量子能谱中的长程关联

从可积系统求迹公式出发,运用Einstein-Brillouin-Keller(EBK)量子化条件,导出了二维无关联振子系统周期轨道作用量量子化条件,由此发现了量子能级与周期轨道之间的对应关系.这种对应关系表明,如果两条能级对应的周期轨道的拓扑相同,这两条能级对回归函数的贡献相干.回归谱中的一个峰是量子能谱中一组与具有相同拓扑的周期轨道相对应的能级之间相干的结果,这一组能级间存在着长程关联.     

周期轨道的量子化与量子能谱中的长程关联

用二维可积系统的半经典量子化方案和二维无关联振子系统的量子能级与周期轨道之间的对应关系,讨论了一组量子能级之间具有长程关联的内在机制,在二维无关联振子系统中,发现了具有相同拓扑M(M1,M2)的周期轨道相对应的量子能级之间存在着长程关联,并以二维4次无关联振子系统为例做了具体说明.     

二维可积系统的求迹公式和周期轨道的量子化

从Berry–Tabor求迹公式出发,导出了二维可积系统周期轨道作用量的半经典量子化条件.利用此量子化条件,考虑周期轨道满足的周期条件,得到了二维无关联四次振子系统周期轨道作用量的半经典量子化条件,并给出了半经典能级公式.对能级与周期轨道的对应关系做了分析.     

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.     

等腰直角三角形的二维量子谱和经典轨道

本文研究了等腰直角三角形中基于其波函数和能级结构的二维量子谱．虽然这个量子台球系统的本征态是无法分离的,但是关于两个变量的问题是完全可解的．通过对二维量子系统的波函数做相应的傅里叶变换,得到了系统的二维量子谱,把得到的结果和经典的二维量子台球轨道做相应性的对比发现：傅里叶变换的量子谱的峰值位置和经典轨道的长度之间存在着很好的对应关系,这说明经典计算的结果和量子计算的结果符合得非常好,从而进一步验证了周期轨道理论的正确性．     

Dicke模型的量子混沌和单粒子相干动力学特性

量子化的Dicke模型在非旋波近似条件下表现为量子混沌动力学特征.利用单粒子一阶时间关联函数,通过数值计算详细考察了Dicke模型中单粒子相干动力学特性.结果表明:当初始相干态处在混沌区域时,一阶时间关联函数曲线衰减较快,而当初始相干态处在规则区域时,一阶时间关联函数曲线衰减较慢,单粒子相干动力学对初态具有较强的敏感性,经典混沌抑制量子相干.考察单粒子相干动力学在相空间的平均演化性质,得到一种较好的量子经典对应关系.最后研究了相空间单粒子相干的整体动力学性质,更好地揭示了相空间的混沌和规则结构.     

磁通量对二维谐振子的量子能级密度的影响

利用周期轨道理论,我们计算了在不同情况下,一个粒子在二维谐振子势中存在和不存在磁通量时的量子能级密度.重点讨论了磁通量对量子能级密度的影响.计算结果表明:当二维谐振子势的频率比值是有理数时,量子能级是分立的,能级密度中的每一条峰正好对应一个量子能级.然而,当频率比是无理数时,能级密度发生振荡,当加上磁通量后,振荡减小.这可以看作是Aharonov-Bohm效应的结果.     

二维拓扑绝缘体退相干效应研究

拓扑绝缘体是当前凝聚态物理研究的热点.退相干效应对该体系的影响的研究不仅有重要的理论意义,而且也是实现未来量子器件的不可或缺的前期工作.文章作者从理论上研究了退相干对二维拓扑绝缘体特别是量子自旋霍尔效应的影响.研究结果表明,作为量子自旋霍尔效应的标志的量子化纵向电阻平台对不破坏自旋记忆的退相干效应(普通退相干)不敏感,但却对破坏自旋记忆的退相干效应(自旋退相干)非常敏感.因此,该量子化平台只能在尺寸小于自旋退相干长度的介观样品中存在,从而解释了量子自旋霍尔效应实验中所观测到的结果(见Science,2007,318:766).同时,文章作者还定义了一个新的物理量,即自旋霍尔电阻,并发现该自旋霍尔电阻也有量子化平台.特别是该量子化平台对两种类型的退相干都不敏感.这说明在宏观样品中也能观测到自旋霍尔电阻的量子化平台,因此更能全面地反映量子自旋霍尔效应的拓扑特性.     

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.     

矩形弹子球中的量子波包分析(英文)

利用波包分析量子力学体系的动力学行为在研究经典和量子的对应关系方面越来越成为一个非常重要的方法.利用高斯波包分析方法,我们计算了矩形弹子球体系的自关联函数,自关联函数的峰和经典周期轨道的周期符合的很好,这表明经典周期轨道的周期可以通过含时的量子波包方法产生.我们还讨论了矩形弹子球的波包回归和波包的部分回归,计算结果表明在每一个回归时间,波包出现精确的回归.对于动量为零的波包,初始位置在弹子球内部的特殊对称点处,出现一些时间比较短的附加的回归.     

The positronium negative ion: Classical properties aaand semiclassical quantization

Properties of collinear and planar periodic orbits for the positronium negative ion are examined with respect to the possibilities for semiclassical quantization. In contrast to other two-electron atomic systems as helium and H^- the relevant orbits for quantization are fully stable and permit a full torus quantization. However, for lower excitations the area of stability in phase-space is too small for a reliable torus quantization. Instead, a quasi-separability of the three-body system is used to apply effective one-dimensional (WKB) quantization. Received 19 January 2001     

Use of harmonic inversion techniques in the periodic orbit quantization of integrable systems

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained. Received 24 February 2000 and Received in final form 22 May 2000     

Unstable periodic orbits and templates of the Rossler system: Toward a systematic topological characterization

The Rossler system has been exhaustively studied for parameter values (a in [0.33,0.557],b=2,c=4). Periodic orbits have been systematically extracted from Poincare maps and the following problems have been addressed: (i) all low order periodic orbits are extracted, (ii) encoding of periodic orbits by symbolic dynamics (from 2 letters up to 11 letters) is achieved, (iii) some rules of growth and of pruning of the periodic orbits population are obtained, and (iv) the templates of the attractors are elaborated to characterize the attractors topology. (c) 1995 American Institute of Physics.     

Periodic orbits and binary collisions in the classical three-body Coulomb problem

In the helium case of the classical three-body Coulomb problem in two dimensions with zero angular momentum, we develop a procedure to find periodic orbits applying two symbolic dynamics for one-dimensional and planar problems. Focusing our attention on binary collisions with these tools, a sequence of periodic orbits are predicted and are actually found numerically. A family of periodic orbits found has regularity in their actions. For this family of periodic orbits, it is shown that thanks to its regularity, a partial summation of the Gutzwiller trace formula with a daring approximation gives a Rydberg series of energy levels.     

Scarring by homoclinic and heteroclinic orbits

In addition to the well-known scarring effect of periodic orbits, we show here that homoclinic and heteroclinic orbits, which are cornerstones in the theory of classical chaos, also scar eigenfunctions of classically chaotic systems when associated closed circuits in phase space are properly quantized, thus introducing strong quantum correlations. The corresponding quantization rules are also established. This opens the door for developing computationally tractable methods to calculate eigenstates of chaotic systems.     

One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

The three-body problem can be traced back to Newton in 1687,but it is still an open question today.Note that only a few periodic orbits of three-body systems were found in 300 years after Newton mentioned this famous problem.Although triple systems are common in astronomy,practically all observed periodic triple systems are hierarchical(similar to the Sun,Earth and Moon).It has traditionally been believed that non-hierarchical triple systems would be unstable and thus should disintegrate into a stable binary system and a single star,and consequently stable periodic orbits of non-hierarchical triple systems have been expected to be rather scarce.However,we report here one family of 135445 periodic orbits of non-hierarchical triple systems with unequal masses;13315 among them are stable.Compared with the narrow mass range(only 10-5)in which stable"Figure-eight"periodic orbits of three-body systems exist,our newly found stable periodic orbits have fairly large mass region.We find that many of these numerically found stable non-hierarchical periodic orbits have mass ratios close to those of hierarchical triple systems that have been measured with astronomical observations.This implies that these stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses quite possibly can be observed in practice.Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses.Note that our approach has general meaning:in a similar way,every known family of periodic orbits of three-body systems with two or three equal masses can be used as a starting point to generate thousands of new periodic orbits of triple systems with distinctly unequal masses.     

Semiclassical Poincare map for integrable systems

The semiclassical Poincare map is applied to integrable systems and in particular to the rectangular billiard. The zeroes of the functional determinant are shown to give EBK quantization. The transfer operator is explicitly unitary and finite, resulting in a finite expansion of the Euler product over periodic orbits.