Quantum Mechanical Time-Delay Matrix in Chaotic Scattering

Abstract

We calculate the probability distnbution of the matnx Q = -ihS 1 3S/3E for a chaotic System with scattenng matnx S at energy E The eigenvalues τ, of Q are the so-called proper delay times, mtroduced by Wigner and Smith to descnbe the time dependence of a scattenng process The distnbution of the inverse delay times turns out to be given by the Laguerre ensemble from random matnx theory [S0031 -9007(97) [8][9][10] evidence that an ensemble of chaotic bilhards contammg a small openmg (through which N modes can propagate at energy E) has a uniform distnbution of S in the group of N X N umtary matrices-restncted only by fundamental symmetnes This universal distnbution is the circular ensemble of random-matnx theory [11], mtroduced by Dyson for its mathematical simphcity [12] The eigenvalues e"^ of S m the circular ensemble are distributed accordmg to The solution of this 40 year old problem is presented here We have found that the eigenvalues of Q are independent of S [26] The distnbution of the inverse delay times y" = l/r" turns out to be the Laguerre ensemble of random-matnx theory,(2) but with an unusual ,/V-dependent exponent (The function P is zero if any one of the T"'S is negative ) The correlation functions of the T"'S consist of senes over (generahzed) Laguene polynomials [27], hence the name "Laguerre ensemble " The eigenvectors of Q are not independent of S, unless β = 2 (which is the case of broken time-reversal symmetry) However, for any β the conelations can be transformed away if we replace Q by the symmetrized matnx (3) which has the same eigenvalues äs Q The matnx of eigenvectors U which diagonahzes QE = U X diag(ri, ,r N )U^ is independent of S and the T"'S, and umformly distributed m the orthogonal, umtary, or symplectic group (for β = l, 2, or 4, respectively) The distnbution (2) confirms the conjecture by Fyodorov and Sommers [19] that the distnbution of tr Q has an algebraic Although the time-delay matnx was interpreted by Smith äs a representation of the "time operator," this Interpretation is ambiguous [19] The ambiguity anses because a wave packet has no well-defmed energy There is no ambiguity m the apphcation of Q to transport Problems where the mcommg wave can be regarded monochromatic, hke the low-frequency response of a chaotic cavity [21,22,28]

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

“…In recent years, matrix models whose weight function has an essential singularity like (1.5) have appeared in several different areas of mathematics and physics; see, e.g., Berry and Shukla [1] in the study of statistics for zeros of the Riemann zeta function, Lukyanov [28] in a calculation of finite temperature expectation values in integrable quantum field theory, and [4,32,38] in the study of the Wigner time delay in quantum transport. Wigner delay time stands for the average time that an electron spends when scattered by an open cavity and is of fundamental importance in the theory of mesoscopic quantum dots.…”

“…In the Laguerre ensemble, the Wigner time delay is given by the sum of random variables τ j such that 1/τ j are distributed like the eigenvalues of matrices; see [4,38]. The partition function, i.e., the quantity defined in (1.2), serves as the moment generating function of the probability density of the Wigner delay time [3,32].…”

Limits of slow light in photonic crystals

“…A formal proof can be given in terms of Wigner-Smith group delay formalism, thus also applying to inhomogeneous structures of finite extension ͑see, e.g., Ref. 23 and references therein͒. Letting g 0 denote the one-dimensional projected density of states along the propagation direction of a homogeneous material with a dielectric function given as the average value of the dielectric function of the photonic crystal, we identify three different regimes of interest: ͑i͒ a longwavelength regime, where the properties of the photonic crystal are independent of the detailed geometrical composition, and thus g͑ ͒Ӎg 0 , ͑ii͒ a slow-light regime, where g͑ ͒ Ͼ g 0 , and ͑iii͒ a "superluminal" regime, where g͑ ͒ Ͻ g 0 .…”

Imaginary potential as a counter of delay time for wave reflection from a one-dimensional random potential

“…It is this 'equilibrated' part of the reflected wave, and not the prompt part that is expected to give universality. Hence the universal 1/τ 2 tail in equation (4). Indeed, the universality of the delay-time distribution directly reflects that of the reflection coefficient given by equation(3) [17][18][19].…”

“…Thus we have the random matrix theory (RMT) for circular ensembles of the S-matrix giving delay times for all the three Dyson Universality classes for the case of a chaotic cavity connected to a single open channel [3]. Generalization to the case of N channels corresponded to the Laguarre ensemble [4] of RMT. The RMT approach has been treated earlier through the supersymmetric technique for the case of a quantum chaotic cavity having a few equivalent open channels [5].…”