We calculate the probability distribution of the matrix Q = -i \hbar S^{-1}
dS/dE for a chaotic system with scattering matrix S at energy E. The
eigenvalues \tau_j of Q are the so-called proper delay times, introduced by E.
P. Wigner and F. T. Smith to describe the time-dependence of a scattering
process. The distribution of the inverse delay times turns out to be given by
the Laguerre ensemble from random-matrix theory.Comment: 4 pages, RevTeX; to appear in Phys. Rev. Let

A. Kudrolli

B. Eckhardt

C. H. Lewenkopf

C. W. J. Beenakker

E. Akkermans

E. Brézin

E. Doron

E. P. Wigner

F. Izrailev

F. J. Dyson

F. T. Smith

H. L. Harney

J. J. M. Verbaarschot

J. Stein

K. M. Frahm

K. Slevin

M. Bauer

M. Büttiker

M. L. Mehta

N. Lehmann

P. W. Brouwer

P. Šeba

R. A. Jalabert

R. Blümel

R. M. Westervelt

T. Nagao

U. Smilansky

V. A. Gopar

V. L. Lyuboshits

Y. V. Fyodorov

English

arXiv

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Beenakker, C.W.J.

Brouwer, P.W.

Frahm, K.M.

Leiden University Scholary Publications

VOLUME 78, NUMBER 25 P H Y S I C A L R E V I E W L E I T E R S 23 JUNE 1997Quantum Mechanical Time-Delay Matrix in Chaotic ScatteringP W Brouwer, K M Prahm,* and C W J BeenakkerInstituut-Lorentz Leiden Umversity, P O Box 9506, 2300 RA Leiden, The Netherlands(Received 21 March 1997)We calculate the probability distnbution of the matnx Q = —ihS 13S/3E for a chaotic Systemwith scattenng matnx S at energy E The eigenvalues τ, of Q are the so-called proper delaytimes, mtroduced by Wigner and Smith to descnbe the time dependence of a scattenng process Thedistnbution of the inverse delay times turns out to be given by the Laguerre ensemble from randommatnx theory [S0031 -9007(97)03411 -X]PACS numbers 05 45 +b, 03 65 Nk, 42 25 Bs 73 23 -bEisenbud [1] and Wigner [2] mtroduced the notion oftime delay m a quantum mechanical scattermg problemWigner's one-dimensional analysis was generahzed to anN X N scattermg matnx S by Smith [3], who studiedthe Hermitian energy derivative Q = —iftS~ldS/dE andmterpreted its diagonal elements äs the delay time for awave packet incident m one of the W scattermg channelsThe matnx Q is called the Wigner-Smith time-delaymatnx and its eigenvalues τι,τζ, , τ Ν are called properdelay timesRecently, interest in the time-delay problem was revivedm the context of chaotic scattermg [4] There is consider-able theoretical [4-7] and expenmental [8-10] evidencethat an ensemble of chaotic bilhards contammg a smallopenmg (through which N modes can propagate at energyE) has a uniform distnbution of S in the group of N X Numtary matrices—restncted only by fundamental symme-tnes This universal distnbution is the circular ensembleof random-matnx theory [11], mtroduced by Dyson for itsmathematical simphcity [12] The eigenvalues e"^ of Sm the circular ensemble are distributed accordmg toΡ(φι,φι Π \e'* ~ e<* \ß, (1)with the Dyson mdex β = l, 2, 4 dependmg on thepresence or absence of time-reversal and spm-rotationsymmetryNo formula of such generahty is known for the time-delay matnx, although many authors have worked on thisproblem [6,13-23] An early result, {tr Q) = TH, is due toLyuboshits [13], who equated the ensemble average of thesum of the delay times tr Q = X^=] r„ to the Heisenbergtime ΤΗ = 2ττ·/ζ/Δ (with Δ the mean level spacmg ofthe closed System) The second moment of trß wascomputed by Lehmann et al [18] and by Fyodorov andSommers [19] The distnbution of Q itself is not known,except for N = l [19,21] The trace of Q deterrmnesthe density of states [24], and is therefore sufficient formost thermodynamic applications [21] For apphcationsto quantum transport, however, the distnbution of allmdividual eigenvalues T„ of Q is needed, äs well äs thedistnbution of the eigenvectors [25]The solution of this 40 year old problem is presentedhere We have found that the eigenvalues of Q are inde-pendent of S [26] The distnbution of the inverse delaytimes y„ = l/r„ turns out to be the Laguerre ensembleof random-matnx theory,Ρ(Ύι, ,Ύκ) «'<} k(2)but with an unusual ,/V-dependent exponent (The functionP is zero if any one of the T„'S is negative ) The corre-lation functions of the T„'S consist of senes over (gen-erahzed) Laguene polynomials [27], hence the name"Laguerre ensemble " The eigenvectors of Q are notindependent of S, unless β = 2 (which is the case ofbroken time-reversal symmetry) However, for any β theconelations can be transformed away if we replace Q bythe symmetrized matnx(3)which has the same eigenvalues äs Q The matnxof eigenvectors U which diagonahzes QE = U Xdiag(ri, ,rN)U^ is independent of S and the T„'S,and umformly distributed m the orthogonal, umtary, orsymplectic group (for β = l, 2, or 4, respectively) Thedistnbution (2) confirms the conjecture by Fyodorov andSommers [19] that the distnbution of tr Q has an algebraicAlthough the time-delay matnx was interpreted bySmith äs a representation of the "time operator," thisInterpretation is ambiguous [19] The ambiguity ansesbecause a wave packet has no well-defmed energy Thereis no ambiguity m the apphcation of Q to transportProblems where the mcommg wave can be regardedmonochromatic, hke the low-frequency response of achaotic cavity [21,22,28] or the Fermi-energy dependenceof the conductance [25] In the first problem, time delayis descnbed by complex reflection (or transmission)coefficients Rmn(a>),f- ιωτ,ηη + Θ(ω2)], (4a)Tmn=lmhS-]ndSmn/dE (4b)Rmn(to) = R,.n fr\\ __ l r· 12^mnvw l ^ m n l »0031-9007/97/78(25)/4737(4)$1000 © 1997 The American Physical Society 4737VOLUME 78, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 23 JUNE 1997The delay time rmn determines the phase shift of the acsignal and goes back to Eisenbud [1]. With respect to asuitably chosen basis, we may require that both the matri-ces Rmn(0) and rm„ are diagonal. Then we haveÄm n(<u) = 8mn[\ a>T (5)where the τ,η (m = l , . . . , ΛΟ are the proper delay times(eigenvalues of the Wigner-Smith time-delay matrix Q).For electronic Systems, the 0(ω) term of Rmn(cu) is thecapacitance. Hence, in this context, the proper delaytimes have the physical interpretation of "capacitanceeigenvalues" [29].We now describe the derivation of our results. WeStart with some general considerations about the invari-ance properties of the ensemble of energy-dependent scat-tering matrices S(E), following Wigner [30], and Gopar,Mello, and Büttiker [21]. The N X N matrix 5 is uni-tary for β = 2 (broken time-reversal symmetry), unitarySymmetrie for β = l (unbroken time-reversal and spin-rotation symmetry), and unitary self-dual for β = 4 (un-broken time-reversal and broken spin-rotation symmetry).The distribution functional P[S(E)] of a chaotic System isassumed to be invariant under a transformationS (E) -* VS(E)V', (6)where V and V' are arbitrary unitary matrices which donot depend on E (V = VT for β = l, V' = VR forβ = 4, where T denotes the transpose and R the dual ofa matrix). This invariance property is manifest in therandom-matrix model for the E dependence of the scatter-ing matrix given in Ref. [22]. A microscopic justificationstarting from the Hamiltonian approach to chaotic scatter-ing [31] is given in Ref. [32]. Equation (6) implies withV = V = iS~1/2 thatP(S,QE) = P(-Ü,QE). (7)Here P(S, QE) is the joint distribution of S and QE,defined with respect to the Standard (flat) measure dQEfor the Hermitian matrix QE and the invariant measure dSfor the unitary matrix S. From Eq. (7) we conclude thatS and QE are statistically uncorrelated; their distributionis completely determined by its form at the special pointS = -1.The distribution of S and QE at 5 = — l is com-puted using established methods of random-matrix theory[11,31]. The N X N scattering matrix S is expressed interms of the eigenvalues Ea and the eigenfunctions ψηα of« i= ithe M X M Hamiltonian matrix 3~[ of the closed chaoticcavity [6],M , , *S =l - iKl iKΔΜ7Γ F —^(8)The Hermitian matrix 3~C is taken from the Gaussian or-thogonal (unitary, symplectic) ensemble [11], P(J-C) <*exp(-yß7T2tr^f2/4A2M). This implies that the eigen-vector elements ψ]α are Gaussian distributed real (com-plex, quaternion) numbers for β = l (2, 4), with zeromean and with variance M""1, and that the eigenvaluesEa have distributionP({Ea}) « Πμ<ν(9)The limit M — » °° is taken at the end of the calculation.The probability P(~^,QE) is found by inspection ofEq. (8) near S = — 1. The case S = — l is special,because S equals — l only if the energy E is an (atleast) N-fold degenerate eigenvalue of 3~C . For matricesS in a small neighborhood of —l, we may restrict thesummation in Eq. (8) to those N energy levels Ea, a =l,..., N, that are (almost) degenerate with E (i.e., \E —Ea\ <5C Δ). The remaining M — N eigenvalues of 3~Cdo not contribute to the scattering matrix. This enormousreduction of the number of energy levels involved providesthe simplification that allows us to compute the completedistribution of the matrix QE.We arrange the eigenvector elements ψηα into anN X N matrix Ψ;β = ψ]αΜι/2. Its distribution Ρ(Ψ) αexp(— β ϋ·ψψν2) is invariant under a transformationΨ — > Ψ Ο, where O is an orthogonal (unitary, symplec-tic) matrix. We use this freedom to replace Ψ by theproduct ΨΟ, and choose a uniform distribution for O.We finally define the W X N Hermitian matrix Hl} =Σα=ι Οια(Εα — E)O*ja. S ince the distribution of the en-ergy levels Ea close to E is given by Υ\μ<ν \Εμ — Ev\&[cf. Eq. (9)], it follows that the matrix H has a uniformdistribution near H = 0. We then find(lOa)(lOb)- 1 isS = -l +QE = TH^-ly~l.Hence the joint distribution of S and QE at Sgiven byThe remaining integral depends entirely on the positive-definite Hermitian matrix Γit is shown that( ί/Ψ/(ΨΨ^ = Γ i/r(detr)(/3"2)/2/(r)0(r),')· (H)In Refs. [27] and [33](12)4738VOLUME 78, NUMBER 25 PHYSICAL REVIEW LETTERS 23 JUNE 1997where Θ(Γ) = l if all eigenvalues of Γ are positive and0 otherwise, and / is an arbitrary function of Γ = ψ ψ tIntegration of Eq (11) with the help of Eq (12) finallyyields the distnbution (2) for the mverse delay times andthe uniform distnbution of the eigenvectors, äs advertisedIn addition to the energy derivative of the scattermgmatnx, one may also consider the derivative with respectto an external parameter X, such äs the shape of theSystem, or the magnetic field [19,20] In random-matnxtheory, the parameter dependence of energy levels andwave functions is described through a parameter depen-dent M X M Hermitian matnx ensemble,(13)where 3~{ and 3~C' are taken from the same Gaussian en-semble We charactenze dS/dX through the symmetnzedderivative_^_ n-1/2 (14)by analogy with the symmetnzed time-delay matrix Qg mEq (3) To calculate the distnbution of Qx, we assumethat the mvanance (6) also holds for the X-dependentensemble of scattermg matrices (A random-matrix modelwith this mvanance property is given m Ref [34] ) Thenit is sufficient to consider the special point S = — lFrom Eqs (l Ob) and (13) we findQx = Ρ(Η') « exp(-/3tr/// 2/16),(15)where//;, = -(TH/n)M~1/2^ ]Φ*μ^'}Φ]ν A calcu-lation similar to that of the distnbution of the time-delaymatrix shows that the distnbution of Qx is a Gaussian,with a width set by QE,j_(16)X ex —XThe fact that delay times set the scale for the sensitivityto an external perturbation in an open System is wellunderstood m terms of classical trajectones [35], inthe sermclassical limit W —> °° Equation (16) makesthis precise in the fully quantum-mechanical regime ofa fimte number of channels N Correlations betweenparameter dependence and delay tmie were also obtamedin Ref s [19,20], for the phase shift derivatives 3<f)j/dXIn summary, we have calculated the distnbution ofthe Wigner-Smith time-delay matrix for the chaoticscattermg This is relevant for expenments on frequencyand parameter-dependent transmission through chaoticmicrowave cavities [9,10] or semiconductor quantum dotswith balhstic pomt contacts [36] The distnbution (1)has been known since Dyson's 1962 paper äs the circularensemble [12] It is remarkable that the Laguerreensemble (2) for the (mverse) delay times was notdiscovered earherWe acknowledge support by the Dutch Science Foun-dation NWO/FOM'"Present address Laboratoire de Physique Quantique,UMR 5626 du CNRS, Universite Paul Sabatier, 31062Toulouse Cedex 4, France[1] L Eisenbud, PhD thesis, Prmceton, 1948[2] E P Wigner, Phys Rev 98, 145 (1955)[3] F T Smith, Phys Rev 118, 349 (1960)[4] U Smilansky, m Chaos and Quantum Physics, editedby M -J Giannom, A Voros, and J Zinn Justin (North-Holland, Amsterdam, 1991)[5] R Blumel and U Smilansky, Phys Rev Lett 60, 477(1988), 64, 241 (1990)[6] C H Lewenkopf and H A Weidenmuller, Ann Phys(N Υ ) 212, 53 (1991)[7] P W Brouwer, Phys Rev B 51, 16878 (1995)[8] E Doron, U Smilansky, and A Frenkel, Phys Rev Lett65, 3072 (1990)[9] J Stein, H -J Stockmann, and U Stoffregen, Phys RevLett 75, 53 (1995)[10] A Kudrolli, V Kidambi, and S Sndhar, Phys Rev Lett75, 822 (1995)[11] M L Mehta, Random Matrices (Academic, New York,1991)[12] F J Dyson, J Math Phys (N Υ ) 3, 140 (1962)[13] V L Lyuboshits, Phys Lett 72B, 41 (1977), Yad Fiz27, 948 (1978) [Sov J Nucl Phys 27, 502 (1978)], YadFiz 37, 292 (1983), [Sov J Nucl Phys 37, 174 (1983)][14] M Bauer, P A Mello, and K W McVoy, Z Phys A 293,151 (1979)[15] H L Harney, F M Dittes, and A Muller, Ann Phys(N Υ ) 220, 159(1992)[16] B Eckhardt, Chaos 3, 613 (1993)[17] F Izrailev, D Saher, and V V Sokolov, Phys Rev E 49,130 (1994)[18] N Lehmann, D V Savin, V V Sokolov, and H -J Som-mers, Physica (Amsterdam) 86D, 572 (1995)[19] Υ V Fyodorov and H -J Sommers, Phys Rev Lett76, 4709 (1996), J Math Phys 38, 1918 (1997), Υ VFyodorov, D V Savin, and H -J Sommers, Phys Rev E55, 4857 (1997)[20] P Seba, K Zyczkowski, and J Zakrewski, Phys Rev E54, 2438 (1996)[21] V A Gopar, P A Mello, and M Buttiker, Phys RevLett 77, 3005 (1996)[22] P W Brouwer and M Buttiker, Europhys Lett 37, 441(1997)[23] E R Mucciolo, R A Jalabert, and J L Pichard, J Phys(France) I (to be pubhshed)[24] E Akkermans, A Auerbach, J E Avron, and B Shapiro,Phys Rev Lett 66, 76 (1991)[25] P W Brouwer, S A van Langen, K M Frahm, M Buttiker, and C W J Beenakker, Phys Rev Lett (to bepubhshed)4739VOLUME 78, NUMBER 25 PHYSICAL REVIEW LEITERS 23 JUNE 1997[26] The absence of correlations between the τ,,'s and the φ „'sis a special property of the proper delay times In con-trast, the derivatives οφη/3Ε considered in Refs [19,20]are correlated with the φ „'s[27] K Slevm and T Nagao, Phys Rev B 50, 2380 (1994),T Nagao and K Slevm, J Math Phys (N Υ ) 34, 2075(1993), T Nagao and P J Forrester, Nucl Phys B435,401 (1995)[28] M Buttiker, A Pretre, and H Thomas, Phys Rev Lett70,4114(1993)[29] For an electronic System (with capacitance C), Cou-lomb mteractions need to be taken mto accountself-consistently, see, e g , Refs [21,28] The resultis Rm„((o) = <5,„„(1 + ιωτ,η) - [ia>Tmrn/(hC/2e2 +Σ,τ,)] + Ο (ω2)[30] E P Wigner, Ann Math 53, 36 (1951), Proc CambridgePhilos Soc 47, 790(1951), Ann Math 55,7(1952)[31] J J M Verbaarschot, H A Weidenmuller, and M RZirnbauer, Phys Rep 129, 367 (1985)[32] P W Brouwer, PhD thesis, Leiden Umversity, 1997[33] E Brezm, S Hikami, and A Zee, Nucl Phys B 464, 411(1996)[34] P W Brouwer and C W J Beenakker, Phys Rev B 54,12705 (1996)[35] R A Jalabert, H U Baranger, and A D Stone, Phys RevLett 65, 2442 (1990)[36] R M Westervelt, in Nano-Science and Technology, editedby G Timp (American Institute of Physics, New York,1997)4740

Quantum mechanical time-delay matrix in chaotic scattering

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