Chaotic Scattering with Localized Losses: S-Matrix Zeros and Reflection Time Difference for Systems with Broken Time Reversal Invariance
CChaotic Scattering with Localized Losses:S-Matrix Zeros and Reflection Time Difference for Systems with Broken TimeReversal Invariance
Mohammed Osman and Yan V. Fyodorov
1, 2 Department of Mathematics, King’s College London, London WC26 2LS, United Kingdom L.D.Landau Institute for Theoretical Physics, Semenova 1a, 142432 Chernogolovka, Russia (Dated:)Motivated by recent studies of the phenomenon of Coherent Perfect Absorption, we developthe random matrix theory framework for understanding statistics of the zeros of the (subunitary)scattering matrices in the complex energy plane, as well as of the recently introduced ReflectionTime Difference (RTD). The latter plays the same role for S − matrix zeros as the Wigner timedelay does for its poles. For systems with broken time-reversal invariance, we derive the n -pointcorrelation functions of the zeros in a closed determinantal form, and study various asymptotics andspecial cases of the associated kernel. The time-correlation function of the RTD is then evaluatedand compared with numerical simulations. This allows to identify a cubic tail in the distributionof RTD, which we conjecture to be a superuniversal characteristic valid for all symmetry classes.We also discuss two methods for possible extraction of S − matrix zeroes from scattering data byharmonic inversion. INTRODUCTION
Wave scattering in cavities with chaotic classical raydynamics has been intensively studied over the last fewdecades[1–4]. The use of random matrix theory (RMT)has allowed for a statistical description of quantities de-rived directly from the M × M , energy-dependent scat-tering matrix S ( E ) , where M is the number of scatter-ing channels; for recent reviews see [5–7]. In an idealflux-conserving system S ( E ) is unitary, but in practiceunavoidable losses, e.g. due to imperfect conductivity ofthe cavity walls, or leaks in connecting microwave waveg-uides [8–15], make the experimentally observed scatteringmatrix subunitary. To that end, a considerable effort hasbeen invested in generalizing the random-matrix basedapproaches to chaotic wave scattering in the presence ofsome form of absorptive loss [16–24]. In recent years theinterest in absorptive scattering has been further muchstimulated by a proposal to construct the so-called coher-ent perfect absorber (CPA), which can be looked at as ascattering system ( e.g. a cavity) with a small amountof loss which completely absorbs a monochromatic waveincident at a particular frequency [25]. Applications fora CPA may include optical filters and switches or logicgates for use in optical computers. Recently, a CPA ina rectangular cavity with randomly positioned scatterersand absorption due to a single antenna has been realisedexperimentally [26], paving the way for the constructionof CPAs based on disordered cavities. In another recentexperiment, a CPA has been realised with a two-portmicrowave graph system, both with and without time-reversal symmetry [27]. In the framework of chaotic scat-tering, a CPA state corresponds to an eigenstate of theS matrix with zero eigenvalue at a real energy. This factnaturally motivates rising interest in a more general ques-tion of characterizing S-matrix complex zeros, which has not been systematically studied for wave chaotic systemswith absorption until very recently [28, 29]. This is insharp contrast with statistics of S-matrix complex poles,known as resonances, whose exact density in the complexplane (and more delicate characteristics) for systems withchaotic scattering has been systematically studied in theframework of random matrix approach[30–42] with someaspects amenable to experimental verification [43–51].As is well-known, an important quantity directly re-lated to resonance poles in a scattering system is theso-called Wigner time delay, the energy derivative of thetotal phase shift [52, 53], which in systems with chaoticscattering can be measured experimentally [54] andwhose various generalizations attracted recently much at-tention [55]. In particular, it has been suggested thatcomplex zeroes of the scattering matrix can manifestthemselves in a very analogous way via a quantity calledthe reflection time difference (RTD), which, in princi-ple, may be measured experimentally [56]. Note that theproblem of characterizing statistics of Wigner time de-lays and related quantities in the RMT framework (andbeyond) keeps attracting considerable interest in the last25 years [57–72], for the reviews see [33] as well as themore recent [73]. It is therefore natural to ask similarquestions about statistics of the Reflection Time Differ-ence in systems with chaotic scattering.The goal of the present paper is to provide some infor-mation on fluctuations of RTD in the framework of theRMT approach. To this end, we mainly consider systemswith broken time-reversal symmetry, where the proper-ties of the complex S − matrix zeroes can be very effi-ciently studied non-perturbatively (in particular, for anylocalized losses as well as any channel coupling) by ad-justing the method suggested for S − matrix poles in [34].This approach allows us to verify (and then exploit) thatthe S-matrix zeros for absorptive system form asymptot- a r X i v : . [ c ond - m a t . d i s - nn ] A p r ically a determinantal process in the complex plane, aslong as the effective dimension N of the Hilbert spacedescribing the cavity Hamiltonian in an appropriate en-ergy range is considered large: N (cid:29) . The explicit formfor the 2-point function is then used to study the RTDcorrelation function along the lines suggested in [56], infull analogy with similar studies of the Wigner time delay[74].To begin with, let us remind that one of the most nat-ural ways of incorporating localized losses into the RMTdescription is to associate them with additional (or “hid-den”) scattering channels. When those channels are nu-merous and weak one can model in this way spatially-uniform absorption, the idea possibly going back to [16],cf. the discussion after equation (12) in the text be-low. For the non-perturbative localized setting the corre-sponding construction has been proposed in a form clos-est to our needs recently in [29]. To this end, considera closed cavity whose internal chaotic wave dynamics ismodelled by an N × N RMT Hamiltonian H , coupled to M scattering and L absorbing channels, the latter rep-resenting the sources of localized loss. The vectors ofcouplings to scattering/absorbing channels are collectedin an N × M (respectively, N × L ) matrix W ( respec-tively, A ). We also define the associated N × N matrices Γ W = πW W † and Γ A = πAA † , of the ranks M and L respectively, and further assume M + L < N . It is alsoconvenient to assume that the columns of W and A aremutually orthogonal: N (cid:88) n =1 W ∗ na A nb = 0 , ∀ a = 1 , . . . , M & b = 1 , . . . , L (1)being in addition orthogonal within each of the channelgroups: N (cid:88) n =1 W ∗ na W nc = γ a δ ac , N (cid:88) n =1 A ∗ nb A nd = ρ b δ bd . (2)The above assumptions lead to the diagonal form ofthe ensemble-averaged scattering matrix, which describesonly the resonant scattering associated with the creationof long-lived intermediate states. The condition (2) canbe easily lifted (see e.g. Appendix A in [24] for a recentdiscussion and further references).The construction of the energy dependent flux-conserving ( M + L ) × ( M + L ) scattering matrix S is donefollowing the standard procedure frequently referred to asthe “Heidelberg approach" and going back to the semi-nal work [75], see also [33]. Adapting it to the presentsituation one gets the following block form, cf. [29]: S = (cid:18) M − πiW † D − W − πiW † D − A − πiA † D − W L − πiA † D − A (cid:19) , (3)where we denoted D ( E ) = E N − H + i (Γ W + Γ A ) . (4) The upper left block S ( E ) := 1 M − πiW † D − W de-scribes the scattering between M “observable” channelsand has the alternative representation: S ( E ) = 1 N − iK A N + iK A , K A = πW † E − H + i Γ A W. (5)Note that due to the presence of hidden/absorbing chan-nels encapsulated via Γ A (cid:54) = 0 the matrix S ( E ) is subuni-tary reflecting the loss of flux injected through the ob-servable channels which escapes via the hidden channels,and as such is treated as irretrievably absorbed. A sim-ilar representation holds for the lower right block S (cid:48) ( E ) after the replacement W ↔ A everywhere.Since S ( E ) is subunitary, positions of its zeros in thecomplex energy plane are no longer conjugates of thecorresponding poles, and thus in principle can be locatedin both half-planes. From 5 one can easily deduce thatthe determinant of S ( E ) has the following form: det S ( E ) = det[ E N − H + i (Γ A − Γ W )]det[ E N − H + i (Γ W + Γ A )] , (6)from which it is clear that the zeros z n of S ( E ) in thecomplex energy plane are the complex eigenvalues of thenon-Hermitian matrix H + i (Γ W − Γ A ) . Writing a similarexpression for det S (cid:48) ( E ) and taking their ratio, we arriveat the following complex number: det S ( E )det S (cid:48) ( E ) = det[ E N − H + i (Γ A − Γ W )]det[ E N − H + i (Γ W − Γ A )] (7) = e iφ ( E ) . (8)which is obviously unimodular for real values of the en-ergy E .Now we follow the proposal of [56] and define the re-flection time difference (RTD) as the energy derivative ofthe phase φ ( E ) : δ T ( E ) := − i ∂∂E log det S ( E )det S (cid:48) ( E ) (9) = N (cid:88) n =1 (cid:61) z n ( E − (cid:60) z n ) + ( (cid:61) z n ) . (10)Such a definition of RTD is inspired by the Wigner timedelay, which has the same form as 10 but with the zeros z n replaced by the complex S − matrix poles located inthe lower half-plane of complex energies. Those are sim-ply complex eigenvalues of another non-Hermitain ma-trix H eff := H − i (Γ W + Γ A ) and will be denoted E n = E n − i Γ n / , with condition (cid:61)E n = − Γ n < dueto non-negativity of Γ W + Γ A . The main difference be-tween the RTD δ T ( E ) and the Wigner time delay is thatthe former can be negative whereas the Wigner time de-lay is always positive in the present model in view of Γ n > . The name reflection time difference comes fromthe fact that the phase of det S ( E ) gives the delay (av-eraged over the scattering channels) in the propagationof a nearly monochromatic wave due to scattering, rel-ative to a perfectly reflecting cavity. Hence the RTD isthe difference in this delay between the first M channels,deemed observable, and the last L channels, deemed ab-sorbing. Let us however stress, that equivalently in anyflux-conserving two-terminal system one can always sim-ply subdivide channels into two groups; most naturally,in a two-terminal scattering setup, via the left/right di-vision. Then S ( E ) and S (cid:48) ( E ) in such a system simplydescribe reflection blocks of the total S − matrix. Notethat RMT-based statistics of entries and eigenvalues ofthe reflection blocks is interesting in itself, and being notunrelated to Wigner time delays have been studied fromvarious viewpoints, e.g. in [18, 63, 71, 76, 77]. STATISTICS OF S MATRIX ZEROS FORSYSTEMS WITH BROKEN TIME REVERSALINVARIANCE
We have thus seen that the zeros z n of S ( E ) in thepresent approach are nothing else but the complex eigen-values of the non-Hermitian N × N random matrix H + i (Γ W − Γ A ) . As the zeroes are the main constituentsof the RTD (10) we briefly analyze statistics of those ze-roes in the complex plane for wave-chaotic systems withbroken time-reversal invariance, when the matrix H istaken from the Gaussian Unitary Ensemble (GUE). Notethat systems of such type can be studied experimentally,see e.g. [78] and references therein.Since we have assumed the orthogonality both in-side and between the two groups of channels, see (1)-(2), exploiting unitary invariance of the GUE part wecan easily check that the matrix Γ := Γ W − Γ A ofrank M + L < N can be chosen to be diagonal:
Γ = diag ( γ , ..., γ M , − ρ , ..., − ρ L , , ..., . In the lossless case L = 0 , Γ is necessarily positive and all n -point corre-lation functions have been derived in [34] in the limit N (cid:29) max( M, n ) . We show in the Appendix which mod-ification are necessary to adapt the method [34] to thelossy case L > . The end result is the following de-terminantal form for the asymptotic n -point correlationfunctions for the eigenvalues z n in the whole complexenergy plane: lim N →∞ N n R n (cid:18) x + z N πν ( x ) , ..., x + z n N πν ( x ) (cid:19) (11) = det[ K ( z i , z ∗ j )] ≤ i,j ≤ n , where x ∈ ( − , , ν ( x ) = π √ − x is the semicirculardensity of real eigenvalues of the GUE matrix H , andthe kernel is given explicitly by: K ( z, w ∗ ) = (cid:112) F ( z ) F ( w ∗ ) (12) × (cid:90) − du e i ( z − w ∗ ) u M (cid:89) a =1 ( g a + u ) L (cid:89) b =1 ( h b − u ) , where for z = (cid:60) z + i (cid:61) z we have F ( z ) = (cid:90) ∞−∞ dk π e − ik (cid:61) z (cid:81) Ma =1 ( g a − ik ) (cid:81) Lb =1 ( h b + ik ) , (13) g a = 12 πν ( x ) (cid:18) γ a + 1 γ a (cid:19) , h b = 12 πν ( x ) (cid:18) ρ b + 1 ρ b (cid:19) . (14)Let us first check the simplest limit of very manyequivalent weakly coupled absorbing channels, namely L → ∞ , ρ b → in such a way that the product Lρ b re-mains a finite constant. Following [16] one expects thatthe absorption becomes spatially uniform across the sam-ple, and that all zeroes z n will be uniformly shifted down-wards in the complex plane by the same amount. Indeed,introducing the notation Lρ b = (cid:15)/ ( πν ( x )) , ∀ b = 1 , . . . , L and assuming it to remain constant as L → ∞ , we seethat we can replace h b ≈ L (cid:15) , and easily verify that thekernel in (12) reduces to the L = 0 case but with theshift (cid:61) z → (cid:61) z + (cid:15) , in full correspondence with the uni-form absorption picture.A less trivial, representative case to be considered nextis that of equivalent scattering channels g a = g, ∀ a =1 , . . . , M , as well of equivalent absorbing channels h b = g , ∀ b = 1 , . . . , L , but both couplings not considered tobe weak. The interesting limiting case arises if we againassume the channels are abundant, so that both M → ∞ and L → ∞ , but in such a way that the ratio LM = p with ≤ p ≤ remaining constant (the case p > followsby replacing p → p and M ↔ L and g ↔ h ). In such alimit the integrals over variables u and k , can be readilyevaluated by the Laplace (saddle-point) method. Thecalculation can be performed for general p, g, g but theresulting expressions are relatively cumbersome; here wepresent explicit formulas only for the special case p = 1 and g = g when they are more elegant. The ensuingasymptotic mean density of complex zeroes is supportedinside the domain (cid:26) ( x, y ) : − ≤ x ≤ , − Mg − ≤ y ≤ Mg − (cid:27) . (15)Note that since g depends on x as g ∼ /ν ( x ) , the widthof the support along the imaginary axis decreases mono-tonically to zero at x → ± . Inside the support, thedensity is constant near the real axis and decays as y − when y = O ( M ) : defining (cid:101) ρ ( x, y ) := ρ ( x,y ) ν ( x ) we then have (cid:101) ρ ( x, y ) = g πM if y = O (1) g πM (cid:101) y (cid:18) √ g (cid:101) y (cid:19) if y = M (cid:101) y . (16)For the general parameters p, g, g the support of thedensity in the complex plane is given for − ≤ x ≤ by − M (cid:18) pg − − g + 1 (cid:19) ≤ y ≤ M (cid:18) g − − pg + 1 (cid:19) . (17)Figures 1a and 1b show the eigenvalues of five × matrices with M = L = 100 and g = g = 1 . and g = 1 . , g = 5 . respectively. (a) M = L = 100 and g = g = 1 . (b) M = L = 100 , g = 1 . and g = 5 . Figure 1:
Eigenvalues of five × matrices with thesolid line indicating equations 15 (top) and 17 (bottom) In particular, we see that if p = 0 then the zeros areall in the upper half-plane, separated from the real axisby the gap M g +1) . Remembering that for p = 0 zeroesare mirror images of poles, this result is simply L = 0 case of [34]. The gap in the poles distribution is thewell-known feature first derived in [31] and [57] in therelated, but slightly different limit M = O ( N ) . It hasprofound consequences for underlying dynamics and alsohas a semiclassic significance, being related to the classic escape time[79, 80]. Moreover, one can see that all theS-matrix zeros will still be in the same half-plane as longas | pg − g | > p (upper half-plane for g − pg > p and lower for pg − g > p ). An illustration of such asituation is given in the figure 1b.When | z − z | = O (1 /N ) , the kernel can be reducedafter some algebraic manipulations to the Ginibre-likeform: | K ( z , z ∗ ) | = (cid:101) ρ ( z ) e − π (cid:101) ρ ( z ) | z − z | , (18)where z = z + z and (cid:101) ρ ( z ) is from 16. Such a kernelwas found in the L = 0 case when M → ∞ [34] andis conjectured to be the universal form of the kernel forstrongly non-Hermitian matrices [39].We also record for completeness the general expressionfor the density of zeros, valid for any M, L : (cid:101) ρ ( x, y ) = 1( g + g ) M + L − (cid:90) − due − uy ( g + u ) M ( g − u ) L × (cid:40) θ ( − y ) e g y L (cid:88) n =1 a n ( L, M ) [ − g + g ) y ] n − Γ( n )+ θ ( y ) e − gy M (cid:88) n =1 a n ( M, L ) [2( g + g ) y ] n − Γ( n ) (cid:41) , (19)where a n ( M, L ) = Γ( M + L − n )Γ( M − n + 1)Γ( L ) . (20)The above mean density in y for x = 0 is comparedwith RMT simulations in figures 2a and 2b. STATISTICS OF REFLECTION TIMEDIFFERENCE FOR SYSTEMS WITH BROKENTIME REVERSAL INVARIANCE
In this section we convert the information about S − matrix zeroes into information about the two-pointconnected correlation function of the Reflection Time dif-ference δ T defined via (10). In doing this we largelyfollow the method proposed in [74] for the Wigner timedelays. We start with recalling that the main micro-scopic energy scale characterizing (real) eigenvalues ofthe Hermitian RMT cavity Hamiltonian H is the asso-ciated mean level spacing ∆ = [ N ν ( E )] − , and introducethe appropriately rescaled RTD via (cid:102) δ T = ∆2 π δ T . Fromthe technical point of view it is more natural to considerthe associated Fourier transform of the two-point corre-lation function in question defined as C M,L ( t ) := 12 π (cid:90) dω e iωt (cid:104) (cid:102) δ T (cid:18) E + ∆2 π ω (cid:19) (cid:102) δ T (cid:18) E − ∆2 π ω (cid:19) (cid:105) c . (21) y p ( y ) (a) N = 100 , M = 4 , L = 2
20 0 20 40 60 80y0.000.010.020.030.040.050.060.070.08 p ( y ) (b) N = 500 , M = 10 , L = 5 Figure 2:
Density of the imaginary parts (cid:61) z n for (cid:60) z n = 0 compared against 19 (solid line). Defining the so-called empirical density of S − matrix ze-roes z n in the complex plane z = (cid:61) z + i (cid:60) z as ρ ( z ) = 1 N N (cid:88) n =1 δ (2) ( z − z n ) , (22)where δ (2) ( z − z n ) := δ ( (cid:60) z − (cid:60) z n ) δ ( (cid:61) z − (cid:61) z n ) , with δ ( x ) being the standard Dirac delta-function, we first rewrite(10) as (cid:102) δ T ( E ) = ∆ N π (cid:90) (cid:61) z ( E − (cid:60) z ) + ( (cid:61) z ) ρ ( z ) d (cid:60) z d (cid:61) z, (23)and then substitute this in (21) and perform the ensembleaveraging. In doing this we exploit the relation betweencovariance functions of empirical densities and the two-point function featuring in (11) for n = 2 , hence the associated kernel (12): (cid:104) ρ ( z ) ρ ( z ) (cid:105) c = (cid:104) ρ ( z ) (cid:105) δ ( z − z ) − Y ( z , z ) , (24)where Y ( z , z ) = | K ( z , z ∗ ) | is the associated two-point “cluster" function. After rescaling and straight-forward manipulations this brings (21) to the form C M,L ( t >
0) = A ( t ) − B ( t ) , A ( t ) = 12 (cid:90) dye − | y | t (cid:101) ρ ( E, y ) , (25)and B ( t ) = 12 π (cid:90) dωdy dy e − itω − ( | y | + | y | ) t (26) ×Y ( E, ω, y , y ) sign ( y y ) , Finally, using the expression 12 for the kernel, we arriveat the following formulas: A ( t )= 14 (cid:90) − du ( g + u ) M ( g − u ) L ( g + g ) M + L × (cid:34) M (cid:88) n =1 a n ( M, L ) (cid:18) g + g g + t + u (cid:19) n + L (cid:88) n =1 a n ( L, M ) (cid:18) g + g g + t − u (cid:19) n (cid:35) , (27) B ( t ) = θ (2 − t )4 (cid:90) − t − du ( g + u ) M ( g − u ) L ( g + g ) M + L ) ( g + t + u ) M × ( g − t − u ) L (cid:34) M (cid:88) n =1 a n ( M, L ) (cid:18) g + g g + t + u (cid:19) n − L (cid:88) n =1 a n ( L, M ) (cid:18) g + g g − u (cid:19) n (cid:35) . (28)where a n ( L, M ) has been defined in (20).A few remarks are due. First, the limit of no absorp-tion is equivalent to sending g → ∞ . The dominantcontribution then obviously comes from the n = M termin the first sum, in both A ( t ) and B ( t ) . The resultingexpression reproduces the well-known correlation func-tion of the Wigner time delay for systems with brokentime-reversal invariance: [33, 59]: C M, ( t ) = 14 (cid:90) max (1 − t, − (cid:18) g + ug + t + u (cid:19) M du, (29)Note that the above correlation function of Wignertime delays decays as t − M for t → ∞ , whereas the cor-responding correlation of RTD for any finite absorption g < ∞ decays in the same limit as t − for any fixedvalue M > , L > . This implies that the variance ofthe RTD which in view of (21) can be found as (cid:28)(cid:104) (cid:102) δ T ( E ) (cid:105) (cid:29) c ∝ (cid:90) ∞ C M,L ( t ) dt, logarithmically diverges for any finite number of ab-sorbing channels. This feature is strikingly differentfrom the Wigner time delay, whose variance is infiniteonly for a single-channel scattering, being finite for any M > . The logarithmic divergence of the variance sug-gests that the distribution of RTD must have the large-tail behaviour P ( (cid:102) δ T ) ∼ (cid:102) δ T − . Although we can notprove this conjecture rigorously in generality, it can bestrongly supported by a perturbative argument (sketchedin the Appendix) valid in the regime of small absorp-tion in a weakly open system with equivalent channels, γ a = γ (cid:28) , ρ a = ρ (cid:28) , when imaginary parts of the S − matrix zeros are much smaller than their separation: (cid:104)|(cid:61) z n |(cid:105) (cid:28) ∆ . Adapting the argument along the linesused in [33] for Wigner time delays, and further assum-ing ρ (cid:38) γ , the probability density for RTD can be shownto have an algebraic decay with universal exponents, P β ( (cid:102) δ T ) ∼ (cid:40) | (cid:102) δ T | − / , (cid:28) | (cid:102) δ T | (cid:28) γ − | (cid:102) δ T | − , | (cid:102) δ T | (cid:29) γ − , (30)and similarly for negative (cid:102) δ T with γ and ρ exchangingtheir roles. This result is “superuniversal”, that is holdingnot only for systems with broken symmetry, but for allstandard symmetry classes of H described by values ofthe Dyson index β = 1 , , and for all M > , L > .We indeed see that the infinite variance is due to thecubic asymptotic decay in the probability density. Notethat in the case of Wigner time delay τ the asymptotictail of the probability density is rather τ − βM/ − makingthe variance finite for M > /β . Anticipating that for ρ = 0 RTD should be indistinguishable from the Wignertime delay, one may consider the parameter range ρ (cid:28) γ and find that in that case the decay (cid:102) δ T − βM/ − alsohappens for RTD in the intermediate asymptotic range /γ (cid:28) (cid:102) δ T (cid:28) /ρ , whereas the cubic tail takes over onlyas (cid:102) δ T (cid:29) /ρ , reconciling the two types of behaviour.One also can be interested in the short-time behaviourof the RTD correlation function. Considering for sim-plicity the simplest case M = L = 1 , we obtain in thislimit: C , ( t (cid:28)
1) = 12 (cid:20) − g − g ) g + g ) (cid:21) + ( g − g ) − g + g − g + g ) t + O ( t ) . (31)The first order term vanishes when g ± = g + 2 ± √ g and vice versa. For < g < √ , the second solution g − = g + 2 − √ g < is not valid since g, g > bydefinition. Thus, the correlator switches from an increas-ing to a decreasing function of time as g passes through g +0 . Reverting to the frequency domain, we find that inthis case correlator decays as ω − , whereas the correla-tor of the Wigner time delay always decays as ω − since ˙ C M, (0) = 1 / .Another curious observation is that due to appearanceof the factor sign ( y y ) in the integrand of (32), the term B ( t ) identically vanishes when Y is even in y and y separately, in which case C ( t ) only depends on the meanglobal density of complex eigenvalues rather than on thetwo-point correlation function. For equivalent channels,a necessary condition for this to happen is M = L and g = g .To compare with numerical simulations, we find thatinstead of directly computing 21, it is more practicalto consider instead the Fourier transform of the RTDweighted by a Gaussian: F ( t, W ) := (cid:90) dE √ πW (cid:102) δ T ( E ) e iEt − E W (32) = (cid:114) π W e − Wt N (cid:88) n =1 sign ( (cid:61) z n ) × (cid:20) erfcx (cid:18) |(cid:61) z n | + W t + i (cid:60) z n √ W (cid:19) + erfcx (cid:18) |(cid:61) z n | − W t − i (cid:60) z n √ W (cid:19)(cid:21) , (33)where erfcx ( z ) = e z erfc ( z ) and erfc ( z ) = √ π (cid:82) ∞ z e − x dx is the complementary error function. Ifwe take N (cid:28) √ W (cid:28) , then we find the approximaterelation: ∆4 π (cid:114) Wπ (cid:104)| F (cid:18) πt ∆ ; W (cid:19) | (cid:105) c ≈ C M,L ( t ) . (34)The advantage of this approach is that the ensembleaverage of F ( t ; W ) converges faster than that of the RTD.Figures 3a and 3b compare the correlator obtained in thisway from RMT simulations ( N = 300 ) with the predic-tion for M = 1 and L = 0 , . ON EXTRACTING S-MATRIX ZEROS FROMSCATTERING DATA
Let us now discuss a possibility of determining the po-sitions of the zeros of a subunitary scattering matrix froman experiment, real or numerical. One may imagine twopossibilities: either one has access to the total unitary S matrix, or only to its subunitary observable/scatteringblock S ( E ) . Indeed, one may consider constructing aCPA in the form of a three port microwave network whereone port plays the role of the attenuator providing ab-sorption. In such a setup, if one disregards imperfectionsin the setup, the output in all three ports is directly ac-cessible, hence the total S matrix.In the usual case of S − matrix poles, one of the mostwell-established methods for determining their positions | F ( t ; W ) | c d e i t ( ) ( ) c (a) M = 1 , L = 0 , g = 1 . | F ( t ; W ) | c d e i t ( ) ( ) c (b) M = 1 , L = 1 , g = g = 1 . Figure 3: C M,L ( t ) of 21 compared with simulations of × random matrices, using 34 as an estimator. Thesmall time discrepancy in the second figure appears to be afinite-size effect which occurs over a smaller window in timeas the dimension of the matrices increases. in the lower half-plane is the “harmonic inversion” [44],which estimates the poles by solving a set of non-linearequations. In a nutshell, one estimates a signal repre-sented in the time domain as a sum of decaying expo-nentials evaluated at times t = nτ , where τ is the samplerate, by relying on the Padé approximation: c n := N (cid:88) k =1 d k e − iz k nτ , (35) f ( z ) := ∞ (cid:88) n =0 c n z − n = N (cid:88) k =1 d k z k z − z k (36) = P K ( z ) Q K ( z ) , (37) where P K /Q K is the order ( K, K ) Padé approximant to f ( z ) . The zeros z k are the poles of Q ( z ) and the ampli-tudes d k are the ratio of the residues and the poles. Theinteger K is chosen as an upper bound to the numberof true zeros N , leading to N − K spurious zeros whichmust be discarded. In general, it is common to introducea cut-off and discard zeros for which the magnitude ofthe residue falls below the cut-off. The procedure for S − matrix poles outlined in [43] used both a cut-off and dis-carding those poles whose imaginary parts were smallerthan the energy spacing (distance between energy sam-ples), as well as with real parts near the boundary of thesampling window.Harmonic inversion can be performed directly on the S − matrix elements or, alternatively, on the Wigner timedelay. When estimating zeros, the signal to be consideredis given by the inverse of the determinant: S ( E ) = N (cid:89) n =1 E − E n + i Γ n / E − (cid:60) z n − i (cid:61) z n . (38)Alternatively, when the whole unitary S can be mea-sured, one can use the expression 10 for the RTD. As-suming additionally a weak uniform absorption (cid:15) (cid:28) in-side the scattering domain, the RTD can be alternativelycomputed from the unitary deficit of the determinant ra-tio, see [56] for a discussion: δ T ( E ) = − (cid:15) (cid:60) log det S ( E + i(cid:15) )det S (cid:48) ( E + i(cid:15) ) + O ( (cid:15) ) . (39)The advantage of using the RTD for extracting posi-tions of complex zeroes is that the Lorentzians in thesum are all normalised to unity, providing us with a wayto distinguish true and spurious zeros by looking at thecorresponding residues. The residues in 38 involve in-stead a product over the remaining zeros and poles whichcan take arbitrary values. It is not a priori clear howto choose an appropriate cut-off, particularly when thezeros/poles appear as Lorentzians with varying ampli-tudes. We compare the performance of both methodsfor extracting the zeros by simulating H from the GUE;parameters affecting the accuracy of the estimates arethe number of samples nE of S ( E ) and the strength ofthe uniform absorption (cid:15) . Since the first step of the har-monic inversion procedure is to take the Fourier trans-form, when using 38 we perform a second transform onthe complex conjugate so that the zeros in both half-planes are accounted for. The detection of spurious zerosdiffers between the two methods. In the first method, wefollow [43] in using a cut-off and removing zeros near theboundaries. The cut-off has been chosen as 1, the valueaccounting well for most of the spurious zeros. In the sec-ond method, the zeros occur in complex conjugate pairswhose residues (after normalising by the energy spacing)should add up to 1. We therefore grouped the zeros intoconjugate pairs and removed those whose residues weresignificantly different from 1. This allowed us to bypassthe need for the removal of zeros near the boundaries.Figure 4 shows the true zeros plotted against those es-timated from 38 and 39, for various parameter configu-rations. In figures 4a and 4c, there are what appear tobe spurious zeros (isolated green crosses) among thoseestimated by the second method. These arise because ofthe appearance of two complex conjugate pairs for thesame zero; in general we observe that as the strength ofthe uniform absorption (cid:15) increases, an increasing numberof zeros are associated with two or more complex conju-gate pairs. This is why these do not appear in figures 4band 4d, where (cid:15) = 10 − . One could attempt to group allpairs in order to bring the sum of resiudes closer to one,or simply discard all but one pair. Note also that in allfour figures there are zeros near the boundaries which arediscarded in the first method but included in the second. z z true zerosdeterminant zerosRTD zeros (a) γ = 0 . , ρ = 0 . , nE = 800 , (cid:15) = 10 − z z true zerosdeterminant zerosRTD zeros (b) γ = 0 . , ρ = 0 . , nE = 800 , (cid:15) = 10 − z z true zerosdeterminant zerosRTD zeros (c) γ = 0 . , ρ = 0 . , nE = 800 , (cid:15) = 10 − z z true zerosdeterminant zerosRTD zeros (d) γ = 0 . , ρ = 0 . , nE = 800 , (cid:15) = 10 − Figure 4:
Estimated zeros using 38 (orange plus signs) andusing 39 (green crosses) compared to true zeros (bluecircles). In all cases, N = 100 , M = 2 , L = 1 . DISCUSSION AND OPEN PROBLEMS.
In conclusion, we have studied the statistics of the ze-ros of the subunitary S matrix for a RMT-based modelof scattering in chaotic cavities with localized absorptionand broken time-reversal symmetry. The n -point corre-lation functions of the zeroes in the complex plane fora system with a finite number of scattering and absorb-ing channels ( M and L , respectively) can be calculatedby extending the method used in [34] for characterizingthe S − matrix poles (formally equivalent to the L = 0 case). The resulting kernel, in the limit of strong non-Hermiticity with at least one of the parameters M and/or L going to infinity, takes the form of a generalised Gini-bre kernel, expected to be universal in this regime [39].For finite M < ∞ and L < ∞ , the kernel was used toobtain the Fourier transform of the correlation function C M,L ( t ) of the RTD, which has been shown to decay as C M,L ( t ) ∼ t − at large times for any L > . This is insharp contrast with the case L = 0 (formally equivalentto replacing RTD with the Wigner time delay), where asimilar tail is known to be M -dependendent: ∼ t − M . Inparticular, this implies that the variance of the RTD isinfinite regardless of the value of M or L . We interpretthis divergence as an evidence for the existence of a fartail P ( (cid:102) δ T ) ∼ δ T ( E ) − in the RTD probability density,and verify this in the regime of weakly open and weaklyabsorbing system. We expect such a tail to be a supe-runiversal feature of the RTD distribition valid for allsymmetry classes.The short time behaviour of the RTD correlation func-tion is again markedly different for L = 0 and L > : C M, ( t ) ∼ t in the former case and C M,L ( t ) ∼ C M,L (0) + ˙ C M,L (0) t in the latter, with C M,L (0) (cid:54) = 0 and the coefficient ˙ C M,L (0) depending on g and g insuch a way that it changes from a positive value throughzero to negative values as g is reduced from infinity to-wards unity. This implies that there must exist a partic-ular value of absorption parameter g (depending on g )such that the large frequency asymptotic of the Fourier-transformed correlation function decays as ω − insteadof the typical decay ω − .We have also examined two methods for extracting thepositions of S − matrix zeros from experimental scatteringdata by harmonic inversion of either the inverse determi-nant of S or from the RTD. The first method is applicablewhen only the scattering part of S is available, while forthe second method the total S matrix must be accessi-ble. The advantage of the second method is that we knowin advance the residue of each zero, which allows us todistinguish more easily between true and spurious zeros.In the recent papers demonstrating the construction of adisordered CPA [26, 27] the S matrix was measured atregular intervals in an energy window, but rather thandetermine the zeros, each S matrix was diagonalised andthe eigenvalue closest to zero selected as a candidate forthe CPA state. Using harmonic inversion one can insteaddirectly estimate the zeros themselves.Finally, let us mention that all non-perturbative treat-ment in our paper has been restricted to systems withbroken time-reversal invariance. Actually, the mean den-sity of complex eigenvalues z n for chaotic systems withpreserved time-reversal invariance, with H being takenfrom the Gaussian Orthogonal Ensemble, can be deducedfor L > from known L = 0 results by an ad hoc analyticcontunuation, see [29]. However its systematic control-lable derivation, not speaking about deducing the formof the two-point (and higher) correlation functions, re-mains an outstanding and challenging problem. Thiscurrently prevents us from any non-perturbative insightsinto statistics of the Reflection Time Difference (apart from its expected superuniversal cubic tail in the proba-bility density) for this important case. Acknowledgments:
We thank Dr. MikhailPoplavskyi for sharing his notes about the derivationof (45) and Prof. Steven Anlage for his interest inthe project and valuable discussions of this and relatedtopics. MO acknowledges funding from the EPSRCCentre for Doctoral Training in Cross-Disciplinary Ap-proaches to Non-Equilibrium Systems (CANES) undergrant EP/L015854/1.
APPENDIXDerivation of the correlation function of S − matrixzeros. We give a brief sketch of the derivation of the exprres-sion 12 for the n -point correlation function. Since thederivation follows mainly the steps of [34] (explained inmore detail in [39]), we omit the detail and point outonly the necessary modification. The object of study isthe spectrum of the perturbed GUE matrix J = H + i Γ ,where Γ = diag ( γ , ..., γ M + L , , ..., is a rank M + L matrix where γ ≥ · · · ≥ γ M > > γ M +1 ≥ · · · γ M + L .It is the fact that the matrix Γ has eigenvalues of bothsigns which makes a difference from [34] and should beproperly accounted for. The joint probability density forthe matrix J induced by H is: P ( J ) dJ ∝ e − N tr (cid:16) J + J † (cid:17) δ (cid:18) J − J † i − Γ (cid:19) , (40)where δ ( A ) = (cid:81) i,j δ ( A ij ) . Making use of the Schurdecomposition J = U ( Z + R ) U † , the integral over theupper triangular matrix R can be performed with thedelta function in the off-diagonal elements. The remain-ing delta functions in the diagonal elements are repre-sented by a Fourier integral over a diagonal matrix K ,leaving a Harish-Chandra/Itzykson/Zuber (HCIZ) inte-gral with K and Γ . Since Γ is not of full rank, the limitof N − M − L eigenvalues going to zero is calculatedby repeated application of l’Hôpital’s rule to the originalHCIZ formula. The result is the following expression forthe density of eigenvalues Z : P ( Z ) ∝ | ∆( Z ) | det N − M − L ( γ )∆( γ ) e − N (cid:60) tr Z − N tr γ (41) × (cid:90) dK (2 π ) N e i tr K (cid:61) Z ∆( K ) D ( K ) where we denoted γ = diag ( γ , ..., γ M + L ) and introduced D ( K ) := (42)0det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − ik γ · · · e − ik γ M + L ( − ik ) N − M − L − · · · ... . . . ... ... . . . ... e − ik N γ · · · e − ik N γ M + L ( − ik N ) N − M − L − · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The difference now from the original derivation is thatthe terms e − ik j γ c are represented by the integral: e − ik j γ c = 12 πi (cid:90) L c e − ik j λ c λ c − k j dλ c , (43)where L c = sign ( γ c )( − R + i . After taking the λ in-tegrals outside the determinant and using the followingidentity: det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ − k · · · λ M − k ( − ik ) N −M− · · · ... . . . ... . . . ... λ − k N · · · λ M − k N ( − ik N ) N −M− · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∝ ∆(Λ)∆( K ) (cid:81) M j =1 (cid:81) Ni =1 ( λ j − k i ) , (44)which follows by elementary row and column operations,the final expression for the density P ( Z ) is: P ( Z ) ∝ N −M ( γ )∆( γ ) | ∆( Z ) | e − N Retr Z − N tr γ × (cid:90) L · · · (cid:90) L M d Λ∆(Λ) e − i (cid:80) M j =1 γ j λ j × N (cid:89) i =1 M (cid:88) j =1 e iλ j Im z i (cid:81) l (cid:54) = j ( λ l − λ j ) θ ( Im λ j Im z i ) , (45)From this point onwards the derivation follows ex-actly along the lines of [39], where in the appendix ofthat paper the n -point correlation function is related tothe average of a product of characteristic polynomialswhich is subsequently evaluated by integrating over anti-commuting variables. Distribution of RTD in the regime of weak couplingand absorption.
In this regime the imaginary part of the zeroes ismuch smaller than the typical separation between thereal parts, and, moreover, imaginary parts of neighbour-ing zeroes are independent to the leading order. Thenthe dominant contribution to RTD at a given real energy E can be estimated by a heuristic argument [33] thattakes into account only the zero whose real part is theclosest to the energy value E in the sum of Lorentzians10. Attributing the index n to this particular zero anddefining u n = βπ ∆ ( E − (cid:60) z n ) and y n = βπ ∆ (cid:61) z n , we have (cid:102) δ T ( E ) (cid:39) y n u n + y n . (46)The real and imaginary parts of z n are independentin the weak coupling regime, with the latter having the density: P βY ( y ) = (cid:90) dk π e iky (cid:81) Ma =1 (1 + 2 ikγ a ) β/ (cid:81) Lb =1 (1 − ikρ b ) β/ . (47)We also assume that u n is uniformly distributed in [ − βπ, βπ ] . The result of these approximations is the fol-lowing estimate for the density of (cid:102) δ T ≡ τ : P β (cid:102) δ T ( τ ) = (cid:90) dyP β ( y ) (cid:90) βπ − βπ duδ (cid:18) τ − yu + y (cid:19) (48) (cid:39) τ (cid:90) min(1 , β π τ )0 (cid:114) y − y (cid:20) θ ( τ ) P βY (cid:18) yτ (cid:19) (49) + θ ( − τ ) P βY (cid:18) − yτ (cid:19)(cid:21) dy. Focusing on positive (cid:102) δ T and equivalent channels γ a = γ (cid:28) , ρ b = ρ (cid:28) , let us first assume that ρ (cid:38) γ . Thenwe can identify two regimes for τ > / ( βπ ) : i) τ γ (cid:28) and ii) τ γ (cid:29) . In the first regime, the dominant termin P βY (2 y/ ( M τ )) is y βM/ − e − yτγ : P β (cid:102) δ T ( τ ) (cid:39) γ βL/ − τ Γ( βM/ γ + ρ ) βL/ (50) × (cid:90) ∞ √ y (cid:18) yτ γ (cid:19) βM/ − e − yτγ dy = γ β ( L +1) / ( γ + ρ ) βL/ τ − / , (51)where we have set the upper limit of integration toinfinity and taken − y (cid:39) , both justified by the ex-ponential damping. In the second regime, the dominantterm is now just e − yτγ , which is approximately unity: P β (cid:102) δ T ( τ ) (cid:39) Γ( β ( M + L ) / − βM/ βL/
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