二维Sinai台球系统的量子混沌研究

研究了二维Sinai台球系统的经典与量子的对应关系,运用定态展开法和Gutzwiller的周期轨道理论对Sinai台球系统的态密度经傅里叶变换得到的量子长度谱进行分析,并把量子长度谱中峰的位置与其所对应的经典体系的周期轨道长度做对比,发现两者之间存在很好的对应关系.观察到了一些量子态局域在短周期轨道附近形成量子scarred态或量子superscarred态.还研究了同心与非同心Sinai台球系统的能级最近邻间距分布,发现同心Sinai台球系统是近可积的,非同心Sinai台球系统在θ=3π/8下,随两中心间距离的增加,能级最近邻间距分布将由近可积向维格那分布过渡.     

Quantum spectra and classical periodic orbit in the cubic billiard

Quantum billiards have attracted much interest in many fields. People have made a lot of researches on the two-dimensional (2D) billiard systems. Contrary to the 2D billiard, due to the complication of its classical periodic orbits, no one has studied the correspondence between the quantum spectra and the classical orbits of the three-dimensional (3D) billiards. Taking the cubic billiard as an example, using the periodic orbit theory, we find the periodic orbit of the cubic billiard and study the correspondence between the quantum spectra and the length of the classical orbits in 3D system. The Fourier transformed spectrum of this system has allowed direct comparison between peaks in such plot and the length of the periodic orbits, which verifies the correctness of the periodic orbit theory. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.     

Correlation of the photodetachment rate of a scarred resonance state with the classical Lyapunov exponent

A metastable resonance state of the periodically driven Gaussian potential well is shown to be scarred on a single unstable periodic orbit of the classical motion. The photodetachment rate of this quantum state is strongly correlated with the Lyapunov exponent of the unstable periodic orbit over a wide range of driving field strengths.     

矩形弹子球中的量子谱和经典周期轨道

利用SU(2)相干态的表示,我们构造了二维矩形弹子球中与经典周期轨道对应的波函数.经典周期轨道和量子波函数之间的关系可以通过物理图像清晰的表示出来.另外,利用周期轨道理论,我们计算了二维矩形弹子球体系的量子谱的傅立叶变换ρ(L).变换谱|ρN(L)|2对L图像中的峰可以和粒子在二维矩形腔中运动的经典轨迹的长度相比较.量子谱中的每一条峰正好对应一条经典周期轨道的长度,表明量子力学和经典力学的对应关系.     

Periodic Orbit Quantization: How to make Semiclassical Trace Formulae Convergent

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of the periodic orbit signal are the semiclassical eigenvalues, and are determined by either linear predictor, Padé approximant, or signal diagonalization. The second method is based upon the direct application of the Padé approximant to the periodic orbit sum. The Padé approximant allows the resummation of the, typically exponentially, divergent periodic orbit terms. Both techniques do not depend on the existence of a symbolic dynamics, and can be applied to bound as well as to open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard.     

Small-scale instabilities in dynamical systems with sliding

We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit remains stable, and its local return map becomes piecewise linear. However, when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.     

Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay

We propose a simple adaptive delayed feedback control algorithm for stabilization of unstable periodic orbits with unknown periods. The state dependent time delay is varied continuously towards the period of controlled orbit according to a gradient-descent method realized through three simple ordinary differential equations. We demonstrate the efficiency of the algorithm with the Rössler and Mackey-Glass chaotic systems. The stability of the controlled orbits is proven by computation of the Lyapunov exponents of linearized equations.     

Painleve property of anharmonic systems with an external periodic field

The Painlevé property of anharmonic systems with an external periodic field is investigated. This property tell us that the only movable singularities exhibited by the solutions are poles. The Painlevé property can serve to distinguish between integrable and nonintegrable systems.     

On a special type of border-collision bifurcations occurring at infinity

In piecewise-smooth dynamical systems, the regions of existence of a periodic orbit are typically parameter sub-spaces confined by border-collision bifurcations of this orbit. We demonstrate that additionally to the usual border-collision bifurcations occurring at finite points in the state space there exist also border-collision bifurcations occurring at infinity.     

Trace Formulas for Stochastic Evolution Operators: Weak Noise Perturbation Theory

Periodic orbit theory is all effective tool for the analysis of classical and quantum chaotic systems. In this paper we extend this approach to stochastic systems, in particular to mappings with additive noise. The theory is cast in the standard field-theoretic formalism and weak noise perturbation theory written in terms of Feynman diagrams. The result is a stochastic analog of the next-to-leading ? corrections to the Gutzwiller trace formula, with long-time averages calculated from periodic orbits of the deterministic system. The perturbative corrections are computed analytically and tested numerically on a simple 1-dimensional system.     

Parametric excitation of a 1D gas in integrable and nonintegrable cases

We study the response of a highly excited 1D gas with pointlike interactions to a periodic modulation of the coupling constant. We calculate the corresponding dynamic structure factors and show that their low-frequency behavior differs dramatically for integrable and nonintegrable models. Nonintegrable systems are sensitive to excitations with frequencies as low as the mean level spacing, whereas much higher frequencies are required to excite an integrable system. This effect can be used as a probe of integrability for mesoscopic 1D systems and can be observed experimentally by measuring the heating rate of a parametrically excited gas.     

Exact Solutions of the Coupled KdV system via a Formally Variable Separation Approach

Most of the nonlinear physics systems are essentially nonintegrable.There in no very doog analytical approach to solve nonintegrable system.The variable separation approach is a powerful method in linear physics.In this letter,the formal variable separation approach is established to solve the generalized nonlinear mathematical physics equation.The method is valid not only for integrable models but also for nonintegrable models.Taking a nonintegrable coupled KdV equation system as a simple example,abundant solitary wave solutions and conoid wave solutions are revealed.     

Bohmian Mechanics,the Quantum-Classical Correspondence and the Classical Limit: The Case of the Square Billiard

Square billiards are quantum systems complying with the dynamical quantum-classical correspondence. Hence an initially localized wavefunction launched along a classical periodic orbit evolves along that orbit, the spreading of the quantum amplitude being controlled by the spread of the corresponding classical statistical distribution. We investigate wavepacket dynamics and compute the corresponding de Broglie-Bohm trajectories in the quantum square billiard. We also determine the trajectories and statistical distribution dynamics for the equivalent classical billiard. Individual Bohmian trajectories follow the streamlines of the probability flow and are generically non-classical. This can also hold even for short times, when the wavepacket is still localized along a classical trajectory. This generic feature of Bohmian trajectories is expected to hold in the classical limit. We further argue that in this context decoherence cannot constitute a viable solution in order to recover classicality.     

On chaotic dynamics in "pseudobilliard" Hamiltonian systems with two degrees of freedom

A new class of Hamiltonian dynamical systems with two degrees of freedom is studied, for which the Hamiltonian function is a linear form with respect to moduli of both momenta. For different potentials such systems can be either completely integrable or behave just as normal nonintegrable Hamiltonian systems with two degrees of freedom: one observes many of the phenomena characteristic of the latter ones, such as a breakdown of invariant tori as soon as the integrability is violated; a formation of stochastic layers around destroyed separatrices; bifurcations of periodic orbits, etc. At the same time, the equations of motion are simply integrated on subsequent adjacent time intervals, as in billiard systems; i.e., all the trajectories can be calculated explicitly: Given an initial data, the state of the system is uniquely determined for any moment. This feature of systems in interest makes them very attractive models for a study of nonlinear phenomena in finite-dimensional Hamiltonian systems. A simple representative model of this class (a model with quadratic potential), whose dynamics is typical, is studied in detail. (c) 1997 American Institute of Physics.     

Quantum integrability is not a trivial consequence of classical integrability

We discuss the construction of quantum mechanical commuting quantities when the classical ones are known. It is shown that the simple correspondence rules proposed so far do not always work. A candidate for a classically integrable quantum mechanically nonintegrable two-dimensional system is given.     

Quantum-classical correspondence in the wave functions of andreev billiards

We present a classical and quantum mechanical study of an Andreev billiard with a chaotic normal dot. We demonstrate that the nonexact velocity reversal and the diffraction at the edges of the normal-superconductor contact render the classical dynamics of these systems mixed indicating the limitations of a widely used retracing approximation. We point out the close relation between the mixed classical phase space and the properties of the quantum states of Andreev billiards, including periodic orbit scarring and localization of the wave function onto other classical phase space objects such as intermittent regions and quantized tori.     

Shape changing and accelerating solitons in the integrable variable mass sine-gordon model

The sine-Gordon model with a variable mass (VMSG) appears in many physical systems, ranging from the current through a nonuniform Josephson junction to DNA-promoter dynamics. Such models are usually nonintegrable with solutions found numerically or perturbatively. We construct a class of VMSG models, integrable at both the classical and the quantum levels with exact soliton solutions, which can accelerate and change their shape, width, and amplitude simulating realistic inhomogeneous systems at certain limits.     

混沌体系中寻找周期轨迹的方法

一个不可积混沌体系，由于扰动而遭到破坏时，存活的周期轨迹体现了体系的本质特征，是 体系的运动骨架.在一定程度上， 可以由周期轨迹来量子化不可积体系，这充分说明了 周期轨迹的重要性.而寻找周期轨迹，也就成为研究混沌体系动力学特性以及对混沌体系进 行量子化的关键问题.结合具体实例，给出了3种常用的寻找周期轨迹方法，并详细探讨了各 种方法的优缺点和适用范围. 关键词： 周期轨迹 数值方法 混沌     

复杂势场量子弹球中疤痕态的量子化条件

量子疤痕是波函数在经典不稳定周期轨道周围反常凝聚的一种量子或波动现象.人们对疤痕态的量子化条件进行了大量研究,对深入理解半经典量子化起到了一定的促进作用.之前大部分研究工作主要集中在硬墙量子弹球上,即给定边界形状的无穷深量子势阱系统.本文研究具有光滑复杂势场的二维量子弹球系统,考察疤痕态的量子化条件及其重复出现的规律,得到了与硬墙弹球不一样的结果,对理解这类现象是一个有益的补充.这些结果将有助于理解具有无规长程杂质分布的二维电子系统的态密度谱和输运行为.