Resonance tunneling in double-well billiards with a point-like scatterer

Coherent tunneling is investigated in rectangular billiards divided into two domains by a classically unclimbable potential barrier. We show that by placing a point-like scatterer inside the billiard, we can control the occurrence and the resonance tunneling rate. The key role of the avoided crossing is stressed.     

正方体量子台球的半经典分析(英文)

近几年,量子台球问题引起人们的广泛兴趣.以前有很多对二维量子台球做过研究,相对于二维台球来说,三维台球更接近实际体系.本文以三维正方体量子台球为例,利用半经典闭合轨道理论计算了正方体量子台球中的经典开轨道,并研究量子谱函数与经典轨道长度之间的对应关系,发现他们之间对应的很好.这将有助于我们分析开放型量子台球中输运性质问题.利用这种方法物理图像清晰,计算量小并且可以帮助理解一些混沌体系的性质.这是半经典理论为联系量子力学与经典力学起桥梁作用的又一证明.     

Scars of periodic orbits in the stadium action billiard

Compact billiards in phase space, or action billiards, are constructed by truncating the classical Hamiltonian in the action variables. The corresponding quantum mechanical system has a finite Hamiltonian matrix. In previous papers we defined the compact analog of common billiards, i.e., straight motion in phase space followed by specular reflections at the boundaries. Computation of their quantum energy spectra establishes that their properties are exactly those of common billiards: the short-range statistics follow the known universality classes depending on the regular or chaotic nature of the motion, while the long-range fluctuations are determined by the periodic orbits. In this work we show that the eigenfunctions also follow qualitatively the general characteristics of common billiards. In particular, we show that the low-lying levels can be classified according to their nodal lines as usual and that the high excited states present scars of several short periodic orbits. Moreover, since all the eigenstates of action billiards can be computed with great accuracy, Bogomolny's semiclassical formula for the scars can also be tested successfully.     

Improved Estimates for Correlations in Billiards

We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich’s stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were sub-optimal (they contained a redundant logarithmic factor). We sharpen those estimates by removing that factor.     

Periodic orbits in magnetic billiards

We propose a simple method to calculate periodic orbits in two-dimensional systems with no symbolic dynamics. The method is based on a line by line scan of the Poincaré surface of section and is particularly useful for billiards. We have applied it to the Square and Sinai's billiards subjected to a uniform orthogonal magnetic field and we obtained about 2000 orbits for both systems using absolutely no information about their symbolic dynamics. Received 21 September 1999 and Received in final form 13 April 2000     

Track Billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of 3-dimensional billiards. Interestingly, we find that for some track billiards, the mechanism generating hyperbolicity is not the defocusing one, which requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary.     

The influence of ions and radiation on tunneling of electrons from the surface of liquid helium

In work previously reported by Saville et al. [1] and Saville [2], we found that electrons tunneling from their shallow potential well above the surface of liquid helium behave as an almost ideal 1D hydrogen atom. Computations [3], without approximation or adjustable parameters, using the time dependent Schrödinger equation are in excellent agreement with measured tunneling rates. The same computation method was used to compute the influence of AC fields on the tunneling rates [3] as preparation for planned experiments. In this paper I discuss phenomena which appear to set upper and lower limits on the observable tunneling rates. Lower limits were determined by photo assisted escape due to thermal radiation and by cosmic ray generation of ions in the bulk liquid helium. Upper limits were set by the creation of excitations in the superfluid film on the top of the sample chamber when electrons impacted with energy greater than 21.6 eV. Through its influence on the tunneling rate, we were also able to measure very small changes in the electron potential at the surface due to a sub-monolayer coverage of³He.     

Nonequilibrium Gas and Generalized Billiards

Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.     

Semi-Focusing Billiards: Hyperbolicity

In this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of ℝ³. We also prove that a related class of semi-focusing billiards has mixed dynamics, i.e., their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other. These billiards are the first rigorous examples of billiards in domains of ℝ³ with divided phase space. The first author was partially supported by the NSF grant #0140165 and the Humboldt Foundation. The second author was partially supported by the FCT (Portugal) through the Program POCTI/FEDER.     

Entanglement distribution statistic in Andreev billiards

We investigate statistical aspects of the entanglement production for open chaotic mesoscopic billiards in contact with superconducting parts, known as Andreev billiards. The complete distributions of concurrence and entanglement of formation are obtained by using the Altland–Zirnbauer symmetry classes of circular ensembles of scattering matrices, which complements previous studies in chaotic universal billiards belonging to other classes of random matrix theory. Our results show a unique and very peculiar behavior: the realization of entanglement in a Andreev billiard always results in non-separable state, regardless of the time reversal symmetry. The analytical calculations are supported by a numerical Monte Carlo simulation.     

Fermi Acceleration in driven relativistic billiards

We show numerical experiments of driven billiards using special relativity. We have the remarkable fact that for the relativistic driven circular and annular concentric billiards, depending on initial conditions and parameters, we observe Fermi Acceleration, absent in the Newtonian case. The velocity for these cases tends to the speed of light very quickly. We find that for the annular eccentric billiard the initial velocity grows for a much longer time than the concentric annular billiard until it asymptotically reach c.     

The influence of confining wall profile on quantum interference effects in etched Ga0.25In0.75As/InP billiards

We present measurements of the potential profile of etched GaInAs/InP billiards and show that their energy gradients are an order of magnitude steeper than those of surface-gated GaAs/AlGaAs billiards. Previously observed in GaAs/AlGaAs billiards, fractal conductance fluctuations are predicted to be critically sensitive to the billiard profile. Here we show that, despite the increase in energy gradient, the fractal conductance fluctuations persist in the harder GaInAs/InP billiards.     

Asymmetric Landau-Zener tunneling in a periodic potential

Using a simple model for nonlinear Landau-Zener tunneling between two energy bands of a Bose-Einstein condensate in a periodic potential, we find that the tunneling rates for the two directions of tunneling are not the same. Tunneling from the ground state to the excited state is enhanced by the nonlinearity, whereas in the opposite direction it is suppressed. These findings are confirmed by numerical simulations of the condensate dynamics. Measuring the tunneling rates for a condensate of rubidium atoms in an optical lattice, we have found experimental evidence for this asymmetry.     

Upgrading the Local Ergodic Theorem for Planar Semi-dispersing Billiards

The Local Ergodic Theorem (also known as the ‘Fundamental Theorem’) gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However, the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of the Local Ergodic Theorem for two dimensional billiards without using the Ansatz.     

Time-reversal symmetry breaking and decoherence in chaotic Dirac billiards

In this work, we perform a statistical study on Dirac Billiards in the extreme quantumlimit (a single open channel on the leads). Our numerical analysis uses a large ensembleof random matrices and demonstrates the preponderant role of dephasing mechanisms in suchchaotic billiards. Physical implementations of these billiards range from quantum dots ofgraphene to topological insulators structures. We show, in particular, that the role offinite crossover fields between the universal symmetries quickly leaves the conductance tothe asymptotic limit of unitary ensembles. Furthermore, we show that the dephasingmechanisms strikingly lead Dirac billiards from the extreme quantum regime to thesemiclassical Gaussian regime.     

Increasing thermoelectric efficiency: a dynamical systems approach

Inspired by the kinetic theory of ergodic gases and chaotic billiards, we propose a simple microscopic mechanism for the increase of thermoelectric efficiency. We consider the cross transport of particles and energy in open classical ergodic billiards. We show that, in the linear response regime, where we find exact expressions for all transport coefficients, the thermoelectric efficiency of ideal ergodic gases can approach the Carnot efficiency for sufficiently complex charge carrier molecules. Our results are clearly demonstrated with a simple numerical simulation of a Lorentz gas of particles with internal rotational degrees of freedom.     

Draining of the sea of chaos: role of resonant transmission and reflection in an array of billiards

We investigate the dynamics of a system of coupled electron billiards by using a magnetic field to dramatically modify the underlying mixed phase space. At specific values of the magnetic field the sea of chaos is drained. At these fields there exist reflected or transmitted orbits associated with maxima and minima in the experimentally observed magnetoresistance. These effects are studied by comparing the classical and quantum-mechanical phase-space dynamics leading to a basic understanding of the role of chaos in the transport in an array of billiards.     

Spectral Gap for a Class of Random Billiards

A random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post-collision distribution of velocities for any given pre-collision velocity. We consider such collision operators for certain random billiards that we call billiards with microstructure. Collisions modeled by these operators can still be thought of as elastic and time reversible. The operators are canonically determined by a second (deterministic) billiard system that models “microscopic roughness” on the billiard table boundary. Our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards. Among the main results, we give geometric conditions for these operators to be Hilbert-Schmidt and relate their spectrum and speed of convergence to stationary Markov chains with geometric features of the microscopic billiard structure. The relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples.     

Chaos in composite billiards

The mechanisms and features of the chaotic behavior in billiards with ray splitting (refraction) are considered. In contrast to ordinary billiards, the law of motion in composite billiards that is coded with a sequence of ray visits to different media is shown to be deterministically chaotic. The analysis is performed in terms of a geometrical-dynamical approach in which a symmetric phase space is used instead of the ordinary Hamiltonian phase space. The chaotization elements in composite billiards of a general position are studied. The dynamics of rays in ring billiards consisting of two concentric media with different refractive indices is considered.     

One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.