Abstract
We study the Wigner–Smith time-delay matrix Q of a ballistic quantum dot supporting N scattering channels. We compute the v-point correlators of the power traces for arbitrary at leading order for large N using techniques from the random matrix theory approach to quantum chromodynamics. We conjecture that the cumulants of the 's are integer-valued at leading order in N and include a MATHEMATICA code that computes their generating functions recursively.
Highlights
The proper delay times t1, 1⁄4, tN are the eigenvalues of the Wigner–Smith time-delay matrix Q, which is defined in term of the scattering matrix S (E) by [19, 40, 46]
The Wigner–Smith matrix provides an overall description of the dynamical aspects of a scattering process in terms of the phase shift between a scattered wavefunction and a freely
Ballistic quantum dots provide a physical realization of systems whose scattering matrix S (E) is effectively modeled by a random unitary matrix [5], where N is the total number of scattering channels
Summary
The proper delay times t1, 1⁄4, tN are the eigenvalues of the Wigner–Smith time-delay matrix Q, which is defined in term of the scattering matrix S (E) by [19, 40, 46]. The joint probability law of the rescaled inverse delay times li = (Nti)-1 was derived in [11] when the scattering matrix S (E) belongs to the Dysons circular ensembles [18] and was generalized in [29] for systems whose symmetries belong to the classification introduced by Altland and Zirnbauer [2, 48]. It is known in random matrix theory (RMT) as the eigenvalue distribution of a Wishart–Laguerre matrix ensemble (equations (31) and (32) below). The paper is complemented by three appendices where we collect a few properties of cumulants (appendix A), we present the derivation of some identities involved in the iterative scheme for the Greens functions (appendix B), and we include a MATHEMATICA code to implement in a systematic way the generating functions (appendix C)
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