Quantum graphs: Applications to quantum chaos and universal spectral statistics

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.     

Symmetry and symmetry breaking in generalized parastatistics

An analysis is made of the characteristics of internal symmetry and symmetry breaking in a quantum field theory with generalized parastatistics, defined by either double commutation relations or single commutation relations. The connection between the two statistics is clarified. We develop a formalism in which statistics is viewed as a dynamical or phase variable of quantum systems. It is shown that the types of Higgs phases possible depend upon statistics. Relationships between physical amplitudes implied by internal symmetry with normal statistics are violated in the case of generalized parastatistics.     

On the effect of fractional statistics on quantum ion acoustic waves

In this paper, I study the effect of a small deviation from the Fermi–Dirac statistics on the quantum ion acoustic waves. For this purpose, a quantum hydrodynamic model is developed based on the Polychronakos statistics, which allows for a smooth interpolation between the Fermi and Bose limits, passing through the case of classical particles. The model includes the effect of pressure as well as quantum diffraction effects through the Bohm potential. The equation of state for electrons obeying fractional statistics is obtained and the effect of fractional statistics on the kinetic energy and the coupling parameter is analyzed. Through the model, the effect of fractional statistics on the quantum ion acoustic waves is highlighted, exploring both linear and weakly nonlinear regimes. It is found that fractional statistics enhance the amplitude and diminish the width of the quantum ion acoustic waves. Furthermore, it is shown that a small deviation from the Fermi–Dirac statistics can modify the type structures, from bright to dark soliton. All known results of fully degenerate and non-degenerate cases are reproduced in the proper limits.     

Braided groups and algebraic quantum field theories

We introduce the notion of a braided group. This is analogous to a supergroup with Bose-Fermi statistics ±1 replaced by braid statistics. We show that every algebraic quantum field theory in two dimensions leads to a braided group of internal symmetries. Every quantum group can be viewed as a braided group.     

量子点腔系统中抽运诱导受激辐射与非谐振腔量子电动力学特性的研究

从量子点和腔相互作用的哈密顿量出发,利用包括非相干抽运,退相干项主方程,并结合量子回归定理,考虑相关初始条件,理论推导和数值计算相结合得出量子点多光子发射光谱精确数值结果.分析多光子发射光谱(费米子统计)和相应单光子情况(玻色子统计)时,得到依据费米子统计,退相干和热浴模型的多光子发射光谱同最近文献[22]量子点——微柱腔实验结果符合得非常好.结合理论和当前实验,显示了量子点腔系统中抽运诱导受激辐射和非谐振腔量子电动力学. 关键词： 量子点 微腔 受激辐射 发射光谱     

Hanbury Brown-Twiss correlations to probe the population statistics of GHz photons emitted by conductors

We present the first study of the statistics of GHz photons in quantum circuits, using Hanbury Brown and Twiss correlations. The super-Poissonian and Poissonian photon statistics of thermal and coherent sources, respectively, made of a resistor and a radio frequency generator, are measured down to the quantum regime at milli-Kelvin temperatures. As photon correlations are linked to the second and fourth moments of current fluctuations, this experiment, which is based on current cryogenic electronics, may become a standard for probing electron/photon statistics in quantum conductors.     

The quantum geometry of spin and statistics

Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin–statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose–Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach.     

On the Quantum Probability Flux Through Surfaces

We remark that the often ignored quantum probability current is fundamental for a genuine understanding of scattering phenomena and, in particular, for the statistics of the time and position of the first exit of a quantum particle from a given region, which may be simply expressed in terms of the current. This simple formula for these statistics does not appear as such in the literature. It is proposed that the formula, which is very different from the usual quantum mechanical measurement formulas, be verified experimentally. A full understanding of the quantum current and the associated formula is provided by Bohmian mechanics.     

Quantum theory for a special kind of hard-core particles with intermediate statistics

Starting from a specific definition of fractional statistics for hard-core particles, we find a set of intermediate statistics systems, in which a single-particle quantum state can be effectively occupied by an integer number of identical particles — M-ons, as in the case for fermions and bosons. A quantum statistical theory of an ideal M-on gas is formulated exactly, and the associated distribution is explicitly represented in a simple analytical form. A possible application to the fractional quantum Hall effect is also briefly discussed.     

Entanglement concentration using quantum statistics

We propose an entanglement concentration scheme which uses only the effects of quantum statistics of indistinguishable particles. This establishes the fact that useful quantum information processing can be accomplished by quantum statistics alone. Because of the basis independence of statistical effects, our protocol requires less knowledge of the initial state than most entanglement concentration schemes. Moreover, no explicit controlled operation is required at any stage.     

Quantum groups and generalized statistics in integrable models

The paper deals with the integrable massive models of quantum field theory. It is shown that generalized statistics of physical particles is closely connected with the invariance under quantum groups. This invariance provides the possibility to construct quasi-local operators (parafermions) possessing generalized statistics which interpolates the physical particles. For the particular examples of SG, RSG models and scaling 3-state Potts model the parafermions are described completely (all their matrix elements in the space of states are presented).     

基于机器视觉的量子点STM形貌图像识别研究

为减轻量子点表面形貌分析过程中的人工工作,使量子点的STM图像分析更加自动化,基于机器视觉对衬底的斜切角及量子点的形貌特性展开研究.利用腐蚀和边缘检测提取台阶形状,并通过反三角变换计算斜切角.利用二值化和阈值下降对量子点的数量与空间坐标进行提取,在此基础上,通过邻域密度计算分析其均匀性,并在解决图像中的粘连问题后找出量子点的尺寸.实验结果显示,与人工统计相比,斜切角、量子点计数及尺寸的平均误差分别为5.02%, 0.7788%及1.12%;并实现量子点均匀性的自动化统计与分析.基于机器视觉算法的自动识别过程,对协助研究者分析量子点表面形貌有实际意义.     

The Fractional Statistics of Generalized Haldane Wave Function in 4D Quantum Hall Effect

Recently, a generalization of Laughlin‘s wave function expressed in Haldane‘s spherical geometry is con-structed in 4D quantum Hall effect. In fact, it is a membrane wave function in CP3 space. In this article, we usenon-Abelian Berry phase to analyze the statistics of this membrane wave function. Our results show that the membranewave function obeys fractional statistics. It is the rare example to realize fractional statistics in higher-dimensional spacethan 2D. And, it will help to make clear the unresolved problems in 4D quantum Hall effect.     

Enhanced electron positron pair creation by the frequency chirped laser pulse

Based on the quantum Vlasov equation, the effect of frequency chirp on electron-positron pair production is investigated. The cycle parameter, which characterizes the laser field cycle degree within the pulse, is also considered. In both supercycle and subcycle laser pulses the frequency chirp can greatly enhance the momentum distribution function of created pairs and the pair number density. The pair number density created by a supercycle laser pulse is larger than that by a subcycle pulse under the same laser frequency and chirping. There exists an optimal cycle parameter corresponding to the maximum value of the created pair number density for different chirp rates. It is found that the pair number density is sensitive/insensitive to chirping rate when the cycle parameter lies below/above the optimal one.     

Rigorous information-theoretic derivation of quantum-statistical thermodynamics. I

In previous publications we have criticized the usual application of information theory to quantal situations and proposed a new version of information-theoretic quantum statistics. This paper is the first in a two-part series in which our new approach is applied to the fundamental problem of thermodynamic equilibrium. Part I deals in particular with informational definitions of equilibrium and the identification of thermodynamic analogs in our modified quantum statistics formalism.Work supported by a grant from Research Corporation to J. L. P.     

Characteristics of level-spacing statistics in chaotic graphene billiards

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.     

Interacting particles as neither bosons nor fermions: A microscopic origin for fractional exclusion statistics

We study a quantum liquid of particles interacting via a long-ranged two-body potential in three dimensions where the original particles are supposed to be either bosons or fermions. We show that such liquids exhibit the nature of a quantum liquid with fractional exclusion statistics. In both quantum liquids enlarged pseudo-Fermi surfaces are formed from bosons and fermions, although with different excitations. Hence, we conclude that the microscopic origin of exclusion statistics comes from the nature of long-ranged two-body interactions between the particles.     

Generalized Fock spaces,new forms of quantum statistics and their algebras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ‘infinite’, Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are: new algebras for infinite statistics,q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ‘doubly-infinite’ statistics, many representations of orthostatistics, Hubbard statistics and its variations.     

Influence of a random telegraph process on the transport through a point contact

We describe the transport properties of a point contact under the influence of a classical two-level fluctuator. We employ a transfer matrix formalism allowing us to calculate arbitrary correlation functions of the stochastic process by mapping them on matrix products. The result is used to obtain the generating function of the full counting statistics of a classical point contact subject to a classical fluctuator, including extensions to a pair of two-level fluctuators as well as to a quantum point contact. We show that the noise in the quantum point contact is a sum of the (quantum) partitioning noise and the (classical) noise due to the two-level fluctuator. As a side result, we obtain the full counting statistics of a quantum point contact with time-dependent transmission probabilities.