Statistical Properties of Response Functions in Quantum Chaotic Systems

研究了量子混沌系统响应函数的统计性质，主要涉及以下两个方面：（1）Berry几何混沌与量子混沌的相关性当哈密顿量中的参数发生绝热变化时，瞬时本征函数获得一个几何原点相位（Berry相位）。这个相位可以用磁通量的形式来表示，磁通量是由绝热本征能在参数空间中的偶然简并所产生的假想“磁场”引起的。量子力学中的偶然简并性对应于经典力学中的不可积性。不可积性反过来又意味着混沌行为。因此，量子Berry相位和经典混沌之间可能存在关联。使用一个简单的模型哈密顿量的数值模拟证明了这种相关性：“磁场”在混沌区域中显示出很大的变化，而在规则区域中可以忽略不计。这是一个有趣 ...更多信息 Berry相在混沌区的行为是否像能级统计那样具有普适性，这是未来的问题。(2)随机矩阵模型在量子混沌散射中的有效性电导（响应函数的一个例子）在介观系统中用散射矩阵表示。用各种随机矩阵代替较复杂的原始哈密顿量，从理论上研究了散射矩阵的统计性质。然而，在散射随机矩阵模型的有效性程度是不清楚的，因为在散射区域的停留时间短的轨迹可能有助于散射矩阵以及长周期遍历轨迹。为了阐明这一点，用准一维紧束缚安德森模型和随机矩阵模型的数值模拟结果进行了比较。比较表明，随着散射区与外部区域之间的有效耦合强度的减弱，计算结果的一致性更好。如何将随机矩阵所描述的普遍行为与单个系统的特定特征分离开来，是一个引人注目的未来问题。少

Statistical properties of response functions in quantum chaotic systems have been investigated concerning the following two points :(1) Corrrelation between Berry's geometric and quantum chaosAn instantaneous eigenfunction acquires a phase of geometrical origin (Berry's phase) during the adiabatic change of parameters included in hamiltonian. This phase can be expressed in a form of flux due to fictious "magnetic fields" originated from acccidental degeneracies of adiabatic eigenenergies in a parameter space. The acccidental degeneracies in quantum mechanics corresponds to non-integrability in classical mechanics. The non-integrability, in turn, implies chaotic behavior. Hence, the possible correlation is expected between quantal Berry's phase and classical chaos. Numerical simulations with use of a simple model hamiltonian demonstrated this correlation : The "magnetic fields" show large variations in a chaotic region while they are negligible in a regular region. It is an interesting … More future problem whether the behavior of Berry's phase in chaotic regions show universalities as the case of energy level statistics.(2) Validity of random matrix models in quantum chaotic scatteringsConductance (one example of response functions) is expressed in terms of scattering matrices in mesoscopic systems. Statistical properties of scattering matrices have been theoretically studied with use of various kind of random matrices in place of more complicated original hamiltonians. However the degree of validity of random matrix models in scaterings is not clear becasue trajectories with short dwell time in a scattering region are likely to contribute to scattring matrices as well as long period ergodic trajectories. In order to clarify this point, results of numercial simulations with use of a quasi-one-dimensional tight binding Anderson model are compared with those with randam matrix models. The comparison shows the better agreement as the effective coupling strength between a scatering region and external regions become weaken. How possible universal behavior described by random matrices can be separated from a specific feature of individual systems are a compelling future problem. Less