Quantum billiards and constrained random wave correlations
W. E. Bies, N. Lepore, E. J. Heller
我们研究了楔形和矩形区域中的混沌特征函数，并推广了Berry猜想。推导了两点关联函数的表达式，并进行了数值验证。
We study chaotic eigenfunctions in wedge-shaped and rectangular regions using a generalization of Berry's conjecture. An expression for the two-point correlation function is derived and verified numerically.
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We illustrate some of the techniques to identify chaos signatures at the quantum level using as a guiding examples some systems where a particle is constrained to move on a radial symmetric, but non planar, surface. In particular, two systems are studied: the case of a cone with an arbitrary contour or dunce hat billiard and the rectangular billiard with an inner Gaussian surface.
The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener-Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well describe…
The paper is devoted to the derivation of random unitary matrices whose spectral statistics is the same as statistics of quantum eigenvalues of certain deterministic two-dimensional barrier billiards. These random matrices are extracted from the exact billiard quantisation condition by applying a random phase approximation for high-excited states. An important ingredient of the method is the calculation of $S$-matrix for the scattering in the slab with a half-plane inside by the Wiener-Hopf method. It appears that these random matrices have the form similar to the one obtained by the author in…
Triggered by a controversy surrounding a universal behaviour of the power spectrum in quantum systems exhibiting regular classical dynamics, we focus on a model of random diagonal matrices (RDM), often associated with the Poisson spectral universality class, and examine how the power spectrum and the form factor get affected by two-sided truncations of RDM spectra. Having developed a nonperturbative description of both statistics, we perform their detailed asymptotic analysis to demonstrate explicitly how a traditional assumption (lying at the heart of the controversy) -- that the power spectr…
Scale Anomaly and Quantum Chaos in the Billiards with Pointlike Scatterers
我们认为，在具有点状散射体的伪可积台球中发现的随机矩阵能谱与经典模拟系统标度不变性的量子破坏有关。结果表明，运行耦合常数的行为，解释了伪可积台球水平统计的关键特征。
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-se…
Recent works have established universal entanglement properties and demonstrated validity of single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians. However, a common property of all quantum-chaotic quadratic Hamiltonians studied in this context so far is the presence of random terms that act as a source of disorder. Here we introduce tight-binding billiards in two dimensions, which are described by non-interacting spinless fermions on a disorder-free square lattice subject to curved open (hard-wall) boundaries. We show that many properties of tight-binding billiar…
Spectral Properties of Three-Dimensional Quantum Billiards with a Pointlike Scatterer
我们研究了三维量子台球的光谱特性，其中有一个单一的点状散射体。研究发现，当形式强度的倒数$\bar{v}^{-1}$位于一个宽度随能量呈抛物线增长的带内时，谱表现出混沌（类随机矩阵）特性。这意味着在非常高的能量下，谱变得像随机矩阵，而与形式强度的值无关。数值实验与矩形框的预测证实。点状散射体的研究结果适用于一个小的，但有限大小的杂质的情况。我们澄清了适当的程序，其零尺寸的限制，其中涉及非平凡的分歧。从这个角度也回顾了以前已知的结果，在一维和二维量子台球内的小杂质。
Given a Dirichlet eigenfunction of a 2D quantum billiard, the boundary domain count is the number of intersections of the nodal lines with the boundary. We study the integer sequence defined by these numbers, sorted according to the energies of the eigenfunctions. Based on a variant of Berry's random wave model, we derive a semi-classical trace formula for the sequence of boundary domain counts. The formula consists of a Weyl-like smooth part, and an oscillating part which depends on classical periodic orbits and their geometry. The predictions of this trace formula are supported by numerical …
Influence of phase space localization on the energy diffusion in a quantum chaotic billiard
研究了具有运动边界的混沌台球的量子动力学。我们发现了一个形状参数的哈密顿展开，使我们能够获得的频谱变形台球的变形如此之大的特征波长。然后，对于一个特定的时间依赖的形状变化，台球内的粒子的量子动力学直接集成。特别地，在Bunimovich体育场台球与振荡边界的能量的色散进行了研究。结果表明，边界第一次振荡（${< Δ^2 E}> =2 D t$）的能量分布是扩散的。我们研究了扩散常数$D$作为边界速度的函数，并发现与基于随机矩阵理论的理论预测的差异。通过从光谱中提取高度相空间局域化的结构，以前的差异显着减少。这一事实提供了第一个数值证据的影响相空间局域化的量子扩散的混沌系统。
We present an extensive numerical study of spectral statistics and eigenfunctions of quantized triangular billiards. We compute two million consecutive eigenvalues for six representative cases of triangular billiards, three with generic angles with irrational ratios with $π$, whose classical dynamics is presumably mixing, and three with exactly one angle rational with $π$, which are presumably only weakly mixing or even only non-ergodic in case of right-triangles. We find excellent agreement of short and long range spectral statistics with the Gaussian orthogonal ensemble of random matrix theo…
The connection of a superconductor to a chaotic ballistic quantum dot leads to interesting phenomena, most notably the appearance of a hard gap in its excitation spectrum. Here we treat such an Andreev billiard semiclassically where the density of states is expressed in terms of the classical trajectories of electrons (and holes) that leave and return to the superconductor. We show how classical orbit correlations lead to the formation of the hard gap, as predicted by random matrix theory in the limit of negligible Ehrenfest time $\tE$, and how the influence of a finite $\tE$ causes the gap to…
Experimental study of coupled quantum billiards with integrable and chaotic classical dynamics and test of a special Rosenzweig-Porter model
本研究通过实验分析双量子弹珠箱耦合系统，测试了罗森茨威格-波特模型的适用性，揭示了耦合强度对谱性质的影响。
Periodic Chaotic Billiards: Quantum-Classical Correspondence in Energy Space
研究了二维周期性波纹台球的本征态和局域态密度的性质，着重研究了它们在能量表象中的量子-经典对应。为了构造LDOS的经典对应物和本征态结构（SES），边界的影响首先被纳入（通过正则变换）到一个有效的潜力，使一个粒子的运动在二维波纹台球相当于两个相互作用的粒子在一维几何。我们表明，经典的SES和LDOS的强混沌运动的情况下，揭示了相当好的对应关系的量子量。我们还表明，SES和LDOS的主要特点可以解释的基本经典动力学，特别是某些周期轨道。另一方面，本征态和LDOS的统计性质与随机矩阵理论所描述的不同。我们讨论的量子效应负责的非遍历字符的本征态和个人LDOS似乎是通用的这种类型的台球与大量的横向通道。
We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. Various approaches t…
Thermodynamic Uncertainty Relations in Chaotic Andreev Billiards
本研究利用随机矩阵理论，在混沌安德烈夫弹球中研究热力学不确定性关系，发现其受对称类和势垒透明度强烈影响，但标准关系始终成立，可作为对称性敏感的普适输运探针。
Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems
我们给出了量子混沌的一些数值方面的介绍。利用标准映射和扰动猫映射，给出了环面上二维保面积映射的经典动力学。讨论了由其生成函数给出的保面积映射的量化，并给出了计算特征值的Python计算机程序。我们说明了两种类型的扰动猫映射的特征值分布，一个导致COE和其他CUE统计。对于量子映射的本征函数，我们研究了本征向量的分布，并与相应的随机矩阵分布进行了比较。Husimi表示允许直接比较相空间中本征态的局部化与相应的经典结构。给出了扰动猫映射和具有不同参数的标准映射的例子。台球系统和相应的量子台球是另一类重要的系统（也与应用有关，例如在介观物理学中）。详细阐述了边界积分方法，它是求解Helmholtz方程本征值和本征函数的一种重要方法。我们讨论了几种方法来确定Fredholm方程的特征值，并说明他们的体育场台球。虚假的解决方案的发生进行了详细讨论，并说明了圆形台球，体育场台球，和环形扇区台球。我们强调正常的导函数的作用，以计算本征函数，动量表示或自相关函数的归一化在一个非常有效和直接的方式。给出了这些量的一些例子并进行了讨论。
Semiclassical Construction of Random Wave Functions for Confined Systems
我们开发了一个统计描述的混沌波函数在封闭系统服从任意边界条件相结合的半经典表达式的空间两点关联函数与治疗的特征函数作为高斯随机场。因此，我们推广Berry的各向同性随机波模型，通过在边界处反射的经典路径将约束效应。我们的方法允许显式计算高度非平凡的统计，如强度分布，在通常很少的短轨道方面，这取决于所考虑的能量窗口。我们比较非洲台球的数值量子结果，并推导出其他突出的限制几何形状的非各向同性随机波模型。
Weakly chaotic or weakly interacting systems have a wide regime where the common random matrix theory modeling does not apply. As an example we consider cold atoms in a nearly integrable optical billiard with displaceable wall ("piston"). The motion is completely chaotic but with small Lyapunov exponent. The Hamiltonian matrix does not look like one taken from a Gaussian ensemble, but rather it is very sparse and textured. This can be characterized by parameters $s$ and $g$ that reflect the percentage of large elements, and their connectivity, respectively. For $g$ we use a resistor network ca…
Gaussian Fluctuations in Chaotic Eigenstates
我们研究的波动预测的自相关函数的能量本征态的混沌，二维台球的猜想（由于贝瑞），本征函数是一个高斯随机变量。我们发现一个明确的公式的根均方振幅的预期波动的自相关函数。这些波动在小$\hbar$（高能量）极限中是$O(\hbar^{1/2})$。为了比较，任何由孤立周期轨道的疤痕引起的修正也是$O(\hbar^{1/2})$。如果自相关函数在分离向量的方向上被平均，则波动呈现特别简单的形式。我们比较我们的各种预测与最近的数值计算李和Robnik的Robnik台球，并找到很好的协议。我们指出我们的结果如何推广到更高的维度。
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