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The M-dimensional unitary matrix S(E), which describes scattering of waves, is a strongly fluctuating function of the energy for complex systems such as ballistic cavities, whose geometry induces chaotic ray dynamics. Its statistical behaviour can be expressed by means of correlation functions of the kind 〈Sij(E + ε)Spq†(E - ε)〉, which have been much studied within the random matrix approach. In this work, we consider correlations involving an arbitrary number of matrix elements and express them as infinite series in 1/M, whose coefficients are rational functions of ε. From a mathematical point of view, this may be seen as a generalization of the Weingarten functions of circular ensembles. [ABSTRACT FROM AUTHOR]

We consider the statistics of time delay in a chaotic cavity having M open channels, in the absence of time-reversal invariance. In the random matrix theory approach, we compute the average value of polynomial functions of the time delay matrix Q = -iħS†dS/dE, where S is the scattering matrix. Our results do not assume M to be large. In a companion paper, we develop a semiclassical approximation to S-matrix correlation functions, from which the statistics of Q can also be derived. Together, these papers contribute to establishing the conjectured equivalence between the random matrix and the semiclassical approaches. [ABSTRACT FROM AUTHOR]

We consider S-matrix correlation functions for a chaotic cavity having M open channels, in the absence of time-reversal invariance. Relying on a semiclassical approximation, we compute the average over E of the quantities Tr[S†(E - ε)S(E + ε)]n, for general positive integer n. Our result is an infinite series in ., whose coefficients are rational functions of M. From this, we extract moments of the time delay matrix Q = -iħS†dS/dE and check that the first 8 of them agree with the random matrix theory prediction from our previous paper [M. Novaes, J. Math. Phys. 56, 062110 (2015)]. [ABSTRACT FROM AUTHOR]

Wave scattering in chaotic systems can be characterized by its spectrum of resonances, z&ldquor; = E&ldquor; -- i f, where E&ldquor; is related to the energy and T&ldquor; is the decay rate or width of the resonance. If the corresponding ray dynamics is chaotic, a gap is believed to develop in the large-energy limit: almost all T&ldquor; become larger than some y. However, rare cases with F < y may be present and actually dominate scattering events. We consider the statistical properties of these supersharp resonances. We find that their number does not follow the fractal Weyl law conjectured for the bulk of the spectrum. We also test, for a simple model, the universal predictions of random matrix theory for density of states inside the gap and the hereby derived probability distribution of gap size. [ABSTRACT FROM AUTHOR]

We investigate the transition from integrability to chaos in a system built of usp(4) elements, both in the quantum case and in its classical limit, obtained using coherent states. This algebraic Hamiltonian consists in an integrable term plus a nonlinear perturbation, and we see that the level spacing distribution for the quantum system is well approximated by the Berry–Robnik–Brody distribution, and accordingly the classical limit displays mixed dynamics. [Copyright &y& Elsevier]

We investigate entanglement production in a generic normal conductor, connected to two single-channel left leads and two single-channel right leads. We consider the joint statistics of the number of entangled pairs produced during a given time and amount of entanglement carried by them. We find a constraint that implies that more entangled states are less likely to be detected. Namely, if C is the concurrence and N is the squared norm of the entangled state, then production occurs only if N(1+C)<1. For the particular case of a chaotic cavity working as quantum entangler, we obtain explicit expressions for the joint distribution of C and N, both for systems with and without time-reversal symmetry. [ABSTRACT FROM AUTHOR]

We consider statistics of electronic transport in chaotic cavities where time-reversal symmetry is broken and one of the leads is weakly nonideal; that is, it contains tunnel barriers characterized by tunneling probabilities Γ i . Using symmetric function expansions and a generalized Selberg integral, we develop a systematic perturbation theory in 1-Γ i valid for an arbitrary number of channels and obtain explicit formulas up to second order for the average and variance of the conductance and for the average shot noise. Higher moments of the conductance are considered to leading order. [ABSTRACT FROM AUTHOR]