Generalization of Wigner Time Delay to Sub-Unitary Scattering Systems
GGeneralization of Wigner Time Delay to Sub-Unitary Scattering Systems
Lei Chen,
1, 2, ∗ Steven M. Anlage,
1, 2, † and Yan V. Fyodorov
3, 4 Quantum Materials Center, Department of Physics,University of Maryland, College Park, MD 20742, USA Department of Electrical and Computer Engineering,University of Maryland, College Park, MD 20742, USA 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: January 22, 2021)We introduce a complex generalization of Wigner time delay τ for sub-unitary scattering systems.Theoretical expressions for complex time delay as a function of excitation energy, uniform and non-uniform loss, and coupling, are given. We find very good agreement between theory and experimentaldata taken on microwave graphs containing an electronically variable lumped-loss element. We findthat time delay and the determinant of the scattering matrix share a common feature in that theresonant behavior in Re[ τ ] and Im[ τ ] serves as a reliable indicator of the condition for CoherentPerfect Absorption (CPA). This work opens a new window on time delay in lossy systems andprovides a means to identify the poles and zeros of the scattering matrix from experimental data.The results also enable a new approach to achieving CPA at an arbitrary energy/frequency incomplex scattering systems. Introduction.
In this paper we consider the generalproblem of scattering from a complex system by meansof excitations coupled through one or more scatteringchannels. The scattering matrix S describes the transfor-mation of a set of input excitations | ψ in i on M channelsinto the set of outputs | ψ out i as | ψ out i = S | ψ in i .A measure of how long the excitation resides in the in-teraction region is provided by the time delay, relatedto the energy derivative of the scattering phase(s) ofthe system. This quantity and its variation with energyand other parameters can provide useful insights into theproperties of the scattering region and has attracted re-search attention since the seminal works by Wigner [1]and Smith [2]. A review on theoretical aspects of timedelays with emphasis to solid state applications can befound in [3]. Various aspects of time delay have recentlybeen shown to be of direct experimental relevance formanipulating wave fronts in complex media [4–6]. Timedelays are also long known to be directly related to thedensity of states of the open scattering system, see dis-cussions in [3] and more recently in [7, 8].For the case of flux-conserving scattering in systemswith no losses, the S -matrix is unitary and its eigenvaluesare phases e iθ a , a = 1 , , ..., M . These phases are func-tions of the excitation energy E and one can then defineseveral different measures of time delay, see e.g. [3, 9],such as partial time delays associated with each channel τ a = dθ a /dE , the proper time delays which are the eigen-values of the matrix ˆ Q = i ~ dS † dE S , and the Wigner-Smithdelay time which is the sum of all the partial time delays( τ W = M P Ma =1 τ a = M T r [ ˆ Q ]).A rich class of systems in which properties of vari-ous time delays enjoyed thorough theoretical attention ∗ [email protected] † [email protected] is scattering of short-wavelength waves from chaotic sys-tems, e.g. billiards with ray-chaotic dynamics. Variousexamples of chaotic wave scattering (quantum or clas-sical) have been observed in nuclei, atoms, molecules,ballistic two-dimensional electron gas billiards, and mostextensively in microwave experiments [10–15]. In suchsystems time delays have been measured starting fromthe pioneering work [16], followed over the last threedecades by measurement of the statistical properties oftime delay through random media [17, 18] and microwavebilliards [19]. Wigner-Smith time delay for an isolatedresonance described by an S -matrix pole at complex en-ergy E − i Γ has a value of Q = 2 ~ / Γ on resonance, hencestudies of the imaginary part of the S -matrix poles probeone aspect of time delay [20–25]. In the meantime, theWigner-Smith operator (WSO) was utilized to identifyminimally-dispersive principal modes in coupled multi-mode systems [26, 27]. A similar idea was used to createparticle-like scattering states as eigenstates of the WSO[4, 28]. A generalization of the WSO allowed maximal fo-cus on, or maximal avoidance of, a specific target insidea multiple scattering medium [6, 29].Time delays in wave-chaotic systems are expected to beextremely sensitive to variations of excitation energy andscattering system parameters, and will display universalfluctuations when considering an ensemble of scatteringsystems with the same general symmetry. Universalityof fluctuations allows them to be efficiently described us-ing the theory of large random matrices [9, 30–37]. Al-ternative theoretical treatments of time delay in chaoticscattering systems successfully adopted a semi-classicalapproach, see [7] and references therein.Despite the fact that standard theory of wave-chaoticscattering deals with perfectly flux-preserving systems,in any actual realisation such systems are inevitably im-perfect, hence absorbing, and theory needs to take thisaspect into account [38]. Interestingly, studying scatter- a r X i v : . [ c ond - m a t . d i s - nn ] J a n ing characteristics in a system with weak uniform (i.e.spatially homogeneous) losses may even provide a possi-bility to extract time delays characterizing idealized sys-tem without losses. This idea has been experimentallyrealized already in [16] which treated the effect of sub-unitary scattering by means of the unitary deficit of the S -matrix. In this case one defines the Q -matrix through S † S = 1 − ( γ ∆ / π ) Q , where γ is the dimensionless ‘ab-sorption rate’ and ∆ is the mean spacing between modesof the closed system. However, this version of time de-lay is always real and positive, and its definition is validonly in the limit of low loss. Various statistical aspects oftime delays in such and related settings were addressedtheoretically in [39–42].Experimental data is often taken on sub-unitary scat-tering systems and a straightforward use of the Wignertime delay definition yields a complex quantity. In ad-dition, both the real and imaginary parts acquire bothnegative and positive values, and they show a systematicevolution with energy/frequency and other parametersof the scattering system. This clearly calls for a detailedtheoretical understanding of this complex generalizationof the Wigner time delay. It is necessary to stress thatmany possible definitions of time delays which are equiv-alent or directly related to each other in the case of alossless flux-conserving systems can significantly differ inthe presence of flux losses, either uniform or spatially lo-calized. In the present paper we focus on a definitionthat can be directly linked to the fundamental character-istics of the scattering matrix - its poles and zeros in thecomplex energy plane, making it useful for fully charac-terizing an arbitrary scattering system. Note that bothpoles and zeros are objects of long-standing theoretical[43–54] and experimental [20–25, 54] interest in chaoticwave scattering. Complex Wigner Time Delay.
In our exposition we usethe framework of the so-called “Heidelberg Approach” towave-chaotic scattering reviewed from different perspec-tives in [55, 56] and [57]. Let H be the N × N Hamil-tonian which is used to model the closed system withray-chaotic dynamics, W denoting the N × M matrix ofcoupling elements between the N modes of H and the M scattering channels, and by A the N × L matrix ofcoupling elements between the modes of H and the L localized absorbers, modelled as L absorbing channels.The total unitary S -matrix, of size ( M + L ) × ( M + L )describing both the scattering and absorption on equalfooting, has the following block form, see e.g. [52]: S ( E ) = (cid:18) M − πiW † D − ( E ) W − πiW † D − ( E ) A − πiA † D − ( E ) W L − πiA † D − ( E ) A (cid:19) , (1)where we defined D ( E ) = E − H + i (Γ W + Γ A ) withΓ W = πW W † and Γ A = πAA † .The upper left diagonal M × M block of S ( E ) is theexperimentally-accessible sub-unitary scattering matrixand is denoted as S ( E ). The presence of uniform-in-space absorption with strength γ can be taken into ac-count by evaluating the S -matrix entries at complex en-ergy: S ( E + iγ ) := S γ ( E ). The determinant of such asubunitary scattering matrix S γ ( E ) is then given by:det S γ ( E ) := det S ( E + iγ ) (2)= det[ E − H + i ( γ + Γ A − Γ W )]det[ E − H + i ( γ + Γ A + Γ W )] (3)= N Y n =1 E + iγ − z n E + iγ − E n , (4)In the above expression we have used that the S -matrix zeros z n are complex eigenvalues of the non-self-adjoint/non-Hermitian matrix H + i (Γ W − Γ A ), whereasthe poles E n = E n − i Γ n with Γ n > H − i (Γ W + Γ A ). Note that when localized absorptionis absent, i.e. Γ A = 0, the zeros z n and poles E n arecomplex conjugates of each other, as a consequence of S -matrix unitarity for real E and no uniform absorption γ = 0. Extending to locally absorbing systems the stan-dard definition of the Wigner time delay as the energyderivative of the total phase shift we now deal with acomplex quantity: τ ( E ; A, γ ) := − iM ∂∂E log det S γ ( E ) (5)= Re τ ( E ; A, γ ) + i Im τ ( E ; A, γ ) , (6)Re τ ( E ; A, γ ) = 1 M N X n =1 (cid:20) Im z n − γ ( E − Re z n ) + (Im z n − γ ) + Γ n + γ ( E − E n ) + (Γ n + γ ) (cid:21) , (7)Im τ ( E ; A, γ ) = − M N X n =1 (cid:20) E − Re z n ( E − Re z n ) + (Im z n − γ ) − E − E n ( E − E n ) + (Γ n + γ ) (cid:21) (8)Equation (7) for the real part is formed by two Lorentzians for each mode of the closed system, poten-tially with different signs. This is a striking differencefrom the case of the flux-preserving system in which theconventional Wigner time delay is expressed as a singleLorentzian for each resonance mode [58]. Namely, thefirst Lorentzian is associated with the n th zero while thesecond is associated with the corresponding pole of thescattering matrix. The widths of the two Lorentziansare controlled by system scattering properties, and whenIm z n → γ ± E = Re z n . Note that thefirst term in Eq. 8 changes its sign at the same energyvalue. These properties are indicative of the “perfect res-onance” condition, with divergence in the real part of theWigner time delay signalling the wave/particle being per-petually trapped in the scattering environment. In dif-ferent words, the energy of the incident wave/particle isperfectly absorbed by the system due to the finite losses.The pair of equations (7, 8) forms the main basis forour consideration. In particular, we demonstrate in theSupplementary Material [59] that in the regime of well-resolved resonances Eqs. (7) and (8) can be used forextracting the positions of both poles and zeros in thecomplex plane from experimental measurements, pro-vided the rate of uniform absorption γ is independentlyknown. We would like to stress that in general the twoLorentzians in (7) are centered at different energies be-cause generically the pole position E n does not coincidewith the real part of the complex zero Re z n .From a different angle it is worth noting that there isa close relation between the objects of our study and thephenomenon of the so called Coherent Perfect Absorp-tion (CPA) which attracted considerable attention in re-cent years, both theoretically and experimentally [60–64].Namely, the above-discussed match between the uniformabsorption strength and the imaginary part of scatteringmatrix zero γ = Im z n simultaneously ensures the deter-minant of the scattering matrix to vanish, see Eq. (4).This is only possible when | ψ out i = 0 despite the factthat | ψ in i 6 = 0, which is a manifestation of CPA, see e.g.[51, 52]. Experiment.
We focus on experiments involvingmicrowave graphs [12, 63, 65, 66] for a number ofreasons. First, they provide for complex scatteringscenarios with well-isolated modes amenable to detailedanalysis. We thus avoid the complications of interactingpoles and related interference effects [67]. Graphs alsoallow for convenient parametric control such as variablelumped lossy elements, variable global loss, and breakingof time-reversal invariance. We utilize an irregulartetrahedral microwave graph formed by coaxial cablesand Tee-junctions, having M = 2 single-mode ports, andbroken time-reversal invariance. A voltage-controlledvariable attenuator is attached to one internal nodeof the graph (see Fig. 1(a)), providing for a variablelumped loss ( L = 1, the control variable Γ A ). Thecoaxial cables and tee-junctions have a roughly uniformand constant attenuation produced by dielectric loss and conductor loss, which is parameterized by the uniformloss parameter γ . The 2-port graph has total length of2.44 m , Heisenberg time τ H = 2 π/ ∆ = 148 ns and equalcoupling on both ports, characterized by a nominal valueof T a = 0 . Comparison of Theory and Experiments.
Figure 1shows the evolution of complex time delay for a singleisolated mode of the M = 2 port tetrahedral microwavegraph as Γ A is varied. The complex time delay is eval-uated as in Eq. 5 based on the experimental S ( f ) data,where f is the microwave frequency, a surrogate for en-ergy E . The resulting real and imaginary parts of thetime delay vary systematically with frequency, adopt-ing both positive and negative values, depending on fre-quency and lumped loss in the graph. The full evolutionanimated over varying lumped loss is available in theSupplemental Material [59]. These variations are well-described by the theory given above.Figure 1(d) and (e) clearly demonstrates that twoLorentzians are required to correctly describe the fre-quency dependence of the real part of the time delay. Thetwo Lorentzians have different widths in general, given bythe values of Im z n − γ and Γ n + γ , and in this case theLorentzians also have opposite sign. The frequency de-pendence of the imaginary part of the time delay alsorequires two terms, with the same parameters as for thereal part, to be correctly described. The data in Fig. 1(b)also reveals that Re[ τ ] goes to very large positive valuesand suddenly changes sign to large negative values at acritical amount of local loss. For another attenuation set-ting of the same mode it was found that the maximumdelay time was 371 times the Heisenberg time, showingthat the signal resides in the scattering system for a sub-stantial time.The measured complex time delay as a function of fre-quency can be fit to Eqs. (7) and (8) to extract thecorresponding pole and zero location for the S -matrix.The method to perform this fit is described in the Supp.Mat. [59] The fitting parameters are Re z n and Im z n − γ for the zero, and E n and Γ n + γ for the pole. Note thatthe Re[ τ ( f )] and Im[ τ ( f )] data are fit simultaneously,and a constant offset C is added to the real part fit.Figure 2 summarizes the parameters required to fit theexperimental complex time delay vs. frequency (shown inFig. 1) as the localized loss due to the variable attenua-tor in the graph is increased. The significant feature hereis the zero-crossing of Im z n − γ at frequency f = f CPA ,which corresponds to the point at which Re[ τ ( f )] changessign. As shown in Fig. 2(a) this coincides with the pointat which | det( S ( f )) | achieves its minimum value at theCPA frequency f CPA . This demonstrates that one ormore eigenvalues of the S -matrix go through a complexzero value precisely as the condition Im z n − γ = 0 and f − Re z n = 0 is satisfied. Associated with this condition | Re[ τ ( f CPA )] | diverges, with corresponding large positiveand negative values of Im[ τ ( f )] occurring just below andjust above f = f CPA . Similar behavior of Re[ τ ( f )] was FIG. 1. a shows a schematic of the graph experimental setup. The lumped loss Γ A is varied by changing the applied voltageto the variable attenuator. b and c show experimental data of both real and imaginary parts of Wigner time delay Re[ τ ]and Im[ τ ] (normalized by the Heisenberg time τ H ) as a function of frequency under different attenuation settings for a singleisolated mode. For each attenuation setting, the data is plotted from 2.645 GHz to 2.665 GHz. For clarity, plots with higherattenuation setting are shifted 0.01 GHz from the previous one. Inset shows the entire range of Re[ τ ] for attenuation setting of2.35 dB. d and e demonstrate the two-Lorentzian nature of the real and imaginary parts of the Wigner time delay as a functionof frequency. The constant C used in the Re[ τ ] fit is C = 0 .
29 in units of τ H . recently observed in a complex scattering system con-taining re-configurable metasurfaces, as the pixels weretoggled [64].Next we wish to estimate the value of uniform atten-uation γ for the microwave graph. Using the unitarydeficit of the S -matrix in the minimum-loss case [16], weevaluate the uniform loss strength γ to be 1 . × − GHz.Figure 2(b) summarizes the locations of the S -matrixpole E n and zero z n of the single isolated mode of themicrowave graph in the complex frequency plane as thelocalized loss is varied. When the S -matrix zero crossesthe Im z n = γ value, one has the traditional signature of CPA. Note from Fig. 2 that the real parts of the zeroand pole do not coincide and in fact move away fromeach other as localized loss is increased. Discussion.
It should be noted that the occurrence ofa negative real part of the time delay is an inevitable con-sequence of sub-unitary scattering, and is also expectedfor particles interacting with attractive potentials [68].The imaginary part of time delay was in the past dis-cussed in relation to changes in scattering unitary deficitwith frequency [69]. Another approach to defining com-plex time delay has been recently suggested to be basedon essentially calculating the time delay of the signalwhich comes out of the system without being absorbed
FIG. 2. a | Fitted parameters Im z n − γ and Γ n + γ for thecomplex Wigner time delay from graph experimental data.Also shown is the evolution of | det( S ) | at the specific fre-quency of interest, f CPA , which reaches its minimum at thezero-crossing point. Inset shows the evolution of Re z n and E n = Re E n with attenuation. b | Evolution of complex zeroand pole of a single mode of the graph in the complex fre-quency plane as a function of Γ A . The black crosses are theinitial state of the zero and pole at the minimum attenuationsetting. Insets show the details of the complex zero and polemigrations. [64]. It should be noted that this ad hoc definition oftime delay is not simply related to the poles and zeros ofthe S -matrix. Moreover, a closer inspection shows that such a definition of complex time delay tacitly assumesthat the real parts of the pole and zero are identical. Ac-cording to our theory such an assumption is incompatiblewith a proper treatment of localized loss.We emphasize that the correct knowledge of the loca-tions of the poles and zeros is essential for reconstruct-ing the scattering matrix over the entire complex energyplane through Weierstrass factorization [70]. Our resultstherefore establish a systematic procedure to find the S -matrix zeros and poles of isolated modes of a complexscattering system with an arbitrary number of couplingchannels, symmetry class, and arbitrary degrees of bothglobal and localized loss. (Graph simulations further sup-port these conclusions [59].)Recent work has demonstrated CPA in disordered andcomplex scattering systems [62, 63]. It has been discov-ered that one can systematically perturb such systems toinduce CPA at an arbitrary frequency [64, 71], and thisenables a remarkably sensitive detector paradigm [64].These ideas can also be applied to optical scattering sys-tems where measurement of the transmission matrix ispossible [72]. Here we have uncovered a general formal-ism in which to understand how CPA can be created inan arbitrary scattering system. In particular this workshows that both the global loss ( γ ), localized loss centers,or changes to the spectrum can be independently tunedto achieve the CPA condition.Future work includes treating the case of overlappingmodes, and the development of theoretical predictions forthe statistical properties of both the real and imaginaryparts of the complex time delay in chaotic and multiplescattering sub-unitary systems. Conclusions.
We have introduced a complex general-ization of Wigner time delay which holds for arbitraryuniform/global and localized loss, and directly relates topoles and zeros of the scattering matrix in the complexenergy/frequency plane. Based on that we developed the-oretical expressions for complex time delay as a functionof energy, and found very good agreement with exper-imental data on a sub-unitary complex scattering sys-tem. Time delay and det( S ) share a common featurethat CPA and the divergence of Re[ τ ] and Im[ τ ] coin-cide. This work opens a new window on time delay inlossy systems, enabling extraction of complex zeros andpoles of the S -matrix from data.We acknowledge Jen-Hao Yeh for early experimentalwork on complex time delay. This work was supportedby AFOSR COE Grant No. FA9550-15-1-0171, NSFDMR2004386, and ONR Grant No. N000141912481. [1] E. P. Wigner, Lower limit for the energy derivative of thescattering phase shift, Physical Review , 145 (1955).[2] F. T. Smith, Lifetime matrix in collision theory, PhysicalReview , 349 (1960).[3] C. Texier, Wigner time delay and related concepts: Ap-plication to transport in coherent conductors, Physica E: Low-dimensional Systems and Nanostructures , 16(2016).[4] S. Rotter, P. Ambichl, and F. Libisch, Generating parti-clelike scattering states in wave transport, Physical Re-view Letters , 120602 (2011).[5] J. Carpenter, B. J. Eggleton, and J. Schr¨oder, Ob- servation of Eisenbud–Wigner–Smith states as princi-pal modes in multimode fibre, Nature Photonics , 751(2015).[6] M. Horodynski, M. K¨uhmayer, B. A, K. Pichler, Y. V.Fyodorov, U. Kuhl, and S. Rotter, Optimal wave fieldsfor micromanipulation in complex scattering environ-ments, Nature Photonics , 149 (2020).[7] J. Kuipers, D. V. Savin, and M. Sieber, Efficient semiclas-sical approach for time delays, New Journal of Physics , 123018 (2014).[8] M. Davy, Z. Shi, J. Wang, X. Cheng, and A. Z. Genack,Transmission eigenchannels and the densities of statesof random media, Physical Review Letters , 033901(2015).[9] Y. V. Fyodorov and H. J. Sommers, Statistics of reso-nance poles, phase shifts and time delays in quantumchaotic scattering: Random matrix approach for systemswith broken time-reversal invariance, Journal of Mathe-matical Physics , 1918 (1997).[10] H.-J. St¨ockmann, Quantum Chaos: An Introduction (Cambridge University Press, 1999).[11] A. Richter, Wave dynamical chaos: An experimental ap-proach in billiards, Physica Scripta
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Lei Chen,
1, 2
Steven M. Anlage,
1, 2 and Yan V. Fyodorov
3, 4 Quantum Materials Center, Department of Physics,University of Maryland, College Park, MD 20742, USA Department of Electrical and Computer Engineering,University of Maryland, College Park, MD 20742, USA 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: January 22, 2021)
We provide some additional details for some of the calculations described in the text of the Letter.
I. EXTRACTING POLES AND ZEROS FROM EXPERIMENTAL DATA
Consider a pair of single terms in the sum over n in Eqs. (7-8)Re τ n ( f ) = (cid:20) Im z n − γ ( f − Re z n ) + (Im z n − γ ) + Γ n + γ ( f − E n ) + (Γ n + γ ) (cid:21) , (S1) − Im τ n ( f ) = (cid:20) f − Re z n ( f − Re z n ) + (Im z n − γ ) − f − E n ( f − E n ) + (Γ n + γ ) (cid:21) (S2)where we ‘relabeled’ the energy parameter E as ‘frequency’ f .Extracting the parameters Re z n , Im z n , E n , Γ n and the uniform absorption strength γ from the experimentallymeasured curves Im τ n ( f ) and Re τ n ( f ) is possible by the following procedure.1. Let us define the “resonance frequency” value f = f ∗ by the conditionIm τ n ( f ∗ ) = 0 (S3)2. Measure from the two above curves the following 3 parameters: s = − ddf Im τ n ( f ) | f = f ∗ , m = Re τ n ( f ∗ ) , δ = − ddf ln [Re τ n ( f )] | f = f ∗ (S4)and use them to build combinations: α − = 12 (cid:16) − sm + m (cid:17) , α + = 12 (cid:16) sm + m (cid:17) (S5)The following relations then can be derived for the positions of resonance poles and zeros.For the real parts: E n = f ∗ − δδ + α − , Re z n = f ∗ − δδ + α (S6)For the imaginary parts: Γ n = − γ + α − δ + α − , Im z n = γ + α + δ + α (S7)Derivation of relations (S6)-(S7)Define for brevity x n := f ∗ − Re z n , r n := Im z n − γ , (cid:15) n := f ∗ − E n and p n := Γ n + γ . a r X i v : . [ c ond - m a t . d i s - nn ] J a n Then condition (S3) implies x n x n + r n = (cid:15) n (cid:15) n + p n (S8)Also we have by the definition m = Re τ n ( f ) | f = f ∗ = r n x n + r n + p n (cid:15) n + p n (S9)On the other hand differentiating (S2) gives with this notation: s = − ddf Im τ n ( f ) | f = f ∗ = r n − x n ( x n + r n ) − p n − (cid:15) n ( p n + (cid:15) n ) (S10)Now using the condition (S8) the above equation simplifies to s = r n ( x n + r n ) − p n ( p n + (cid:15) n ) = (cid:20) r n ( x n + r n ) − p n ( p n + (cid:15) n ) (cid:21) (cid:20) r n ( x n + r n ) + p n ( p n + (cid:15) n ) (cid:21) (S11)which after using (S9) gives s/m = (cid:20) r n ( x n + r n ) − p n ( p n + (cid:15) n ) (cid:21) (S12)Now the pair (S9)-(S12) implies α + := 12 (cid:16) sm + m (cid:17) = r n x n + r n , α − := 12 (cid:16) − sm + m (cid:17) = p n p n + (cid:15) n (S13)Finally, let us consider − ddf Re τ n ( f ) | f = f ∗ = r n x n ( x n + r n ) + p n (cid:15) n ( p n + (cid:15) n ) (S14)which after first using (S8) and then (S9) can be further rewritten as= (cid:15) n p n + (cid:15) n (cid:20) r n x n + r n + p n p n + (cid:15) n (cid:21) = (cid:15) n p n + (cid:15) n Re τ n ( f ) | f = f ∗ implying finally δ := − ddf ln [Re τ n ( f )] | f = f ∗ = (cid:15) n p n + (cid:15) n = r n x n + r n (S15)The pair of equations (S13) and (S15) can be easily solved and gives (S6)-(S7).So the only parameter which remains to be found from independent measurement is the uniform absorption γ . Note:
Note that generically both s = − ddf Im τ n ( f ) | f = f ∗ = 0 and m = Re τ n ( f ) | f = f ∗ = 0 so | α + | 6 = | α − | . Wethen see from Eq. (S6) that to have E n = Re z n is only possible if δ = 0, which in turn is only possible if ddf [Re τ n ( f )] | f = f ∗ = 0. The latter condition would mean that the maximum or minimum on the curve Re τ n ( f )happens exactly at the same frequency f ∗ where the imaginary part vanishes. Generically this never happensdue to the two-Lorentzian nature of Re τ n ( f ). II. SIGN CONVENTION FOR THE PHASE EVOLUTION OF THE S -MATRIX ELEMENTS It should be noted that there are two widely-used conventions for the evolution of the phase of the complex S -matrix elements with increasing frequency. Microwave network analyzers utilize a convention in which the phase of thescattering matrix elements decreases with increasing frequency. Here we adopt the convention used in the theoreticalliterature that the phase of S -matrix elements increases with increasing frequency. III. FURTHER DETAILS ABOUT CPA AND COMPLEX TIME DELAY
We note that at CPA both the peak in | Re τ | and the point at which Im τ changes its sign coincide in energy, butaway from CPA they may occur at different energies. Note that the real and imaginary parts share the same forms inthe denominator of the Lorentzians. This leads to a synchronous evolution of their shapes with energy and scatteringcharacteristics. This property is clearly demonstrated in both the data shown in the main text and in the simulationsbelow. IV. CONNECTIONS TO EARLIER WORK ON NEGATIVE REAL TIME DELAY AND IMAGINARYTIME DELAY
In the past, the concept of a complex transmission delay was developed in the context of principal modes in multi-mode waveguides [1], and a similar quantity was later used in the experimental realization of particle-like scatteringstates [2]. Complex dwell time was defined for a multiple scattering medium with lossy resonant absorbers [3].An early study of Gaussian pulse propagation through an anomalously dispersive medium predicted negative delay[4], and later measurements confirmed the theory [5]. The observed negative delay of the pulse was attributedto the fact that the leading edge of the pulse is attenuated less than the later parts, and the Gaussian shape isapproximately preserved, under appropriate circumstances. Negative real parts of reflection delay time have beenmeasured experimentally in a one-dimensional Levy-flight system but were dismissed as an artefact due to promptreflections [6].Our findings show that in general the real part of the delay as a function of frequency is not symmetric about theresonance as a consequence of the differences in the real parts of z n and E n , and that it has a distinctive two-Lorentziancharacter that had not been appreciated until now. V. SIMULATIONS OF GRAPHS AND EVALUATION OF COMPLEX TIME DELAY
We have simulated the microwave graphs using CST Microwave Studio utilizing an idealized simulation model. Inthis model, an irregular tetrahedral graph similar to that used in the experiment is considered. The graph nodesare represented by Tee-junctions which are set to be ideal (point-like with an ideal scattering matrix), and the onlysource of loss in the setup comes from uniform dielectric loss of the coaxial cables which can be conveniently variedby changing the dielectric loss parameter tan δ . The 2-port graph simulation model has a total length of 6.00 m ,Heisenberg time τ H = 364 ns, equal coupling on both ports, characterized by a nominal value of T a = 0 .
75, and nolumped loss (i.e. Γ A = 0).The new insight created by the simulation comes from the ability to systematically vary the uniform loss γ whilekeeping all other parameters fixed. In this way we can show how CPA can be accomplished by simply changing theuniform loss. Results are shown in Figs. S1 and S2, which are similar to the experimental results Figs. 1 and 2 inthe main text. The CPA condition is achieved for a single isolated mode at f CPA = 6.15265 GHz when the uniformloss tangent value is tuned through a value of tan δ = 7 . × − .We note that a small constant C = 0 . τ H is required to fit the Re[ τ ] data, presumably because one of theLorentzians is quite wide, and because of neighboring modes outside this frequency window. VI. ANIMATIONS OF TIME DELAY EVOLUTION WITH VARIATION OF SYSTEM PARAMETERS
Animation of experimental Re[ τ ( f )] and Im[ τ ( f )] as lumped loss is varied through the CPA condition.Animation of simulation Re[ τ ( f )] and Im[ τ ( f )] as uniform loss γ is varied through the CPA condition. [1] S. Fan and J. M. Kahn, Principal modes in multimode waveguides, Optics Letters , 135 (2005).[2] J. B¨ohm, A. Brandst¨otter, P. Ambichl, S. Rotter, and U. Kuhl, In situ realization of particlelike scattering states in amicrowave cavity, Physical Review A , 021801 (2018).[3] M. Durand, S. M. Popoff, R. Carminati, and A. Goetschy, Optimizing light storage in scattering media with the dwell-timeoperator, Physical Review Letters , 243901 (2019). FIG. S1.
Complex Wigner time delay from graph simulation with varying uniform loss γ . a shows a schematicof the graph simulation setup. The uniform loss γ is varied by changing the dielectric loss parameter tan δ of the coaxialcables. b and c show simulation data of both real and imaginary parts of Wigner time delay Re[ τ ] and Im[ τ ] (normalizedby the Heisenberg time τ H ), as a function of frequency under different uniform loss settings for a single isolated mode near6.1526 GHz. Inset in b shows the entire range of Re[ τ ] for tan δ = 7 . × − , and inset in c shows the zoom-in view of Im[ τ ]for tan δ = 10 − . d and e demonstrate the two-Lorentzian nature of the real and imaginary parts of the Wigner time delay(normalized by the Heisenberg time τ H ) as a function of frequency. The constant C used in the Re[ τ ] fit is C = 0 .
097 in unitsof τ H .[4] C. G. B. Garrett and D. E. McCumber, Propagation of a gaussian light pulse through an anomalous dispersion medium,Physical Review A , 305 (1970).[5] S. Chu and S. Wong, Linear pulse propagation in an absorbing medium, Physical Review Letters , 738 (1982).[6] L. A. Razo-L´opez, A. A. Fern´andez-Mar´ın, J. A. M´endez-Berm´udez, J. S´anchez-Dehesa, and V. A. Gopar, Delay time ofwaves performing l´evy walks in 1d random media, Scientific reports , 20816 (2020). FIG. S2.
Evolution of scattering matrix zero and pole for a single mode in the graph simulation with varyinguniform loss γ . Plot shows the fitted parameters Im z n and Γ n for the complex Wigner time delay from graph simulationdata. Also shown is the evolution of | det( S ) | at the specific frequency of interest, f CPA , which reaches its minimum at thecrossover point where Im z n − γ = 0. Inset a shows the zoom-in details of the crossover when Im z n matches with γ , and inset b shows the evolution of Re z n and E n = Re E n with varying tan δδ