We investigate the low-frequency dynamics for transmission or reflection of a
wave by a cavity with chaotic scattering. We compute the probability
distribution of the phase derivative phi'=d phi/d omega of the scattered wave
amplitude, known as the single-mode delay time. In the case of a cavity
connected to two single-mode waveguides we find a marked distinction between
detection in transmission and in reflection: The distribution P(phi') vanishes
for negative phi' in the first case but not in the second case.Comment: 10 pages including 3 figures; to be published in Physica Scripta
  (proceedings Nobel Symposium on Quantum Chaos

Beenakker, C. W. J.

Schomerus, H.

van Bemmel, K. J. H.

English

arXiv

Beenakker, C.W.J.

Bemmel, K.J.H. van

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Single-mode delay time statistics for scattering by a chaotic cavity

K. J. H. van Bemmel

H. Schomerus

C. W. J. Beenakker

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Single-Mode Delay Time Statistics for Scattering by a Chaotic Cavity

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Leiden University Scholary Publications

* Physica Scnpta T90, 278-282, 2001*Single-Mode DelayTime Statistics for Scattering by a Chaotic CavityK J H van Bemmel, H Schomerus and C W J BeenakkerInstituut-Lorentz Universität Leiden, PO Box 9506 2300 RA Leiden The NetherlandsReceived Septembei l 2000PACS Ref 05 45 Mt, 42 25 Dd, 42 25 HzAbstractWe mvestigate the low-frequency dynamics for transmission or reflection of awave by a cavity with chaotic scattermg We compute the probabihty distn-bution of the phase derivative φ = άφ/άω of the scattered wave amphtude,known äs the smgle-mode delay time In the case of a cavity connectedto two smgle-mode waveguides we find a marked distinction between detec-tion m transmission and m reflection The distribution Ρ(φ') vamshes fornegative φ' m the first case but not in the second caseβ = l (2) mdicates the presence (absence) of time-reversalsymmetry We assume a smgle polanzation for simphcity,äs m the microwave expenments in a planar cavity [2]The dynamical observable is the correlator p of an elementof the scattermg matnx S (ω) at two nearby frequencies,0)1. IntroductionMicrowave cavities have proven to be a good expenmentaltestmg ground for theones of chaotic scattermg [1] Muchwork has been done on staue scattermg properties, butrecently dynamic features have been measured äs well [2]A key dynamical observable, introduced by Genack andcoworkers [3-5], is the frequency derivative φ' = άφ/άωof the phase of the wave amphtude measured in a smglespeckle of the transmitted or reflected wave Because onespeckle corresponds to one element of the scattermg matnx,and because φ' has the dimension of time, this quantity iscalled the smgle-channel or smgle-mode delay time It isa linear superposition of the Wigner-Smith delay times intro-duced m nuclear physics [6,7]The probabihty distribution of the Wigner-Smith delaytimes for scattermg by a chaotic cavity is known [8] Thepurpose of this paper is to denve from that the distributionΡ(φ') of the smgle-mode delay time The calculation followsclosely our previous calculation of Ρ(φ') for reflection froma disordered waveguide in the locahzed regime [9] Theabsence of locahzation m a chaotic cavity is a sigmficantsimplification For a small number of modes N connectmgthe cavity to the outside we can calculate Ρ(φ') exactly, whilefor 7V ̂ > l we can use perturbation theory in l /N Thelarge-jV distribution has the same form äs that followmgfrom diffusion theory m a disordered waveguide [4,5], butfor small N the distribution is quahtatively different In par-ticular, there is a marked distinction between the distri-bution m transmission and in reflectionThe mdices n and m mdicate the detected outgoing mode andthe mcident mode, respectively The smgle-mode delay timeφ' is defmed by [3-5]φ' = hmImpΌ δωΐ'(2)with 7 = \Snm((o)\2 the mtensity of the scattered wave mmode n for unit mcident mtensity m mode m If we writethe scattermg amphtude S„m = -v/Je1* m terms of amphtudeand phase, then φ' = άφ/άω We will mvestigate the distri-bution of φ' m an ensemble of chaotic cavities havmg shghtlydifferent shape, at a given mean frequency mterval Δbetween the cavity modes For notational convemence,we choose umts of time such that 2π/Δ = lThe smgle-mode delay times are Imearly related to theWigner-Smith [6,7] delay times τ \ , τ ζ , ., IN, which are theeigenvalues of the matnx(3)To see this, we first expand the scattermg matnx Imearly inδω,l, .,τΝ)ϋ (4)Smce S is Symmetrie for β = l, one then has V = U Forβ = 2, V and U are statistically mdependent Combmation2. Formulation of the problemThe geometry studied is shown schematically m Fig l Itconsists of an TV-mode waveguide connected at one endto a chaotic cavity Reflections at the connection betweenwaveguide and cavity are neglected (ideal impedancematching) The N modes may be divided among differentwaveguides, for example, N = 2 could refer to twosmgle-mode waveguides The cavity may contain a fern-magnetic element äs m Refs [10,11], m which casetime-reversal symmetry is broken The symmetry mdexFig l Sketch of a chaotic cavity coupled to 7V propagatmg modes via one ormore waveguides The shape of the cavity is the quartered Sinai bilhard usedm recent microwave expenments [2]Physica Scnpta T90 Physica Scnpta 2001of Eqs. (1), (2), and (4) leads to [9]/ = 2^iT'UiV>V, M,V,ί—ij J JSingle-Mode Delay Time Statistics for Scattering by a Chaotic Cavity 219*The result is(5) d^ocsin2y dy]~|da,. (14)u, = Um, v, = Vm. (6) The joint distribution function Ρ(τ+, τ_) follows fromEq. (7). For β = l one hasThe distribution of the Wigner-Smith delay times for achaotic cavity was calculated in Ref. [8]. It is a Laguerre D, Λ ι , ,/ 2 2\-4" τ τ- — τ - τ ~ τ ,Λ _,(15),. . ,. . , .ensemble in the rates μ, — l /τ,,Ρ(μ, , ..., μΝ) α Π ΙΛ - μ/ Π, Λ ι , ,/ 2 2 ντ+> τ ι  Ϊ9 Ιτ-Κτ+ ~ τ-χexp(- (7)- Ιτ-Ι)·(16)The step function ö(x) = l for χ > 0 and ö(z) = 0 for χ < 0.It follows from Eq. (7) that (Σ, τ,) = l, a result that wasknown previously [12].Το calculate the joint distribution P(I, φ') from Eq. (5), wealso need the distribution of the coefficients u, and vt. Thisfollows from the Wigner conjecture [13], proven in Ref. [8], «mtanty of tf one has |v,| = |M2| and|v2 = | M , | . There-^ = ° and Φ = τ+> s° f l s mdependent of 7. IntegrationEq" (15) over τ" results mwhile for β = 2Ρ(τ+,τ.) = ίτ2_(τ2+-τ2_Τ6χ εχρ(-2τ+(τ2+ - τ2,)'1)First we consider thi . Because of theaccording to which the matrices U and F are uniformly dis-tributed in the unitary group. The calculation for smallN is now a straightforward Integration, see Section 3. » „ „ Λ *For large N we can use perturbation theory, see Section 4. Ρ(Φ') = \Φ' (Φ' + 2Φ' + 2) ε\ρ(-2/φ')θ(φ').Because of the uniform distribution of U and V, indepen-dent of the T,'S, we can evaluate the average of φ' directly In this case (as wel1 as in the case Ν = ̂ Φ can take onfor anv N(17)positive values, but this is atypical, as we will see shortly.From Eqs. (10) and (12) we find the relation between /1. (8)oi3-al-a2). (18)The distribution of I resulting from the measure (14) isΡ(Γ) = \Γ1Ι2Θ(Ι)Θ(\ - /), (19)We define the rescaled variable φ' = φ'/ (φ') — N φ', that wewill use in the next sections3. Small number of modesT-, ,r , .. . ,.„. , . ,, „r ,-, ·., in agreement with Refs. [16,171For N = l there is no draerence between the Wigner-Smith _ ° . ,T ~ „ ,, , ... , , . . . j i ^ T .11 4.delay time and the single-mode delay time. In that caseT ! , ΐ, , , · ,· . ·, , , ,· . r i ^ nn7 = 1 and φ = φ is distnbuted according to [14,15] For the case 7 v = 2 , o = l, n = m we use that u\ = v\., , · , ·«2 = ^2 and obtam the parametnzationv--€βφ'-2-^εχρ(-ίβ/φ')θ(φ'). (9)The normalization coefficient Cß equals (2π)~1/2 for β = land l for β = 2.Now we turn to the case N = 2. By writing out the sum-mation in Eq. (5) for 7 and φ', one obtains φ' — τ+ + ατ_with τ± = =(τι ± 12) and(10)- «i - «2),« = (cos2y)/7.(20)(21)The distnbution P(I, «) resulting from the measure (14) is-̂2π7)~1/2(1 - 7α2)-1/2θ(7)0(1 - 7)0(1 - 7α2).(22)α = (iHil 2^! 2 - |M2|2|v2|2)/7. (11) The joint distribution of 7 and φ' = 2φ' takes the formTo find the joint distribution P(I, a) we parametrize U inp,j ΐ/^ _ fterms of 4 independent angles, Λsin j exp(-i«i - icosyexp(-ioc3)(12)y°° άτ+ Ρ(τ+, τ-)ρ(ΐ, α =~(23)with α, e (Ο, 2π) and y e (0, π/2). The invariant measured,u α |Det g\dy f], da, in the unitary group follows from The distribution of 7 following from Integration of P(7, a)the metric tensor g, defined by over a is given by Eq. (19) with 7 -> l - 7, as it should.The integrations over τ+, τ_, and 7, needed to obtainTr dt/dt/ t = '^gljdxldxj, {x,} — {γ, «ι, α2, «3}· (13) /»(i '̂) can be evaluated numerically, see Fig. 2. Notice that'>/ ^(φ') has a tail towards negative values of φ'.Physica Scnpta 2001 Physica Scripta T90280 K J H van Bemmel, H Schomerus and C W J Beenakkei2-0.5 0 05 l 15 2 2.5Φ'Fig 2 Distribution of the single-mode delay time m the case of preservedtime-reversal symmetry (ß = 1) The curves for 7V = 1,2 follow from Eqs (9),(17), and (23) The curve for ./V S> l follows from Eq (36), and is the same forn = m and n ̂  m The delay time φ — Φ'/(Φ')ls rescaled such that the meanis l for all curves Data pomts are the result of a Monte Carlo calculationm the Laguerre ensemble (with N = 400 n φ m representmg the large-Nhmit)l· 50.50l l Γ-0.5 Ο 0.5 l 1.5 2 2.5φ>Fig 3 Same äs in Fig 2, for broken time-reversal symmetry (ß = 2) Thecurves for 7V = l 2 and for N » l follow from Eqs (9), (23), and (36) Thereis no difference between n = m and n φ m for any N The large 7V-resultfor ß = 2 is the same äs for ß = lFor N — 2, ß = 2 it doesn't matter whether n and m areequal or not Parametnzation of both U and K leads toeigenvalues,cos η,(24)α = (l _ jti _ χ2)/7, (25)with a measure άμ oc dxidx2d?7 and XL jca e (0, 1), j? e (Ο, π)The jomt distnbution P(I, a) is now given byP(I, a) - i 71/20(7)0(1 - 7)0(1 - 7a2) (26)Integration over α leads to [16,17]P(/) = 0(7)0(1 - 7) (27)The distnbution P(I, φ') follows upon msertion of Eqs (16)and (26) mtq Eq (23) Numerical Integration yields the dis-tnbution Ρ(φ') plotted in Fig 3 As in the previous case, thereis a tail towards negative φ'4. Large number of modesWe now calculate the jomt distnbution P(I, φ') for N » lFirst the case n φ m will be considered, when there is no dis-tmction between ß = l and ß = 2 In the large /V-hmit thevectors H and v become uncorrelated and their elementsbecome independent Gaussian numbers with zero meanand vanance l /N We first average over v, followmgRef [9] We mtroduce the weighted delay time W = Ιφ'The Founer transform of P(I, W) is given by χ(ρ, q) —(exp(ipl + iqW)) The average over v is a GaussianIntegration, that gives), (28)H =/?H*KT +^q(u*ur + H*MT),where ü, = η,τ, The matnx 77 has only two nonzeroPhysica Scnpta T90^ +p ±= > U,(30)(31)Performing the mverse Founer transforms and returnmg tothe variables φ' and 7 leads to, φ') = (τΥ37/π)1/2 exp(-7V7)χ (B2 -(32)The averages over u, and τ, still have to be performedUp to now the denvation is the same äs for the disorderedwaveguide in the localized regime [9], the only differencebemg the different distnbution of the Wigner-Smith delaytimes τ, The absence of localization in a chaotic cavitygreatly simplifies the subsequent calculation in our presentcase While m the localized waveguide anomalously largeτ/s lead to large fluctuations in B\ and ΒΊ, m the chaoticcavity the term μ^Ν/2 m Eq (7) suppresses large delay timesFluctuations in B^ are smaller than the mean by a factorl /VN For N 3> l we may therefore replace Bk m Eq (32)by (Bk)To calculate the average of B\ and ΒΊ we need the densityρ(τ) = (Σι δ(τ - τ,)} of the delay times It is given by [8]v (τ+ - τ-), τ± = ·3±V8Ν(33)for τ mside the interval (τ_, τ+) The density is zero outsidethis interval The resultmg averages are (B\) = N ' and© Physica Scnpta 2001Single-Mode Delay Time Statistics for Scattermg by a Chaotic Cavitv 281^(B2) = 2N'2, which leads toP(I, φ') = (7V37/7i)1/2 exp(-7V7[l + (φ' - l)2])ö(7) (34)(Recall that φ' = φ' /(φ') = Ν φ' ) Integration over φ' or /gives7>(7) = 7Vexp(-7V7)Ö(7), (35)(36)This distnbution of / and φ' has the same form äs that of adisordered wavegmde m the diffusive regime [4,5]We next turn to the case n = m and β = l (For β = 2 thereis no difference between n = m and n ̂  m ) Smce u, = v, inEq (5) we have(37)The jomt distnbution P(Co, C\) has the Founer transformimagmary part of the off-diagonal elements are mdependently Gaussian distnbuted with zero mean and unitvanance The real diagonal elements are mdependentlyGaussian distnbuted with zero mean and variance 2 Wediagonahze H, order the eigenvalues from large to small,and multiply the n-th normahzed eigenvector by a randomphase factor e'a", with cc„ chosen umformly from (0,2π)The resulting matnx of eigenvectors is umformly distnbutedin the umtary groupThe Laguerre ensemble (7) for the rates μ, = l/τ, can begenerated by a random matnx of the Wishart type [18,19]Consider a N χ (2Ν — l + 2/ß) matnx X, where X is realfor β = l and complex for β = 2 (The matnx X is neitherSymmetrie nor Hermitian) The matnx elements areGaussian distnbuted with zero mean and vanance(\xnm\2) = l The jomt probabihty distnbution of theeigenvalues of the matnx XX^ is then given by Eq (7)The results of our numencal check are mcluded in Figs2 and 3 The large-7V hmit is represented by 7V = 400,n Φ m The analytical curves agree well with the numencaldata= (exp(y?oReCo + +(38)(39)] (40)Fluctuations in L are smaller than the average by a factorl/7V We may therefore approximate (exp(—L)} «aAveragmg over H we findχ(ρο,ρι, qo, qi) = (exp(-L)),L =exp(-L) Because + ρ\τι~)2 + N~2(q0 is oforder l /N, we may expand the loganthm m Eq (40)The average follows upon Integration with the density (33),Va + Va + gf +N2Inverse Founer transformation givesN46. ConclusionWe have investigated the statistics of the single-mode delaytime φ' for chaotic scattenng For a large number 7V ofscattermg channels the distnbution has the same form äsfor diffusive scattenng [4,5], but for small 7V the distnbutionis different The case N = 2 is of particular interest,representing a cavity connected to two single-modewaveguides For preserved time-reversal symmetry anddetection in transmission (ß — l, η φ m), we find that φ'can take on only positive values, similarly to the Wigner-Smith delay times In contrast, for detection in reflection(or for broken time-reversal symmetry) the distnbutionacquires a tail towards negative φ' These theoreticalpredictions are amenable to expenmental test in themicrowave cavities of current interest [2]P(C0, Q) =(2π)2: + 7V2Re CoC*) Acknowledgements(42)ΛThe resulting distnbution of φ' and 7 is, φ') = (7ν3//2π)1/2exp(-l/V/[l + (φ' -We thank P W Brouwer and M Patra for valuable advice This researchwas supported by the "Nederlandse orgamsatie voor WetenschappehjkOnderzoek" (NWO) and by the "Stichtmg voor Fundamenteel Onderzoekder Materie" (FOM)(43) ReferencesIt is the same äs the distnbution (34) for n Φ m, apart fromthe rescalmg of 7 by a factor of 2 äs a result of coherentbackscattermgThe distnbution (36) of 0' for 7V » l is mcluded m Figs 2and 3 for comparison with the small 7V-results5. Numerical checkWe can check our analytical calculations by performmg aMonte Carlo average over the Laguerre ensemble for theT,'S and the umtary group for the u,'s and v,'s For the aver-age over the umtary group we generate a large numberof complex Hermitian 7V χ 7V matrices H The real and1 Stockmann, H -J , "Quantum Chaos — An Introduction", (CambridgeUmversity, Cambridge, 1999)2 Persson, E , Rotter, I , Stockmann, H -J and Barth, M , Phys RevLett 85, 2478 (2000)3 Sebbah, P, Legrand, O and Genack, A Z , Phys Rev E 59, 2406(1999)4 Genack, A Z, Sebbah, P , Stoytchev, M and van Tiggelen, B A ,Phys Rev Lett 82, 715 (1999)5 van Tiggelen, B A , Sebbah, P, Stoytchev, M and Genack, A Z ,Phys Rev E 59, 7166 (1999)6 Wigner, E P, Phys Rev 98, 145 (1955)7 Smith, F T , Phys Rev 118, 349 (1960)8 Brouwer, P W , Frahm, K M and Beenakker, C W J , Phys RevLett 78, 4737 (1997), Waves Random Media 9, 91 (1999)9 Schomerus, H , van Bemmel, K J H and Beenakker, C W J , cond-mat/0004049ι Physica Scnpta 2001 Physica Scnpta T90* 282 K J H van Bemmel, H Schomerus and C W J Beenakker" 1 0 So, P , Anlage, S M , Ott, E and Oerter, R N , Phys Rev Lett 74, 15 Gopar, V A Mello, P A and Buttiker, M , Phys Rev Lett 77, 3005' 2662 (1995) (1996)11 Stoffregen, U , Stein, J , Stockmann, H -J , Kus, M and Haake, F, 16 Baranger, H U and Mello, P A , Phys Rev Lett 73, 142 (1994)Phys Rev Lett 74, 2666 (1995) 17 Jalabert, R A , Pichard, J-L and Beenakker, C W J , Europhys Lett12 Lyuboshitz, V L, Phys Lett B 72, 41 (1977) 27, 255 (1994)13 Wigner, E P , Ann Math 53, 36 (1951) 55, 7 (1952). 18 Edelman, A , Linear Algebra Appl 159, 55 (1991)14 Fyodorov, Υ V and Sommers, H-J , Phys Rev Lett 76, 4709 19 Baker, T H , Forrester, P J and Pearce, P A , J Phys A 31, 6087(1996) (1998)Physica Scnpta T90 © Physica Scnpta 2001

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Single-mode delay time statistics for scattering by a chaotic cavity

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