J. Phys. I France
Volume 5, Numéro 7, July 1995
Page(s) 877 - 906
DOI: 10.1051/jp1:1995111
J. Phys. I France 5 (1995) 877-906

Brownian Motion Ensembles and Parametric Correlations of the Transmission Eigenvalues: Application to Coupled Quantum Billiards and to Disordered Wires

Klaus Frahm and Jean-Louis Pichard

Service de Physique de l'État Condensé, CEA Saclay, 91191 Gif-sur-Yvette, France

(Received 20 November 1994, accepted 7 March 1995)

The parametric correlations of the transmission eigenvalues Ti of a N-channel quantum scatterer are calculated assuming two different Brownian motion ensembles. The first one is the original ensemble introduced by Dyson and assumes an isotropic diffusion for the S-matrix. We derive the corresponding Fokker-Planck equation for the transmission eigenvalues, which can be mapped for the unitary case onto an exactly solvable problem of N non-interacting fermions in one dimension with imaginary time. We recover for the Ti the same universal parametric correlation than the ones recently obtained for the energy levels, within certain limits. As an application, we consider transmission through two chaotic cavities weakly coupled by a n-channel point contact when a magnetic field is applied. The S-matrix of each chaotic cavity is assumed to belong to the Dyson circular unitary ensemble (CUE) and one has a $2\times {\rm CUE}\to$ one CUE crossover when n increases. We calculate all types of correlation functions for the transmission eigenvalues Ti and we get exact finite N results for the averaged conductance $\langle g \rangle$ and its variance $\langle \delta g^2 \rangle$, as a function of the parameter n. The second Brownian motion ensemble assumes for the transfer matrix M an isotropic diffusion yielded by a multiplicative combination law. This model is known to describe a disordered wire of length L and gives another Fokker-Planck equation which describes the L-dependence of the Ti. An exact solution of this equation in the unitary case has recently been obtained by Beenakker and Rejaei, which gives their L-dependent joint probability distribution. Using this result, we show how to calculate all types of correlation functions, for arbitrary L and N. This allows us to get an integral expression for the average conductance which coincides in the limit $N\to \infty$ with the microscopic non linear $\sigma$-model results obtained by Zirnbauer et al., establishing the equivalence of the two approaches. We review the qualitative differences between transmission through two weakly coupled quantum dots and through a disordered line and we discuss the mathematical analogies between the Fokker-Planck equations of the two Brownian motion models.

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