Random Matrix Theory in Chaotic Scattering: Symplectic Symmetry and Time Reversal Invariance Breaking

Scattering theory is a key tool for the investigation of quantum systems. As analytical methods are unfeasible in many cases, universal behaviour is of interest and a statistical approach is called for. This is commonly carried out within the Heidelberg approach, which models the Hamilton operator of the scattering target by a random matrix. In this work, we derive exact expressions for the distributions of the real and imaginary parts of the off-diagonal scattering matrix elements and cross sections in systems, which are time reversal invariant and have half-integer spin but no further symmetries. The corresponding Random Matrix Theory ensemble is the Gaussian Symplectic Ensemble. Historically, such systems have been the least accessible out of Dyson's threefold way. This is no longer the case, and a multitude of recent experiments have been able to observe scattering phenomena in systems described by the Gaussian Symplectic Ensemble. Our results complete the derivation of the distributions for all classes of Dyson's threefold way. Remarkably, we find that all spin orientations and the real and imaginary parts are equally distributed. Furthermore, these results unveil the exact parameter dependencies of such systems. We provide a modernised and constructive approach to some of the mathematical aspects necessary for such calculations, and draw connections to the physical properties of the systems. Up until now, measurements of off-diagonal scattering matrix elements or cross sections for the Gaussian Symplectic Ensemble have not been carried out. In this work, we present an experimental setup capable of measuring such data. We find that the experimental data does not fit our analytical results, and show that these discrepancies are a result of the experimental setup. Hence, we also propose changes to the setup and subsequently find great agreement of numerical simulations and our exact results. Additionally, we further validate our results through comparison with Monte-Carlo simulations. We find excellent agreement in all cases. In many applications, systems neither display intact nor fully broken time reversal invariance. The study of these systems has a long history in nuclear physics and Random Matrix Theory. We introduce transitional ensembles, which interpolate between the Gaussian Orthogonal Ensemble or Gaussian Symplectic Ensemble and the Gaussian Unitary Ensemble. Using these ensembles, we obtain exact results, which enable the investigation of time reversal invariance breaking at the level of the statistics of a scattering matrix element. Thus, it not only allows for a more sophisticated analysis of experiments, as previously only the first few moments or correlators were derived exactly, but also does not require the testing for detailed balance. For systems corresponding to the Gaussian Symplectic Ensemble, we also break time reversal invariance in the channels through a Zeeman-like splitting of the spin orientations. We again derive expressions for the aforementioned distributions in the case of weak splitting. As a consequence, the spin orientations are no longer distributed equally.