In many situations, the statistical properties of wave systems with chaotic classical limits are well described by random matrix theory. However, applications of random matrix theory to scattering problems require introduction of system-specific information into the statistical model, such as the introduction of the average scattering matrix in the Poisson kernel. Here, it is shown that the average impedance matrix, which also characterizes the system-specific properties, can be expressed in terms of classical trajectories that travel between ports and thus can be calculated semiclassically. Theoretical results are compared with numerical solutions for a model wave chaotic system.

Abstract:Statistical fluctuations in the eigenvalues of the scattering, impedance and admittance matrices of 2-Port wave-chaotic systems are studied experimentally using a chaotic microwave cavity. These fluctuations are universal in that their properties are dependent only upon the degree of loss in the cavity. We remove the direct processes introduced by the non-ideally coupled driving ports through a matrix-normalization process that involves the radiation-impedance matrix of the two driving ports. We find good agreement between the experimentally obtained marginal probability density functions (PDFs) of the eigenvalues of the normalized impedance, admittance and scattering matrix and those from Random Matrix Theory (RMT). We also experimentally study the evolution of the joint PDF of the eigenphases of the normalized scattering matrix as a function of loss. Experimental agreement with the theory by Brouwer and Beenakker for the joint PDF of the magnitude of the eigenvalues of the normalized scattering matrix is also shown. Of particular interest is the case when the ray trajectories within the enclosure show chaotic dynamics in the classical limit. This interest has spawned the field of "wave chaos" (or "quantum chaos"), and has attracted much theoretical and experimental work [6, 7] to understand its nature. On account of the small wavelength of the scattered waves, as compared to the characteristic length-scale of the enclosure, the response of these systems exhibit extreme sensitivity to small changes in configuration, driving frequency, nature of driving ports, ambient conditions such as temperature, etc. Thus, an intimate knowledge of the response of any such system for a given well-defined stimulus or system configuration will not provide any foresight in predicting the response of a similar system when the stimulus or system configuration is slightly altered. This calls for a statistical approach to quantify the nature of such wave-chaotic systems. PACSIn this regard, Random Matrix Theory (RMT) [8] has proved to be an integral tool in predicting universal statistical aspects of wave chaotic systems. It has been conjectured that in the short-wavelength regime, RMT can be used to model wave-chaotic systems. In particular, the statistics of systems that show Time-Reversal Symmetry are conjectured to be described by the Gaussian Orthogonal Ensemble (GOE) of random matrices, while the statistics of systems showing Broken Time-Reversal Symmetry are conjectured to be described by the Gaussian Unitary Ensemble (GUE) of random matrices. There is also a third random matrix ensemble corresponding to certain systems with spininteractions (Gaussian Symplectic Ensemble). RMT provides a potential framework for uncovering universal statistical properties of short-wavelength wave chaotic systems (e.g. Ericson fluctuations in nuclear scattering [7, 9] and universal conductance fluctuations (UCF) in quantum-transport systems [10] ).Since the applicability of RMT and the concomitant universal statistics is conjectural ra...

Prediction of the statistics of scattering in typical wave-chaotic systems requires combining system-specific information with universal aspects of chaotic scattering as described by random matrix theory. This Rapid Communication shows that the average impedance matrix, which characterizes such system-specific properties, can be semiclassically calculated in terms of ray trajectories between ports. Theoretical predictions are compared with experimental results for a microwave billiard, demonstrating that the theory successfully uncovered universal statistics of wave-chaotic scattering systems.

Predicting the statistics of realistic wave-chaotic scattering systems requires, in addition to random matrix theory, introduction of system-specific information. This paper investigates experimentally one aspect of system-specific behavior, namely, the effects of short ray trajectories in wave-chaotic systems open to outside scattering channels. In particular, we consider ray trajectories of limited length that enter a scattering region through a channel (port) and subsequently exit through a channel (port). We show that a suitably averaged value of the impedance can be computed from these trajectories and that this can improve the ability to describe the statistical properties of the scattering systems. We illustrate and test these points through experiments on a realistic two-port microwave scattering billiard.

Abstract:The mathematical equivalence of the time-independent Schrödinger equation and the Helmholtz equation is exploited to provide a novel means of studying universal conductance fluctuations in ballistic chaotic mesoscopic systems using a two-dimensional microwave-cavity. The classically chaotic ray trajectories within a suitably-shaped microwave cavity play a role analogous to that of the chaotic dynamics of non-interacting electron transport through a ballistic quantum dot in the absence of thermal fluctuations. The microwave cavity is coupled through two single-mode ports and the effect of non-ideal coupling between the ports and cavity is removed by a previously developed method based on the measured radiation impedance matrix. The LandauerBüttiker formalism is applied to obtain the conductance of a corresponding mesoscopic quantum-dot device. We find good agreement for the probability density functions (PDFs) of the experimentally derived surrogate conductance, as well as its mean and variance, with the theoretical predictions of Brouwer and Beenakker. We also observe a linear relation between the quantum dephasing parameter and the cavity ohmic loss parameter.