Random matrix theory of quantum transport in chaotic cavities with nonideal leads

Abstract

We determine the joint probability density function (JPDF) of reflection eigenvalues in three Dyson's ensembles of normal-conducting chaotic cavities coupled to the outside world through both ballistic and tunnel point contacts. Expressing the JPDF in terms of hypergeometric functions of matrix arguments (labeled by the Dyson index β), we further show that reflection eigenvalues form a determinantal ensemble at β = 2 and a new type of a Pfaffian ensemble at β = 4. As an application, we derive a simple analytic expression for the concurrence distribution describing production of orbitally entangled electrons in chaotic cavities with tunnel point contacts when time reversal symmetry is preserved.

Distribution of the Wigner–Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases

“…Our starting point is the joint distribution of the reflection eigenvalues {R n } n=1,...,N for a chaotic cavity perfectly coupled to N channels and N φ > N other channels via a tunnel barrier T [39]: (53) where N t = βN/2 and R n = 1 for n > N . There is a duality between the cases β = 1 and β = 4: the distribution of the reflection eigenvalues for β = 1 is given in terms of an integral for β = 4, and vice-versa.…”

“…In Section 2.2 we obtain the joint distribution of eigenvalues (4), valid in the three symmetry classes, starting from the results of Refs. [39,67]. We show in Section 2.3 how the Coulomb gas method can be adapted to handle the distribution (4) in the limit of many open channels.…”

“…Besides providing an alternative derivation to the one given in Refs. [39,67], the procedure described in Section 2.1 has the advantage to consider the full reflection matrix r (eigenvalues and eigenvectors), and could in principle be extended to a situation where the N real channels are not equivalent.…”

Semiclassical treatment of quantum chaotic transport with a tunnel barrier

“…In the perturbative M 1 setting, average and variance of the conductance were obtained in [34], while the average shot-noise was considered in [36] (see also [37]). Exact formulas for the eigenvalue distribution of T , valid for arbitrary M , were derived [38,39] (see also [40]) in terms of hypergeometric functions of matrix argument. This theory was then used in [41] to derive finite-M results for transport moments.…”

Tratamento semiclássico do transporte quântico caótico com uma barreira de tunelamento

“…M arbitrário e apenas uma guia não ideal, foram derivadas em [50,51] por Vidal e Kanzieper em termos de matrizes com elementos sendo funções hipergeométricas, no caso sem simetria de reversão temporal.…”