Statistics of time delay and scattering correlation functions in chaotic systems. I. Random matrix theory

Novaes, Marcel

doi:10.1063/1.4922746

Cited by 26 publications

(28 citation statements)

References 33 publications

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“…It is worth emphasizing the different scalings of the variances with N : A systematic expression of the moments τ n W of the Wigner time delay for β = 2 was given more recently by Novaes [152]. We see that the variance diverges for N β 2, and the third cumulant for N β 4.…”

Section: Wigner Time Delay τ Wmentioning

confidence: 68%

Wigner time delay and related concepts: Application to transport in coherent conductors

Texier

2016

Physica E: Low-dimensional Systems and Nanostructures

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The concepts of Wigner time delay and Wigner-Smith matrix allow to characterize temporal aspects of a quantum scattering process. The article reviews the statistical properties of the Wigner time delay for disordered systems ; the case of disorder in 1D with a chiral symmetry is discussed and the relation with exponential functionals of the Brownian motion underlined. Another approach for the analysis of time delay statistics is the random matrix approach, from which we review few results. As a pratical illustration, we briefly outline a theory of non-linear transport and AC transport developed by Büttiker and coworkers, where the concept of Wigner-Smith time delay matrix is a central piece allowing to describe screening properties in out-of-equilibrium coherent conductors.The published version has been updated.

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Section: Wigner Time Delay τ Wmentioning

confidence: 68%

Wigner time delay and related concepts: Application to transport in coherent conductors

Texier

2016

Physica E: Low-dimensional Systems and Nanostructures

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“…(other references on time delay correlations are [31,43,16], and can also be found in the review papers quoted above). The joint distribution for the partial time delaysτ i 's and the diagonal matrix elements Q ii 's are still unknown.…”

Section: Chaotic Scattering In Quantum Dots and Random Matricesmentioning

confidence: 99%

Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots

2017

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Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues {λ 1 , · · · , λ N }. We study the distribution of truncated linear statistics of the formL = p i=1 f (λ i ) with p < N . This problem has been considered by us in a previous paper when the p eigenvalues are further constrained to be the largest ones (or the smallest). In this second paper we consider the same problem without this restriction which leads to a rather different analysis. We introduce a new ensemble which is related, but not equivalent, to the "thinned ensembles" introduced by Bohigas and Pato. This question is motivated by the study of partial sums of proper time delays in chaotic quantum dots, which are characteristic times of the scattering process. Using the Coulomb gas technique, we derive the large deviation function forL. Large deviations of linear statistics L =

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“…The matrix generalisation of the latter approach to arbitrary N > 1 was presented in the influential work [36,37] by Brouwer, Frahm, and Beenakker (BFB) who showed that the proper time delays (more precisely, their inverses) are distributed according to the Laguerre ensemble of random matrices. This provided a route for applying powerful RMT techniques (like orthogonal polynomials and the Coulomb gas method) to study various densities, moments and correlators built on the Wigner-Smith matrix [21,44,45,46,47,48,49,50,51,52,53,54]. We refer to Ref.…”

Section: Perfect Coupling T =mentioning

confidence: 99%

Wigner–Smith time-delay matrix in chaotic cavities with non-ideal contacts

Grabsch¹,

Savin²,

Texier³

2018

J. Phys. A: Math. Theor.

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We consider wave propagation in a complex structure coupled to a finite number N of scattering channels, such as chaotic cavities or quantum dots with external leads. Temporal aspects of the scattering process are analysed through the concept of time delays, related to the energy (or frequency) derivative of the scattering matrix S. We develop a random matrix approach to study the statistical properties of the symmetrised Wigner-Smith time-delay matrix Q s = −i S −1/2 ∂ ε S S −1/2 , and obtain the joint distribution of S and Q s for the system with non-ideal contacts, characterised by a finite transmission probability (per channel) 0 < T 1. We derive two representations of the distribution of Q s in terms of matrix integrals specified by the Dyson symmetry index β = 1, 2, 4 (the general case of unequally coupled channels is also discussed). We apply this to the Wigner time delay τ W = (1/N ) tr Q s , which is an important quantity providing the density of states of the open system. Using the obtained results, we determine the distribution P N,β (τ ) of the Wigner time delay in the weak coupling limit N T 1 and identify the following three regimes. (i) The large deviations at small times (measured in units of the Heisenberg time) are characterised by the limiting behaviour P N,β (τ ) ∼ τ −βN 2 /2−3/2 exp − βN T /(8τ ) for τ T .(ii) The distribution shows the universal τ −3/2 behaviour in some intermediate range T τ 1/(T N 2 ). (iii) It has a power law decay P N,β (τ ) ∼ T 2 N 3 (T N 2 τ ) −2−βN/2 for large τ 1/(T N 2 ).

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Statistics of time delay and scattering correlation functions in chaotic systems. I. Random matrix theory

Cited by 26 publications

References 33 publications

Wigner time delay and related concepts: Application to transport in coherent conductors

Wigner time delay and related concepts: Application to transport in coherent conductors

Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots

Wigner–Smith time-delay matrix in chaotic cavities with non-ideal contacts

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