Semiclassical approximation for transmission through open Sinai billiards

We use a semiclassical approximation to study the transport through the weakly open chaotic Sinai quantum billiards which can be considered as the schematic of a Sinai mesoscopic device,with the diffractive scatterings at the lead openings taken into account.The conductance of the ballistic microstructure which displays universal fluctuations due to quantum interference of electrons can be calculated by Landauer formula as a function of the electron Fermi wave number,and the transmission amplitude can be expressed as the sum over all classical paths connecting the entrance and the exit leads.For the Sinai billiards,the path sum leads to an excellent numerical agreement between the peak positions of power spectrum of the transmission amplitude and the corresponding lengths of the classical trajectories,which demonstrates a good agreement between the quantum theory and the semiclassical theory.     

Quantum transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.     

A new computation scheme for triangular quantum billiards

A new scheme for computing the eigenvalues and eigenstates of the Laplacian with Dirichlet boundary conditions on arbitrary triangular domains is presented. Its reliability is tested by comparing numerical results with analytical ones whenever possible. The computation of eigenvalues shows a good agreement with analytical results. The procedure is shown to give accurate results also in the case of eigenfunctions computation. Finally, the sensitivity of our scheme to the geometry of the domain is discussed and the algorithm is shown to detect small changes in the shape of the domain.     

Rashba billiards

We study the energy levels of non-interacting electrons confined to move in two-dimensional billiard regions and having a spin-dependent dynamics due to a finite Rashba spin splitting. The free space Green's function for such Rashba billiards is constructed analytically and used to find the area and perimeter contributions to the density of states, as well as the corresponding smooth counting function. We show that, in contrast to systems with spin-rotational invariance, Rashba billiards always possess a negative energy spectrum. A semi-classical analysis is presented to interpret the singular behavior of the density of states at certain negative energies for circular Rashba billiards. Our detailed analysis of the spin structure of circular Rashba billiards reveals a finite out-of-plane spin projection for electron eigenstates.     

Periodic orbits in polygonal billiards

We review some properties of periodic orbit families in polygonal billiards and discuss in particular a sum rule that they obey. In addition, we provide algorithms to determine periodic orbit families and present numerical results that shed new light on the proliferation law and its variation with the genus of the invariant surface. Finally, we deal with correlations in the length spectrum and find that long orbits display Poisson fluctuations.     

Evolution of classical projected phase space density in billiards

The classical phase space density projected on to the configuration space offers a means of comparing classical and quantum evolution. In this alternate approach that we adopt here, we show that for billiards, the eigenfunctions of the coarse-grained projected classical evolution operator are identical to a first approximation to the quantum Neumann eigenfunctions. Moreover, there exists a correspondence between the respective eigenvalues although their time evolutions differ.     

Numerical exploration of a family of strictly convex billiards with boundary of classC 2

We are interested in the possible existence of strictly convex ergodic billiards. Such billiards are searched for by means of numerical investigation. The boundary of a billiard is built with four arcs of classC . Adjacent arcs have equal curvatures at connecting points. The surface of section of the billiards is explored. It seems as if symmetric billiards always have invariant curves (islands). Asymmetric billiards have been found which look ergodic. They are built with an arc of an ellipse, two arcs of circles, and one-half of a Descartes oval.     

The K-property of 4D billiards with nonorthogonal cylindric scatterers

The K-property of cylindric billiards given on the 4-torus is established. These billiards are neither orthogonal, where general necessary and sufficient conditions were obtained by D. Szász, nor isomorphic to hard-ball systems, where the connecting path formula of N. Simányi is at hand.This paper is dedicated to Philippe Choquard on the occasion of his 65th birthday.     

开放量子台球体系散射的赝路径半经典近似

本文讨论正方形量子台球的输运性质，考虑电子以费米能量穿过台球区域，在台球出口和入口处对入射和出射波函数采用基尔霍夫散射．采用微扰论的Dyson方程得到半经典格林函数，并把赝路径半经典近似作微扰展开得到体系的传输矩阵元．比较了传输矩阵元的傅立叶变换谱的峰位置与腔内自由电子经典轨道长度，发现在精度允许范围内它们符合的很好．     

开放量子台球体系散射的赝路径半经典近似

本文讨论正方形量子台球的输运性质,考虑电子以费米能量穿过台球区域,在台球出口和入口处对入射和出射波函数采用基尔霍夫散射.采用微扰论的Dyson方程得到半经典格林函数,并把赝路径半经典近似作微扰展开得到体系的传输矩阵元.比较了传输矩阵元的傅立叶变换谱的峰位置与腔内自由电子经典轨道长度,发现在精度允许范围内它们符合的很好.     

Quantum renormalization group approach to geometric phases in spin chains

A relation between geometric phases and criticality of spin chains are studied using the quantum renormalization-group approach. I have shown how the geometric phase evolve as the size of the system becomes large, i.e., the finite size scaling is obtained. The renormalization scheme demonstrates how the first derivative of the geometric phase with respect to the field strength diverges at the critical point and maximum value of the first derivative, and its position, scales with the exponent of the system size.     

Origin of conductance oscillations in square electron billiards: A semi-classical approach

We perform semi-classical and quantum mechanical calculations on square billiards and provide a semi-classical interpretation of the conductance oscillations. We outline its relation to the Gutzwiller's picture of periodic orbits. The frequencies of the conductance oscillations are shown to be due to interference of pairs of long trajectories, which in the phase space are typically situated near the corresponding periodic orbit. We identify the pair of trajectories causing the pronounced peak in a recent experiment and from this directly extract the phase coherence length.     

Model of a nonhomogeneous medium conducting light

We consider a ray of light propagating within a system of infinitely many adjoining rectangles in a plane with passages between any pair of neighboring ones. The ray is assumed to be reflected by the sides of the rectangles, and is refracted while passing from one rectangle to its neighbor. We prove that if the sizes of the rectangles or the coefficients of refraction inside them are random, then with probability 1 the ray reaches arbitrarily remote rectangles.     

具有三体相互作用海森堡自旋链模型的纠缠与几何失协

近年来有研究发现量子失协可以度量非纠缠的量子关联,而且非纠缠的量子关联在量子通信和量子计算中起到很重要的作用.研究了磁场,两种三体相互作用,各向异性参数,耦合常数,温度等参数对同时具有两种三体相互作用海森堡XXZ自旋链系统的量子纠缠,几何失协的影响以及与量子相变的关系.研究表明:量子纠缠和几何失协都可以清晰的表征本模型系统的量子相变现象; 随着XZX+YZY型三体相互作用的增加量子纠缠和几何失协即使在高温时也可达到最大值;几何失协比量子纠缠更全面地描述了量子关联; XZY-YZX型三体相互作用的增加对量子纠缠有抑制作用;XZY-YZX型三体相互作用在一定区域内的增加对几何失协有抑制作用,在另一区域的增加可使几何失协增大到一个稳定的非零值。增大磁场和自旋耦合常数,减小各向异性参数会使纠缠的临界温度变大; 调节自旋耦合常数可更有效的使量子纠缠和几何失协在高温时仍有一个较大的值.同时发现,在磁场 和各向异性参数 的区域两种量子关联都可以维持在最大值.     

在冲力作用下台球运动的力学分析

作为刚体运动的例证,讨论了在冲力作用下台球的运动情形,计算了在不同的击打条件下,台球进入无滑滚动状态的绕质心转动的速度.     

具有三体相互作用海森堡自旋链模型的纠缠与几何失协

近年来有研究发现量子失协可以度量非纠缠的量子关联,而且非纠缠的量子关联在量子通信和量子计算中起到很重要的作用.本文研究了磁场,两种三体相互作用,各向异性参数,耦合常数,温度等参数对同时具有两种三体相互作用海森堡XXZ自旋链系统的量子纠缠,几何失协的影响以及与量子相变的关系.研究表明:量子纠缠和几何失协都可以清晰的表征本模型系统的量子相变现象;随着XZX+YZY型三体相互作用的增加量子纠缠和几何失协即使在高温时也可达到最大值;几何失协比量子纠缠更全面地描述了量子关联;XZY-YZX型三体相互作用的增加对量子纠缠有抑制作用;XZY-YZX型三体相互作用在一定区域内的增加对几何失协有抑制作用,在另一区域的增加可使几何失协增大到一个稳定的非零值.增大磁场和自旋耦合常数,减小各向异性参数会使纠缠的临界温度变大;调节自旋耦合常数可更有效的使量子纠缠和几何失协在高温时仍有一个较大的值.同时发现,在磁场0B5和各向异性参数-1J_Z10的区域两种量子关联都可以维持在最大值.     

Geometry effects in the magnetoconductance of normal and Andreev Sinai billiards

We study the transport properties of low-energy (quasi)particles ballistically traversingnormal and Andreev two-dimensional open cavities with a Sinai-billiard shape. We considerfour different geometrical setups and focus on the dependence of transport on the strengthof an applied magnetic field. By solving the classical equations of motion for each setupwe calculate the magnetoconductance in terms of transmission and reflection coefficientsfor both the normal and Andreev versions of the billiard, calculating in the latter thecritical field value above which the outgoing current of holes becomes zero.     

Magnetoconductance of a few-electron open quantum dot

Magnetoconductance of a small open lateral dot is studied both theoretically and experimentally for the conditions when the dot contains down to 15 electrons. We confirm the existence of a new regime for open dots in which the transport through the structure occurs through individual eigenstates of the corresponding closed dot. In particular, at low magnetic fields the characteristic features in the conductance are related to the underlying eigenspectrum shells. When the number of modes in the leads is reduced more detailed structures within the shells due to single eigenlevels becomes discernible. At higher fields Landau level condensation is evident as well as the crossing of levels collapsing to the different Landau levels.