a mechanism to circumvent the Sturm-Liouville theorem.
Spectral Properties of Three-Dimensional Quantum Billiards with a Pointlike Scatterer  [PDF]
T. Shigehara,Taksu Cheon
Physics , 1997, DOI: 10.1103/PhysRevE.55.6832
Abstract: We examine the spectral properties of three-dimensional quantum billiards with a single pointlike scatterer inside. It is found that the spectrum shows chaotic (random-matrix-like) characteristics when the inverse of the formal strength $\bar{v}^{-1}$ is within a band whose width increases parabolically as a function of the energy. This implies that the spectrum becomes random-matrix-like at very high energy irrespective to the value of the formal strength. The predictions are confirmed by numerical experiments with a rectangular box. The findings for a pointlike scatterer are applied to the case for a small but finite-size impurity. We clarify the proper procedure for its zero-size limit which involves non-trivial divergence. The previously known results in one and two-dimensional quantum billiards with small impurities inside are also reviewed from the present perspective.
Wave Chaos in Quantum Pseudointegrable Billiards  [PDF]
T. Shigehara,H. Mizoguchi,T. Mishima,Taksu Cheon
Physics , 1997,
Abstract: We clarify from a general perspective, the condition for the appearance of chaotic energy spectrum in quantum pseudointegrable billiards with a point scatterer inside.
Transitions to quantum chaos in a generic one-parameter family of billiards  [PDF]
Sunghwan Rim,Soo-Young Lee,Eui-Soon Yim,C. H. Lee
Physics , 1998,
Abstract: Generic one-parameter billiards are studied both classically and quantally. The classical dynamics for the billiards makes a transition from regular to fully chaotic motion through intermediary soft chaotic system. The energy spectra of the billiards are computed using finite element method which has not been applied to the euclidean billiard. True generic quantum chaotic transitional behavior and its sensitive dependence on classical dynamics are uncovered for the first time. That is, this sensitive dependence of quantum spectral measures on classical dynamics is a genuine manifestation of quantum chaos.
Relevance of Chaos in Numerical Solutions of Quantum Billiards  [PDF]
Baowen Li,Marko Robnik,Bambi Hu
Physics , 1998, DOI: 10.1103/PhysRevE.57.4095
Abstract: In this paper we have tested several general numerical methods in solving the quantum billiards, such as the boundary integral method (BIM) and the plane wave decomposition method (PWDM). We performed extensive numerical investigations of these two methods in a variety of quantum billiards: integrable systens (circles, rectangles, and segments of circular annulus), Kolmogorov-Armold-Moser (KAM) systems (Robnik billiards), and fully chaotic systems (ergodic, such as Bunimovich stadium, Sinai billiard and cardiod billiard). We have analyzed the scaling of the average absolute value of the systematic error $\Delta E$ of the eigenenergy in units of the mean level spacing with the density of discretization $b$ (which is number of numerical nodes on the boundary within one de Broglie wavelength) and its relationship with the geometry and the classical dynamics. In contradistinction to the BIM, we find that in the PWDM the classical chaos is definitely relevant for the numerical accuracy at a fixed density of discretization $b$. We present evidence that it is not only the ergodicity that matters, but also the Lyapunov exponents and Kolmogorov entropy. We believe that this phenomenon is one manifestation of quantum chaos.
Quantum and Wave Dynamical Chaos in Superconducting Microwave Billiards  [PDF]
B. Dietz,A. Richter
Physics , 2015, DOI: 10.1063/1.4915527
Abstract: Experiments with superconducting microwave cavities have been performed in our laboratory for more than two decades. The purpose of the present article is to recapitulate some of the highlights achieved. We briefly review (i) results obtained with flat, cylindrical microwave resonators, so-called microwave billiards, concerning the universal fluctuation properties of the eigenvalues of classically chaotic systems with no, a threefold and a broken symmetry; (ii) summarize our findings concerning the wave-dynamical chaos in three-dimensional microwave cavities; (iii) present a new approach for the understanding of the phenomenon of dynamical tunneling which was developed on the basis of experiments that were performed recently with unprecedented precision, and finally, (iv) give an insight into an ongoing project, where we investigate universal properties of (artificial) graphene with superconducting microwave photonic crystals that are enclosed in a microwave resonator, i.e., so-called Dirac billiards.
Is there relevance of chaos in numerical solutions of quantum billiards?  [PDF]
Baowen Li,Marko Robnik
Physics , 1995,
Abstract: In numerically solving the Helmholtz equation inside a connected plane domain with Dirichlet boundary conditions (the problem of the quantum billiard) one surprisingly faces enormous difficulties if the domain has a problematic geometry such as various nonconvex shapes. We have tested several general numerical methods in solving the quantum billiards. Following our previous paper (Li and Robnik 1995) where we analyzed the Boundary Integral Method (BIM), in the present paper we investigate systematically the so-called Plane Wave Decomposition Method (PWDM) introduced and advocated by Heller (1984, 1991). In contradistinction to BIM we find that in PWDM the classical chaos is definitely relevant for the numerical accuracy at fixed density of discretization on the boundary $b$ ($b$ = number of numerical nodes on the boundary within one de Broglie wavelength). This can be understood qualitatively and is illustrated for three one-parameter families of billiards, namely Robnik billiard, Bunimovich stadium and Sinai billiard. We present evidence that it is not only the ergodicity which matters, but also the Lyapunov exponents and Kolmogorov entropy. Although we have no quantitative theory we believe that this phenomenon is one manifestation of quantum chaos. PACS numbers: 02.70.Rw, 05.45.+b, 03.65.Ge, 03.65.-w
Resonance Tunneling in Double-Well Billiards with a Pointlike Scatterer  [PDF]
Taksu Cheon,T. Shigehara
Physics , 1996, DOI: 10.1016/S0375-9601(97)00119-9
Abstract: The coherent tunneling phenomenon is investigated in rectangular billiards divided into two domains by a classically unclimbable potential barrier. We show that by placing a pointlike scatterer inside the billiard, we can control the occurrence and the rate of the resonance tunneling. The key role of the avoided crossing is stressed. Keywords: chaotic tunneling, quantum billiard, delta potential, diabolical degeneracy PACS: 3.65.-w, 4.30.Nk, 5.45.+b, 73.40.Gk
Chaos in Andreev Billiards  [PDF]
Ioan Kosztin,Dmitrii L. Maslov,Paul M. Goldbart
Physics , 1995, DOI: 10.1103/PhysRevLett.75.1735
Abstract: A new type of classical billiard - the Andreev billiard - is investigated using the tangent map technique. Andreev billiards consist of a normal region surrounded by a superconducting region. In contrast with previously studied billiards, Andreev billiards are integrable in zero magnetic field, {\it regardless of their shape}. A magnetic field renders chaotic motion in a generically shaped billiard, which is demonstrated for the Bunimovich stadium by examination of both Poincar\'e sections and Lyapunov exponents. The issue of the feasibility of certain experimental realizations is addressed.
Quantum Chaos of Bogoliubov Waves for a Bose-Einstein Condensate in Stadium Billiards  [PDF]
Chuanwei Zhang,Jie Liu,Mark G. Raizen,Qian Niu
Physics , 2004, DOI: 10.1103/PhysRevLett.93.074101
Abstract: We investigate the possibility of quantum (or wave) chaos for the Bogoliubov excitations of a Bose-Einstein condensate in billiards. Because of the mean field interaction in the condensate, the Bogoliubov excitations are very different from the single particle excitations in a non-interacting system. Nevertheless, we predict that the statistical distribution of level spacings is unchanged by mapping the non-Hermitian Bogoliubov operator to a real symmetric matrix. We numerically test our prediction by using a phase shift method for calculating the excitation energies.
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