On-demand coherent perfect absorption in complex scattering systems: time delay divergence and enhanced sensitivity to perturbations
Philipp del Hougne, K. Brahima Yeo, Philippe Besnier, Matthieu Davy
OOn-demand coherent perfect absorption in complex scattering systems:time delay divergence and enhanced sensitivity to perturbations
Philipp del Hougne, K. Brahima Yeo, Philippe Besnier, and Matthieu Davy Univ Rennes, INSA Rennes, CNRS, Institut d’Electronique et desTechnologies du num´eRique, UMR6164, F-35000 Rennes, France (Dated: October 14, 2020)Non-Hermitian photonic systems capable of perfectly absorbing incident radiation recently at-tracted much attention both because fundamentally they correspond to an exotic scattering phe-nomenon (a real-valued scattering matrix zero) and because their extreme sensitivity holds greattechnological promise. The sharp reflection dip is a hallmark feature underlying many envisionedapplications in precision sensing, secure communication and wave filtering. However, a rigorous linkbetween the underlying scattering anomaly and the sensitivity of the system to a perturbation isstill missing. Here, we develop a theoretical description in complex scattering systems which quan-titatively explains the shape of the reflection dip. We further demonstrate that coherent perfectabsorption (CPA) is associated with a phase singularity and we relate the sign of the diverging timedelay to the mismatch between excitation rate and intrinsic decay rate. We confirm our theoreticalpredictions in experiments based on a three-dimensional chaotic cavity excited by eight channels.Rather than relying on operation frequency and attenuation inside the system to be two free pa-rameters, we achieve “on-demand” CPA at an arbitrary frequency by tweaking the chaotic cavity’sscattering properties with programmable meta-atom inclusions. Finally, we theoretically prove andexperimentally verify the optimal sensitivity of the CPA condition to minute perturbations of thesystem.
I. INTRODUCTION
The scattering of waves as they interact with matter isthe basis of countless experimental methods, a prominentexample being imaging. Recently, many exotic scatter-ing phenomena such as perfect absorption, exceptionalpoints or bound states in the continuum have been ex-tensively studied due to their disruptive potential in ar-eas such as sensing and computing; fundamentally, theycan be understood in terms of analytical properties ofthe associated scattering matrix [1]. Indeed, any wavescattering process is fully characterized by the distribu-tion of poles and zeros of the scattering matrix in thecomplex frequency plane [1, 2]. Poles and zeros are spec-tral singularities associated with outgoing and incomingboundary conditions, respectively. By including the ap-propriate amount of gain in the system, a pole can bepulled up onto the real frequency axis: the scatteringmatrix will have an infinite eigenvalue and lasing occurs.Conversely, by including the appropriate amount of loss,a zero can be pulled down onto the real frequency axis,resulting in a zero eigenvalue of the scattering matrix andcoherent perfect absorption (CPA) [3, 4]. Incident radi-ation corresponding to the eigenvector associated withthe zero eigenvalue will be perfectly absorbed. CPA canbe interpreted as a generalization of the critical cou-pling condition [5] and can be understood as the timereverse operation of a laser (anti-laser) [6, 7]. The gen-erality of these concepts implies that they also apply torandom scattering media. Indeed, a random laser reso-nantly enhances light by multiple scattering inside a dis-ordered medium [8, 9]. Recently, the feasibility of realiz-ing CPA in random scattering media and chaotic cavitieshas been studied theoretically and demonstrated exper- imentally [10–13], overcoming the immense difficulty ofbalancing excitation and decay rate of a random system.A hallmark signature of a system with a scattering ma-trix zero on the real frequency axis is a very pronounceddip of the energy reflected off the system as a function offrequency or any other local or global system parameter,evidencing an extreme sensitivity to tiny perturbations.This feature is at the heart of the concept’s technolog-ical relevance: for regular systems, it was, for instance,leveraged to demonstrate coherent modulation of lightwith light, i.e. without any non-linearity [14–16]; for ran-domly scattering systems, the extreme sensitivity is thebasic ingredient of a recently demonstrated physically se-cure wireless communication scheme [17] as well as ofenvisioned precision-sensing applications [13, 17]. Intu-itively, one may explain this extreme sensitivity with thefact that the incident radiation is trapped for an infi-nite time [3] when a real-valued scattering matrix zero isaccessed: the longer the wave’s lifetime inside a chaoticsystem, the more likely it is that its evolution is impactedeven by a tiny perturbation. Nonetheless, a rigorous ex-planation of the physical origins of this extreme sensitiv-ity and its link to the underlying scattering anomaly isto date missing.In our work, we fill this gap by studying the time delayof waves at the CPA condition inside a random medium.Our theoretical model quantitatively explains the shapeof the reflection dip and relates the sign of the divergingtime delay to the difference between the systems excita-tion and decay rate. We further analytically demonstratethe CPA conditions optimal sensitivity to minute per-turbations, irrespective of their location inside a chaoticsystem. Our theoretical findings are corroborated withrandom matrix simulations as well as experiments in the a r X i v : . [ phy s i c s . c l a ss - ph ] S e p microwave domain involving a chaotic cavity. To facili-tate accessing a real-valued zero experimentally, we tuneour disordered system’s scattering properties with pro-grammable meta-atom inclusions to a state in which azero of the scattering matrix hits the horizontal axis [17].This procedure enables the “on-demand” observation of aCPA condition at any desired frequency and without re-liance on attenuation being dominated by a localized andtunable loss center. Our results pave the way for sensorswith optimal sensitivity to minute perturbations of dis-ordered matter such as tiny intrusions, defaults, changesin temperature or concentration. II. THEORETICAL MODEL OF TIME DELAYSIN RANDOM CPAA. Dip in Frequency-Dependent ReflectionCoefficient
The wave-matter interaction in a complex scatteringsystem is fully characterized by its scattering matrix S ( ω )which relates an incoming field ψ in to the correspondingoutgoing field ψ out via ψ out = S ( ω ) ψ in . In a systemwithout any absorption or loss, S ( ω ) is unitary and itszeros ( z m = ω m + i Γ m /
2) and poles (˜ ω m = ω m − i Γ m / ω m and Γ m denote central frequency and linewidth of the system’s m th resonance. In non-Hermitian systems, the presenceof attenuation (or gain) Γ a moves the zeros in the com-plex plane: z m = ω m + i (Γ m − Γ a ) /
2. When attenuationlosses exactly balance dissipation through the channelsfor a given zero (labelled with the subscript n in thefollowing), that is Γ a = Γ n , the zero crosses the real fre-quency axis such that z n = ω n . Then, S ( ω n ) has a zeroeigenvalue such that the corresponding eigenvector ψ CPA satisfies S ( ω n ) ψ CPA = 0 and the multi-channel reflectioncoefficient R ( ω n ) = (cid:107) ψ out (cid:107) vanishes: R ( ω n ) → M × M Hamiltonian H describes the internal system, itscoupling to the N channels is characterized by a matrix V , and S ( ω ) = − iV T [ ω − H + i ( V V T + Γ a ) / − V [18–20]. The scattering matrix can also be decomposedin terms of the system’s natural resonances (the poles)as [18, 21, 22] S ( ω ) = − i Σ Mm =1 W m W Tm ω − ω m + i (Γ m + Γ a ) / . (1)The complex eigenfrequencies ˜ ω m = ω m − i Γ m / H eff = H − iV V T / W m are the projec-tions of the eigenfunctions φ m of H eff onto the chan-nels: W m = V T φ m . Using the completeness of theeigenfunctions, it can be demonstrated (see SM) that theeigenstate corresponding to a CPA condition at ω = ω n , S ( ω n ) ψ CPA = 0, is the time-reverse of the modal wave-front: ψ CPA = W ∗ n / (cid:107) W n (cid:107) . ψ CPA hence provides maximalexcitation of the selected mode [22, 23]. The interpreta-tion of W ∗ n as the time-reversed output of a lasing mode(if loss mechanisms were replaced by gain mechanisms ofequal strength) led to the term “anti-laser” [3, 7, 12], ananalogy that should be used with caution since it neglectsessential nonlinear processes in laser operation.Any realistic experimental observation of CPA is how-ever inevitably confronted with multiple practical imper-fections: (i) noise corrupts the measurement of S suchthat one measures S + ∆ S , (ii) a small mismatch ∆Γbetween the modal linewidth and losses: Γ a = Γ n + ∆Γ,(iii) a frequency shift ∆ ω = ω − ω n . In the vicinity of theCPA condition, the resonance associated with the real-valued zero dominates the sum in Eq. (1). The latterimplies that ψ CPA is still an eigenvector of S with a re-flection coefficient which is therefore not zero but (seeSM for algebraic details) R ( ω ) = 4∆ ω + (∆Γ) ω + (2Γ n + ∆Γ) + (cid:107) ∆ S (cid:107) F N . (2)We will show below that this equation explains the char-acteristic shape of the reflection dip and can be exploitedto extract the experimental parameters by fitting themeasured data with this model.
B. Time Delay
Having an analytical description of the reflection dipat the CPA condition in a random system, we can moveon to study the associated time delays. The time de-lay of an incoming wavefront scattered in a multichan-nel system is determined via the Wigner-Smith operator Q ( ω ) = − iS ( ω ) † ∂ ω S ( ω ) which involves the derivative of S ( ω ) with angular frequency: τ ( ω ) = ψ † in Q ( ω ) ψ in ψ † in S ( ω ) † S ( ω ) ψ in . (3)The real part of the complex-valued τ ( ω ) is related tothe frequency derivative of the scattering phase and canhence be interpreted as the delay of reflected intensity foran incoming pulse with vanishing bandwidth. The imag-inary part of τ ( ω ) is related to the variation of reflectedintensity with frequency [24–26]. Our model also allowsus to estimate the real and imaginary parts of τ ( ω ) atthe CPA condition, ψ in = ψ CPA , (see SM):Re[ τ ( ω )] = 1 R ( ω ) 4Γ n (4∆ ω − ∆Γ(2Γ n + ∆Γ))[4∆ ω + (2Γ n + ∆Γ) ] (4)andIm[ τ ] = − n ω + ∆Γ ω (Γ n + ∆Γ)4∆ ω + (2Γ n + ∆Γ) , (5)where R ( ω ) is given by Eq. (S9). Obviously both thereal and imaginary parts of τ ( ω ) diverge as ∆ ω → →
0, which confirms the intuition that the waveinjected into the system is trapped for an infinitely longtime at the CPA condition.Surprisingly, we observe a phase transition of Re[ τ ( ω = ω n )] as the amount of losses increases and surpasses themodal linewidth. When the zero z n = ω n + i (Γ n − Γ a ) / a < Γ n ), the time delay Re[ τ ( ω )] is positive. Atthe crossover of the real frequency axis (Γ a = Γ n ), asingularity occurs and | Re[ τ ( ω )] | diverges. When lossesdominate the coupling to channels, Γ a > Γ n , Re[ τ ( ω )]becomes negative. We note here that negative time de-lays have previously been found for wavefronts that arestrongly absorbed by lossy resonant targets within multi-ply scattering media [26–28]. Negative time delays arisedue to the distortion of the incident pulse for which theintensity is more strongly absorbed at long lifetimes thanat early times (see SM). C. Sensitivity to Perturbations
We now seek to demonstrate that the strong enhance-ment of the delay time provides an extreme sensitivity ofthe outgoing field to tiny perturbations within the sys-tem. To establish this link, let us begin by considering the generalized
Wigner-Smith operator Q α = − iS † ∂ α S [29]defined with respect to a change of a parameter α of thesystem; α can be a local or a global parameter. In sys-tems where S is close to unitarity, the optimal eigenstatesof Q α provide the optimal wavefronts to locally manip-ulate a perturber [29, 30]. In analogy with the delaytime in Eq. (3), the variations of the outgoing field for achange of α are encapuslated within the complex param-eter τ α = ψ † in Q α ψ in /R . The relation between the delaytime τ ( ω ) and τ α at the CPA condition can be estab-lished using the modification of the system’s resonancesdue to the perturbation.First, we note that upon injection of the CPA wave-front ( ψ in = ψ CPA ), the variation of the outgoing fieldin the vicinity of the CPA condition results from thechange of a single eigenstate (labelled n ) in Eq. (1).This property yields a linear relation between the projec-tion of the CPA wavefront on Q α and on the generalized Wigner-Smith operator applied to a perturbation of ω n , Q ω n = − iS † ∂ ω n S , ψ † CPA Q α ψ CPA = [ ∂ α ω n ] ψ † CPA Q ω n ψ CPA . (6)The perturbation-induced shift in ω n is given by ∂ α ω n = β ∇| φ ( r ) | for local perturbations, where ∇| φ ( r ) | is the gradient of the energy density taken in the direction ofthe displacement and β depends on the geometry of theperturber [31].Second, in the approximation of a single resonance con-tribution, the projection on the operators Q = − iS † ∂ ω S and Q ω n yields the same result except for a global mi-nus sign. This provides the sought-after relation between τ ( ω ) and τ α for the CPA condition: τ α = − [ ∂ α ω n ] τ ( ω ) . (7)The divergence of τ ( ω ) hence leads to an extreme sensi-tivity of the outgoing field. In our case, the CPA wave-front does not provide focusing on a perturber but is theoptimal wavefront to detect a tiny variation anywherewithin the cavity. We emphasize that ψ CPA is also theoptimal eigenvector of the operator − iS − ∂ α S (see SM).This property results from the vanishing eigenvalue of S leading to a pseudo-inverse matrix S − dominated by asingle eigenstate.Equation (7) directly provides the sensitivity of thereflection coefficient at the CPA condition. Using that ∂ α R = − ψ † CPA Q α ψ CPA ], we obtain1
R ∂ α R = [ ∂ α ω n ]Im[ τ ( ω )] . (8)The derivative of the logarithm of the reflection coeffi-cient ( ∂ α log( R ) = [ ∂ α R ] /R ) hence increases extremelyrapidly in the vicinity of the CPA condition. Thisunique feature makes it possible to finely characterizethe strength of the perturbation from the shape of thereflection dip. Because the change in R is proportionalto the change in ω n , this dip is anew given by Eq. (S9) inwhich ∆ ω has to be replaced with ∆ ω n ∼ ∆ α [ ∂ α ω n ] =∆ α [ β ∇| φ ( r ) | ].We note that another slightly different operator hasbeen introduced in recent related work to maximize themeasurement precision of an observable parameter [32].The eigenstates of the operator F α = ( ∂ α S ) † ∂ α S haveindeed been identified as “maximum information states”maximizing the Fisher-information related to the param-eter α . F α coincides with Q α only for a unitary scat-tering matrix. The results in Ref. [32] however differsharply from our present work in two important ways:first, while Ref. [32] considers a “random” configurationof a disordered medium, we operate under the very spe-cial CPA condition for which S has a real-valued zero;second, while Ref. [32] identifies a so-called “maximuminformation state” that is specific to the observable ofinterest (e.g. location of the pertuber), our CPA con-dition yields an optimal sensitivity to any perturbationirrespective of its location. d I m ( ω ) Re( ω )CPA z n b ω n a Zeroes of the scattering matrix A n t e n n a A r r a y PerturberMetasurfaces c
10 20 30 40Iterations-100-80-60-40-20 R e fl e c t i o n ( d B ) Random configurations 5.12 5.13 5.14 5.15 5.16 5.17 5.18-100-80-60-40-200 R e fl e c t i o n ( d B ) Frequency (GHz) f CPA = 5.137 GHzf
CPA = 5.142 GHzf
CPA = 5.147 GHzf
CPA = 5.152 GHzf
CPA = 5.157 GHz
Semispheres Optimization
FIG. 1. “On-demand” realization of CPA in a programmable complex scattering enclosure. a,
Experimental setupconsisting of a three-dimensional electrically large irregular metallic enclosure equipped with two arrays of 1-bit reflection-programmable meta-atoms to tune the system’s scattering properties. The system is connected to eight channels via waveguide-to-coax transitions. Small perturbations of the system can be induced by rotating a metallic rod placed on a metallic platform. b, Illustration of operation principle in the complex frequency plane. By tuning the system’s scattering properties with theprogrammable meta-atoms, a zero of the scattering matrix is moved onto the real frequency axis at a target horizontal position(here 5.147 GHz). c, Dynamics of an example iterative optimization of the meta-atom configurations. d, Spectrum of the multi-channel reflection coefficient R ( ω ) for five optimized systems targeting five distinct regularly spaced nearby target frequenciesbetween 5.137 GHz and 5.157 GHz. III. EXPERIMENTAL MEASUREMENT OFTIME DELAYS
Having established a theoretical model for the char-acteristic reflection dip and delay time associated withCPA, we now seek to verify its validity in experiments.Experimentally realizing CPA in a random medium is avery challenging task that was only mastered recently forthe first time [12, 13]. These early realizations relied onboth ω and Γ a being freely tunable parameters to iden-tify a setting in which one zero of S ( ω ) lies on the realfrequency axis. For our experiments, we consider a morerealistic three-dimensional complex scattering enclosurewith fixed homogeneously distributed losses and we fixthe working frequency to 5 .
147 GHz. In order to real-ize CPA “on demand” without control over ω and Γ a ,we dope our system with reflection-programmable meta-atoms (see Appendix). In our experiment illustrated inFig. 1a and as detailed in the Appendix, N = 8 channelsare connected to a chaotic cavity and an iterative algo-rithm is used to optimize the configuration of the 304 meta-atoms. The latter allow us to tweak S such thatone of its zeros hits the real frequency axis [17], as il-lustrated schematically in Fig. 1b. Moreover, our setupincludes an irregular metallic structure attached to a ro-tation stage; we can thus identify different realizationsof CPA by rotating this “mode-stirrer” to different po-sitions and optimizing the meta-atom configurations foreach position.Random configurations of the meta-atoms yield re-flection values between −
20 dB and −
40 dB. Startingfrom a configuration corresponding to roughly −
40 dB,our optimization algorithm which minimizes the reflec-tion R = (cid:107) Sψ in (cid:107) eventually identifies a setting with R ∼ . × − = −
93 dB, as depicted in Fig. 1c. Wethen repeat the experiment for other predefined frequen-cies. In each case, the corresponding reflection spectrumin Fig. 1d displays the expected very narrow and deepdip at the desired working frequency.We can now test our theoretical model’s prediction forthe reflection and delay time at the CPA condition. Be-fore confronting it with our experimental data, we per-form a random matrix simulation for which the inter- -100-90-80-70-60-50 R e fl e c t i o n ( d B ) Frequency (GHz) r e a l ( τ ) [ μ s ] Optim. 1TheoryOptim. 2 -100 -80 -60 -40
Reflection (dB) -40-20020406080 r e a l ( τ ) [ μ s ] Frequency (GHz) d e f × ∆ ω / Γ n -4-2-20 r e a l ( τ ) × × -3 × ∆ ω / Γ n ∆Γ / Γ n c Experiment Experiment Experiment -5 0 5 ∆Γ/Γ n × -4 -100-90-80-70-60 R e fl e c t i o n ( d B ) -5 0 5 × -4 -505 r e a l ( τ ) × ∆Γ/Γ n a b Simulations Simulations
ModelTheory × -3 n FIG. 2.
CPA signature on time delays.
Multichannel reflection coefficient R ( ω ) ( a ) and time delay Re[ τ ( ω )] ( b ) foundin random matrix simulations with an effective Hamiltonian model (red lines) in the vicinity of a CPA condition at ω n fora mode with linewidth Γ n , plotted as a function of linewidth detuning ∆Γ / Γ n = (Γ a − Γ n ) / Γ n . The time delay has beennormalized by its value in absence of absorption. These numerical results are in excellent agreement with our theoreticalpredictions (black dashed lines) given by Eq. (S9) and Eq. (S13). c, Variations of the time delay with ∆Γ / Γ n and frequencydetuning ∆ ω/ Γ n = ( ω − ω n ) / Γ n are visualized as three-dimensional surface. The experimental results in d and e as a functionof frequency detuning are also perfectly fitted with our theory for two different realizations of the CPA condition. f, Evolutionof time delay during an example optimization. The absolute time delay | Re[ τ ( ω )] | increases as R ( ω ) is minimized but the signof Re[ τ ( ω )] jumps from negative to positive after 40 iterations. This is a signature of the time-delay singularity. The color-codeindicating the iteration index is the same as in Fig. 1c. nal Hamiltonian H is a a real symmetric matrix drawnfrom the Gaussian orthogonal ensemble and V is a realrandom matrix with Gaussian distribution [33]. We se-lect a resonance and explore the variations of R ( ω ) andRe[ τ ( ω )] near the CPA condition as a function the nor-malized linewidth mismatch ∆Γ / Γ n and frequency de-tuning ∆ ω/ Γ n . As shown in Fig. 2a-c, the dip in thereflection coefficient and the singularity of the time de-lay found in simulations are perfectly reproduced byEqs. (S9) and (S13). In particular, Fig. 2b highlights thedivergence of Re[ τ ( ω )] as ∆Γ → n , ∆Γ and (cid:107) ∆ S (cid:107) F , by fitting our model to the spectraof R ( ω ) and Re[ τ ( ω )]. We thereby obtain a linewidth γ CPA = Γ
CPA / (2 π ) ∼
51 MHz, a mismatch ∆Γ / (2 π ) ∼ . (cid:107) ∆ S (cid:107) F / (cid:107) S (cid:107) F ∼ × − . As seen in Fig. 2d,e, our model exactly fits our experimental data on a frequency range smaller than themean level spacing ∆ = (cid:104) ω n +1 − ω n (cid:105) / (2 π ) given by Weyl’slaw ∆ = c / (8 πV f ) ∼ .
54 MHz. The very small valueof ∆Γ evidences that our optimized system is extremelyclose to the CPA condition. In Fig. 2e we observe a strongenhancement of the time delay for two representative re-alizations of CPA; | Re[ τ ( ω )] | reaches values as high as80 µ s. For comparison, wavefronts that are orthogonalto ψ CPA yield an average time delay of 10.9 ns, almostfour orders of magnitude smaller. The two CPA realiza-tions in Fig. 2d,e associated with a positive (negative)time delay correspond to a zero of S just above (below)the real frequency axis. During the optimization of themeta-atom configurations, the zero can even cross thereal frequency axis, an example thereof being shown inFig. 2f for which Re[ τ ( ω )] suddenly jumps from − µ sto 41 µ s after a single iteration. ec d Unoptimized wavefront CPA wavefront a b
Rotation angle -100-80-60-40-200 R e fl e c t i o n ( d B ) CPA wavefrontunoptimized wavefront θ=7.7θ=3.5θ=0.7θ=0 -60 -50 -40 -30 -20 -10
Reflection (dB) P r o b a b i l i t y d i s t r i b u t i o n
115 pixels5 pixels1 pixel -16 -14 -12 -10 -8 -6
Reflection (dB) P r o b a b i l i t y d i s t r i b u t i o n
115 pixels5 pixels1 pixel
Rotation angle C o rr e l a t i o n c o e ffi c i e n t CPA wavefrontunoptimized wavefront
Number of detuned meta-atoms -80-60-40-200 R e fl e c t i o n ( d B ) CPA wavefrontunoptimized wavefrontModel
FIG. 3.
Enhanced sensing with the time-delay singularity at the CPA condition. a,
Magnitude of the correlationcoefficient C ( ψ in , θ ) of the outgoing field as a function of the size of the perturbation (angle of rotation θ ) for the case ofinjecting the CPA wavefront or an unoptimized wavefront. b, Corresponding multichannel reflection coefficient. The rightpanel shows a histogram of observed reflection values for different perturbation strengths, based on 18 CPA realizations withdifferent initial orientations of the perturber. c, d,
Distributions of R shown for a random incoming wavefront ( c ) and theCPA wavefront ( d ) for detuning 1, 5 and 115 meta-atoms. e, Variations of the reflection coefficient on a logarithmic scale atthe CPA condition (blue crosses) and for a random incoming wavefront (black circles), as a function of the number of detunedmeta-atoms p . The bars give the corresponding 90 percentiles for R . For small perturbations, the result obtained with theCPA wavefront is well explained by Eq. (S9) upon substituting ∆ ω = 2 πKp , with K = 0 .
35 MHz (see main text for details).
IV. TIME DELAY SINGULARITY FOROPTIMAL SENSITIVITY
Now that we have have established and experimentallyconfirmed the physical origin of the time-delay singular-ity at the CPA condition in complex scattering media, wego on to investigate experimentally how this singularitycan enhance the sensitivity of measurements to paramet-ric perturbations, which is essential in sensing applica-tions. The term “sensitivity” in the present work refersto a transduction coefficient of the sensor from the quan-tity to be measured (a perturbation ∆ α of a parameter α ) to an intermediate output quantity. This should notbe confused with the smallest measurable change of theinput quantity which pivotally depends on the measure-ment noise [34]. In the following, first, we show that atthe CPA condition, the ability to detect tiny perturba-tions is enhanced due to a rapid field decorrelation, thelatter being intimately linked to the time delay. Then,second, we investigate the extent to which the CPA con-dition also enables a characterization of the perturbation in terms of its strength.Wave chaos is generally known to be quite sensitiveto perturbations; techniques known as diffuse wave spec-troscopy (DWS) study the decorrelation of the outgoingfield due to a perturbation in order to quantify the lat-ter [35–37]. Purely relying on the sensitivity of wavechaos corresponds (in our system tuned to have a real-valued zero) to injecting an unoptimized random wave-front ψ in = ψ rand . However, because of the divergence ofthe delay time, the perturbation-induced decorrelation ofthe wave may be dramatically enhanced by injecting theoptimized wavefront ψ in = ψ CPA . We begin by defininga correlation coefficient based on the outgoing field as C (∆ α ) = ψ † out ( α + ∆ α ) ψ out ( α ) (cid:112) R ( α ) R ( α + ∆ α ) . (9)To confirm the predicted rapid decorrelation of the outgo-ing field at the CPA condition experimentally, we grad-ually perturb the system by rotating a small metallicstructure (a metallic pillar located on a metallic plat-form) in steps of ∆ θ = 0 . ◦ (see Fig. 3a). At eachstep, we measure S ( θ ) and evaluate the outgoing field ψ out ( θ ) = S ( θ ) ψ in for unoptimized and optimized wave-fronts. We repeat this procedure for 18 realizations of dif-ferent initial positions of the rotating object. In Fig. 3awe plot | C (∆ α ) | as a function of the size of the perturba-tion (here the angle of rotation θ ). For ψ in = ψ rand , thefield barely decorrelates even after 100 rotation steps. Inother words, DWS-based sensing would not be capableof detecting or quantifying the perturbation that we con-sider. In contrast, for ψ in = ψ CPA , the field stronglydecorrelates even for the smallest step: C (∆ θ ) ∼ R ( α + ∆ α ) increaseswith ∆ α so that it may be possible to discriminate be-tween two perturbations ∆ α and ∆ α based on mea-surements of R (∆ α ) and R (∆ α ). The reflection coeffi-cient R ( θ, ψ CPA ) shown in Fig. 3b increases rapidly with θ but then saturates for θ > ◦ . Note that the plateaureached for large angles is 20 dB below R ( θ, ψ rand ) as theperturbation is small.In wave-chaotic systems as our complex scattering en-closure, the energy density (and hence ∂ α ω n and R ) aredistributed quantities which fluctuate for different real-izations of the system; an example are the histograms of R ( θ, ψ CPA ) shown in Fig. 3b. To investigate the depen-dence of R ( α + ∆ α ) on the perturbation strength ∆ α ,a statistical analysis is hence required. We convenientlyachieve this by considering a different type of perturba-tion: the detuning of p meta-atoms away from the CPAconfiguration [17]. We successively determine R for 200realizations of p detuned meta-atoms in 15 CPA realiza-tions, with p varying from p = 1 to p = 115.As expected, for random wavefronts, R ( p, ψ rand )is statistically independent of p : the distributions P ( R ( p, ψ rand )) found for p = 1, 5 and 115 detuned meta-atoms hence completely overlap, as seen in Fig. 3c. Incontrast, Fig. 3d reveals that P ( R ( p, ψ CPA )) strongly de-pends on p because the variations of ω n due to localchanges of the boundary conditions increase with thenumber of detuned meta-atoms. We find that in theregime of small perturbations (here p ≤ (cid:104) R ( p, ψ CPA ) (cid:105) is in good agreement with Eq. (S9) upon replacing thefrequency shift ∆ ω = ω − ω n with 2 πKp , where K =0 .
35 MHz is a constant depending on the scattering cross-section of the meta-atoms and the volume of the cavity.Our model’s validity is confirmed by its faithful fit to theexperimental data for p ≤ R being a distributedquantity results in fundamental limits on the precisionwith which the strength of any given perturbation canbe determined. Indeed, the distributions P ( R ( p, ψ CPA ))partially overlap for close values of p , as shown in Fig. 3d.This limitation may be understood as the price for theenhanced sensitivity to any perturbation within the sys-tem irrespective of its location. V. DISCUSSION AND CONCLUSION
The discussed CPA condition can be interpreted as aspecial case of two distinct more general scattering phe-nomena: coherently enhanced absorption (CEA, [38])and virtual perfect absorption (VPA, [39]).CEA is a route to achieving very low (but finite) reflec-tion values over an extended frequency range. Similarlyto CPA, CEA relies on injecting the incoming wavefrontgiving the smallest reflection, which is the eigenstate ofthe matrix S † ( ω ) S ( ω ) with minimal eigenvalue. Unlessa zero of S happens to lie on the real-frequency axis at ω , this eigenstate generally does not correspond to aneigenstate of S (CPA condition) so that the reflectioncoefficient does not vanish. Multiple resonances of thesystem are then involved and the reflection dip associatedwith CEA is not as pronounced as for CPA. In the caseof CEA, the reflection coefficient decorrelates on a scaleinversely proportional to the absorption mean free path[38]. The time delay of waves and the sensitivity of themedium to a perturbation are hence generally boundedfor CEA.The idea of CPA is to bring a zero onto the real-frequency axis such that it can be accessed with amonochromatic excitation oscillating at a real frequency.If, however, the zero is not on the real frequency axis, itcan still be accessed in the transient regime using a non-monochromatic signal oscillating at a complex frequency.This VPA concept was recently studied for regular (al-most) lossless systems for which the zero always lies inthe upper half of the complex frequency plane [39, 40].Consequently, the excitation signal has to exponentiallyincrease in time to interfere destructively with the wavesreflected off the system. The interaction of the incidentpulse with the scattering medium then provides idealenergy storage until the interruption of the exponentialgrowth of the injected signal. Given the generality ofthe scattering matrix formalism, this concept can be ex-tended to disordered lossy matter such as complex scat-tering enclosures. The zeros may then lie anywhere inthe complex frequency plane, implying that the neces-sary excitation is not always an exponentially increasingone. Interesting links with the sign of the time delay asdiscussed in the present work then arise.To summarize, we have proposed a theoretical descrip-tion of the hallmark sign of CPA in complex scatteringsystem and we verified our theory experimentally. Ourwork rigorously explains the divergence of the time de-lay at the CPA condition and how this singularity jus-tifies the optimal sensitivity of the CPA condition fordetecting minute perturbations. This feature will enablenovel precision sensing tools but also impact other areassuch as filter applications and secure wireless communi-cation [13, 17]. Furthermore, our experiments demon-strated how a CPA condition can be accessed “on de-mand” at an arbitrary frequency without controlling thelevel of attenuation in the system. Finally, we note thatour results are very general in nature and apply to othertypes of wave phenomena, too. Note added. — In the process of finalizing thismanuscript, we became aware of related work [41] thatalso generalizes the concept of “on-demand” access to areal-valued scattering matrix zero in a complex scatter-ing enclosure from single-channel [17] to multi-channelexcitation.
APPENDIX: EXPERIMENTAL METHODSA. Experimental Setup
Our complex scattering enclosure (depicted in Fig. 1a)is a metallic cuboid (50 × ×
30 cm ) with two hemi-spheres on the inside walls to create wave chaos. Inshort, the difference between the trajectories of two rayslaunched from the same position in slightly different di-rections will increase exponentially in time. The systemis excited via eight antennas (waveguide-to-coax transi-tions designed for operation in the 4 < ν < × ν = 5 .
147 GHz, we mea-sure the spectra with an extremely small frequency step(∆ ν = 1 kHz) and subsequently fit them with a linearregression in the Argand diagram to further reduce theimpact of noise.In order to tune our random system’s scattering ma-trix, two programmable metasurfaces [42] are placed ontwo neighboring walls of the cavity [43]. Each metasur-face consists of an array of 152 1-bit programmable meta-atoms. Each meta-atom has two digitalized states, “0”and “1”, with opposite electromagnetic responses. Theworking principle relies on the hybridization of two reso-nances of which one is tunable via the bias voltage of a pindiode; details can be found elsewhere [44]. The two statesare designed to mimic Dirichlet and Neumann boundaryconditions and their detailed characteristics are providedin the SM. B. Optimization of Meta-Atom Configurations
Identifying a configuration of the programmable meta-surfaces that yields CPA at the targeted frequency is anon-trivial task since there is no forward model describ-ing the impact of the metasurface configuration on S .Hence, we opt for an iterative optimization algorithmsimilar to the one in Ref. [45]. We begin by measuring S for 200 random configurations. We then use the configu-ration for which the smallest eigenvalue λ of S ( ν ) is thelowest as starting point. For each iteration, we randomlyselect z meta-atoms and flip their state. If the resulting S ( ν ) has a lower λ , we keep the change. We graduallyreduce the number of meta-atoms whose state is flippedper iteration, according to z = max(int(50 e − . k ) , k is the iteration index. C. Characterization of Chaotic Cavity
We characterize our system’s linewidth associated withthe N attached channels, N γ c , and its linewidth associ-ated with global absorption effects, γ a , in the following.To that end, we measure S ( ω ) for 300 random meta-surface configurations and repeat the measurements fora number of channels connected to the cavity varyingfrom N = 3 to N = 8. In each case, we estimatethe average linewidth (cid:104) γ (cid:105) by fitting the exponential de-cay of average reflected intensities in the time domain, I ( t ) = e − π (cid:104) γ (cid:105) t (see SM for details). Using the variationsof (cid:104) γ (cid:105) with respect to the number of attached channels, (cid:104) γ (cid:105) = (cid:104) γ a (cid:105) + N (cid:104) γ c (cid:105) , we obtain an average absorptionstrength (cid:104) γ a (cid:105) = 11 . (cid:104) γ n (cid:105) = 8 (cid:104) γ c (cid:105) = 0 .
33 MHz. Comparedwith the linewidth γ CPA found at the CPA condition itappears at first sight surprising that γ CPA exceeds (cid:104) γ a (cid:105) and (cid:104) γ n (cid:105) by more than one order of magnitude. The ap-parent paradox is resolved by noting that the resonancewidths in wave-chaotic systems are not normally dis-tributed; instead they have a distribution skewed towardlower values with a long tail for larger values [46, 47].Such a distribution is not fully characterized by its aver-age, and observing values well above the average, as inthe case of γ CPA , is by no means impossible.
ACKNOWLEDGMENTS
This publication was supported by the French “AgenceNationale de la Recherche” under reference ANR-17-ASTR-0017, by the European Union through the Euro-pean Regional Development Fund (ERDF), and by theFrench region of Brittany and Rennes M´etropole throughthe CPER Project SOPHIE/STIC & Ondes. The meta-surface prototypes were purchased from Greenerwave.The authors acknowledge P. E. Davy for the 3D renderingof the experimental setup in Fig. 1a. M. D. acknowledgesthe Institut Universitaire de France. [1] A. Krasnok, D. Baranov, H. Li, M.-A. Miri, F. Monti-cone, and A. Al´u, Adv. Opt. Photon. , 892 (2019).[2] V. Grigoriev, A. Tahri, S. Varault, B. Rolly, B. Stout,J. Wenger, and N. Bonod, Phys. Rev. A , 011803(2013).[3] Y. Chong, L. Ge, H. Cao, and A. D. Stone, Phys. Rev.Lett. , 053901 (2010).[4] D. G. Baranov, A. Krasnok, T. Shegai, A. Al, andY. Chong, Nat. Rev. Mater. , 17064 (2017).[5] M. Cai, O. Painter, and K. J. Vahala, Phys. Rev. Lett. , 74 (2000).[6] W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, andH. Cao, Science , 889 (2011).[7] Z. J. Wong, Y.-L. Xu, J. Kim, K. O’Brien, Y. Wang,L. Feng, and X. Zhang, Nat. Photon. , 796 (2016).[8] H. Cao, Waves Random Media , R1 (2003).[9] D. S. Wiersma, Nat. Phys. , 359 (2008).[10] Y. V. Fyodorov, S. Suwunnarat, and T. Kottos, J. Phys.A , 30LT01 (2017).[11] H. Li, S. Suwunnarat, R. Fleischmann, H. Schanz, andT. Kottos, Phys. Rev. Lett. , 044101 (2017).[12] K. Pichler, M. K¨uhmayer, J. B¨ohm, A. Brandst¨otter,P. Ambichl, U. Kuhl, and S. Rotter, Nature , 351(2019).[13] L. Chen, T. Kottos, and S. M. Anlage, arXiv:2001.00956(2020).[14] J. Zhang, K. F. MacDonald, and N. I. Zheludev, LightSci. Appl. , e18 (2012).[15] R. Bruck and O. L. Muskens, Opt. Express , 27652(2013).[16] S. M. Rao, J. J. Heitz, T. Roger, N. Westerberg, andD. Faccio, Opt. Lett. , 5345 (2014).[17] M. F. Imani, D. R. Smith, and P. del Hougne, Adv.Funct. Mater. TBA , 2005310 (2020).[18] I. Rotter, Phys. Rev. E , 036213 (2001).[19] Y. V. Fyodorov, D. Savin, and H. Sommers, J. Phys. A , 10731 (2005).[20] I. Rotter, J. Phys. A , 153001 (2009).[21] F. Alpeggiani, N. Parappurath, E. Verhagen, andL. Kuipers, Phys. Rev. X , 021035 (2017).[22] M. Davy and A. Z. Genack, Nat. Commun. , 4714(2018).[23] P. del Hougne, R. Sobry, O. Legrand, F. Mortessagne,U. Kuhl, and M. Davy, arXiv preprint arXiv:2001.04658(2020).[24] J. B¨ohm, A. Brandst¨otter, P. Ambichl, S. Rotter, andU. Kuhl, Phys. Rev. A , 021801 (2018).[25] S. Fan and J. M. Kahn, Opt. Lett. , 135 (2005).[26] M. Durand, S. M. Popoff, R. Carminati, andA. Goetschy, Phys. Rev. Lett. , 243901 (2019).[27] J. Muga and J. Palao, Ann. Phys. , 671 (1998).[28] H. Tanaka, H. Niwa, K. Hayami, S. Furue, K. Nakayama,T. Kohmoto, M. Kunitomo, and Y. Fukuda, Phys. Rev.A , 053801 (2003).[29] P. Ambichl, A. Brandst¨otter, J. B¨ohm, M. K¨uhmayer,U. Kuhl, and S. Rotter, Phys. Rev. Lett. , 033903(2017).[30] M. Horodynski, M. K¨uhmayer, A. Brandst¨otter, K. Pich-ler, Y. V. Fyodorov, U. Kuhl, and S. Rotter, Nat. Pho-tonics, 1 (2019). [31] M. Barth, U. Kuhl, and H.-J. St¨ockmann, Phys. Rev.Lett. , 2026 (1999).[32] D. Bouchet, S. Rotter, and A. P. Mosk, arXiv preprintarXiv:2002.10388 (2020).[33] U. Kuhl, O. Legrand, and F. Mortessagne, Fortschr.Phys. , 404 (2013).[34] W. Langbein, Phys. Rev. A , 023805 (2018).[35] D. Pine, D. Weitz, P. Chaikin, and E. Herbolzheimer,Phys. Rev. Lett. , 1134 (1988).[36] G. Maret, Curr. Opin. Colloid & Interface Sci. , 251(1997).[37] J. de Rosny, P. Roux, M. Fink, and J. Page, Phys. Rev.Lett. , 094302 (2003).[38] Y. D. Chong and A. D. Stone, Phys. Rev. Lett. ,163901 (2011).[39] D. G. Baranov, A. Krasnok, and A. Al`u, Optica , 1457(2017).[40] G. Trainiti, Y. Ra’di, M. Ruzzene, and A. Al`u, Sci. Adv. , eaaw3255 (2019).[41] B. W. Frazier, T. M. Antonsen Jr., S. M. Anlage, andE. Ott, arXiv preprint arXiv:2009.05538 (2020).[42] T. J. Cui, M. Q. Qi, X. Wan, J. Zhao, and Q. Cheng,Light Sci. Appl. , e218 (2014).[43] M. Dupr´e, P. del Hougne, M. Fink, F. Lemoult, andG. Lerosey, Phys. Rev. Lett. , 017701 (2015).[44] N. Kaina, M. Dupr´e, M. Fink, and G. Lerosey, Opt.Express , 18881 (2014).[45] P. del Hougne, M. Davy, and U. Kuhl, Phys. Rev. Ap-plied , 041004 (2020).[46] T. Kottos, J. Phys. A , 10761 (2005).[47] U. Kuhl, R. H¨ohmann, J. Main, and H.-J. St¨ockmann,Phys. Rev. Lett. , 254101 (2008). SUPPLEMENTAL MATERIALI. THEORETICAL MODEL
In this section, we provide additional details and in-termediate algebraic steps for our theoretical results re-ported in the main text.
A. Effective Hamiltonian approach
We analyze the properties of the scattering matrix S ( ω ) using a non-perturbative approach based on theeffective Hamiltonian. This model was initially devel-oped to characterize open quantum systems (see Ref. [1])but also qualitatively describes the transport of classi-cal waves in chaotic systems [2]. The coupling of thechannels to a system is analyzed in terms of an effec-tive Hamiltonian H eff expressed as H eff = H − iV V T / H characterizes the closedsystem and V is a real matrix of dimensions M × N de-scribing the coupling of the N channels to the M modesof the closed system. To compare experimental resultswith quantum-mechanical predictions, we use the anal-ogy between the experimental eigenfrequencies ω n andlinewidths Γ n with their quantum-mechanical counter-parts E n = ω n and ˜Γ n = ω n Γ n [2]. Under the assump-tion of small linewidths, Γ n /ω (cid:28)
1, we can express thescattering matrix giving the field coefficient between in-coming and outgoing channels as S ( ω ) = − iV T ω − H + i ( V V T + Γ a ) V. (S1)Here, we have introduced the linewidth Γ a to take intoaccount the losses within the internal system. One canalternatively express the scattering matrix in terms ofthe eigenfunctions of the non-effective Hamiltonian. Inabsence of absorption, the poles of S are found at thecomplex eigenvalues ˜ ω m = ω m − i Γ m /
2. Here ω m isthe central frequency and Γ m is the associated linewidth.The scattering matrix S ( ω ) can be decomposed as thesuperposition of natural resonances: S ( ω ) = − i Σ Mm =1 W m W Tm ω − ω m + i (Γ m + Γ a ) / . (S2)The vectors W m are the projection of the eigenfunctions φ n onto the channels coupled to the system: W m = V T φ m .The zeroes z m of S ( ω ) are the complex frequenciessatisfying det S ( z m ) = 0. They are found using the rela-tion [3] det S ( ω ) = det( ω − H + i Γ a − i Γ)det( ω − H + i Γ a + i Γ) , (S3) where Γ is the matrix Γ = V T V /
2. Eq. (S3) shows thatin absence of absorption (Γ a = 0) the zeroes and polesof the scattering matrix are symmetrically placed in theupper and lower complex plane with z m = ˜ ω ∗ m . Whenabsorption within the medium is added (Γ a (cid:54) = 0), thezeroes and poles move in the complex plane with z m = ω m + i (Γ m − Γ a ) / ω m = ω m − i (Γ m + Γ a ) / S ( ω ) is locatedon the real frequency axis. For notational ease, we labelthe m th resonance (out of the M resonances contributingto the sum in Eq. (S2)) that is associated with this spe-cial real-valued zero with the subscript n . This notationis applied to all quantities associated with individual res-onances, such as ω m or Γ m . Eq. (S3) demonstrates thatCPA is hence found when the dissipation within the cav-ity fully compensates losses through channels for a givenpole: Γ n = Γ a . The frequency of this CPA condition isthen ω = ω n . B. Perfectly absorbed incoming wavefront
When a zero of the scattering matrix crosses the realfrequency axis, there exists a vector ψ CP A which satis-fies S ( ω n ) ψ CP A = 0. In analogy with random lasing,this vector has been identified as the time-reverse of thelasing wavefront. We now demonstrate that ψ CP A in-deed is the phase-conjugate of the modal vector, ψ CP A = W ∗ n / (cid:107) W n (cid:107) . Using Eq. (S2), we find that the outgoingvector ψ out = Sψ CP A for ω = ω n and Γ a = Γ n is ψ out = 1 (cid:107) W n (cid:107) (cid:20) W ∗ n − i Σ Mm =1 W m W Tm W ∗ n ˜ ω ∗ n − ˜ ω m (cid:21) . (S4)The eigenfunctions φ m , and hence the vectors W m , aregenerally non-orthogonal in open systems: φ † m φ m (cid:54) = δ m m . They are, however, bi-orthogonal and the degreeof correlation of the vectors W m is related to the corre-lation between eigenfunctions φ Tm φ ∗ m as φ Tm φ ∗ m = i W Tm W ∗ m ˜ ω ∗ m − ˜ ω m . (S5)Note that φ Tm φ ∗ m is an element of the Bell-Steinbergernon-orthogonality matrix U giving the correlation be-tween each pair of eigenfunctions of the system. By in-corporating Eq. (S5) within Eq. (S4) we get ψ out = 1 (cid:107) W n (cid:107) [ W ∗ n − V T (Σ Mm =1 φ m φ Tm ) φ ∗ n ] . (S6)Using the completeness of the eigenfunctions,Σ Mm =1 φ m φ Tm = , and the relation V T φ ∗ n = W ∗ n ,we finally verify that ψ out = 0, as expected.1 -100-80-5-60 R e fl e c t i o n ( d B ) -40-20 × -3 ∆ ω / Γ n
05 -0.01 ∆Γ / Γ n FIG. S1. Simulation results of the reflection coefficient R ( ω )in the vicinity of a CPA condition at ω n are presented as afunction the linewidth detuning ∆Γ a / Γ n and the frequencydetuning ∆ ω/ Γ n . The results in excellent agreement withthe theoretical predictions (black lines) shown for ∆ ω = 0and ∆Γ = 0. C. Reflection coefficient
In a realistic experiment we never perfectly achieve theCPA condition with ω = ω n and Γ a = Γ n . Therefore, wenow set ω = ω n + ∆ ω and Γ a = Γ n + ∆Γ. We use a per-turbation approach to derive the multichannel reflectioncoefficient R ( ω ) = (cid:107) S ( ω ) ψ CP A (cid:107) . Eq. (S4) yields ψ out = 1 (cid:107) W n (cid:107) (cid:20) W ∗ n − i Σ m W m W Tm W ∗ n ˜ ω ∗ n − ˜ ω m + ∆ ω + i ∆Γ / (cid:21) . (S7)In the vicinity of the CPA condition (∆ ω (cid:28) ( ω n +1 − ω n )and ∆Γ (cid:28) Γ n ) we assume that the perturbation onlyaffects the n th resonance. Eq. (S7) can then be simplifiedto ψ out ∼ W ∗ n (cid:107) W n (cid:107) (cid:20) − i Γ n ∆ ω + i (Γ n + ∆Γ / (cid:21) . (S8)Straightforward calculations finally lead to R ( ω ) = (cid:107) ψ out (cid:107) = 4∆ ω + (∆Γ) ω + (2Γ n + ∆Γ) . (S9)Finally, the noise level in the experiment corrupts themeasurement of S as S → S +∆ S , where ∆ S is a randommatrix with complex random elements with Gaussian dis-tribution. This mismatch increases the reflection coeffi-cient by an additional level of (cid:107) ∆ S (cid:107) F /N , where (cid:107) . . . (cid:107) F denots the Frobenius norm and N is the number of chan-nels coupled to the system. We hence obtain Eq. (2) ofthe main text. -101 r ea l ( τ ) -5 0 5 Δω/ Γ n x -3 -101 i m ag ( τ ) x104 ab x104 FIG. S2. Simulation results (blue lines) of the real (a) andimaginary (b) parts of the time delay in the vicinity of aCPA condition at ω n as a function of the frequency detuning∆ ω/ Γ n , in excellent agreement with the theoretical predic-tions (red dashed lines). We now compare Eq. (S9) to numerical simulations ofour model using Eq. (S1). H is modelled as a real sym-metric matrix of dimensions 500 ×
500 drawn from theGaussian orthogonal ensemble; the coupling matrix V tothe N = 8 channels is modelled as a real random matrixwith Gaussian distribution. The channels are assumedto be fully coupled to the system. We select the reso-nance which is nearest to the middle of the band andthen explore the reflection coefficient R = (cid:107) S ( ω ) ψ CP A (cid:107) as a function of frequency detuning ω = ω n + ∆ ω andlinewidth detuning Γ a = Γ n + ∆Γ. The results shownin Fig. S1 demonstrate an excellent agreement betweenthe random matrix simulations and our theoretical pre-dictions. D. Time delay
The time delay can be expressed in terms of theWigner-Smith (WS) operator Q ( ω ) = − iS ( ω ) † ∂ ω S ( ω ): τ ( ω ) = ψ † CP A Q ( ω ) ψ CP A ψ † CP A S ( ω ) † S ( ω ) ψ CP A . (S10)Note that the projection of the incoming wavefront ontothe Wigner-Smith operator is normalized by the reflec-tion coefficient so that τ ( ω ) can be interpreted as phasederivative.Obviously a singularity may appear at the CPA con-dition with ω = ω n and Γ a = Γ n as R ( ω n ) = ψ † CP A S ( ω n ) † S ( ω n ) ψ CP A = 0. However, this singular-ity is removed for ω (cid:54) = ω n and/or Γ a (cid:54) = Γ n . We now2estimate τ ( ω ) in the vicinity of the CPA using the sameapproach as for R ( ω ). We first calculate the derivativeof S ( ω ) with respect to ω : ∂ ω S ( ω ) = i Σ Mm =1 W m W Tm [ ω − ω m + i (Γ m / a / . (S11)This leads to ψ † Q ( ω ) ψ = 4Γ n ω + (2Γ n + ∆Γ) (2∆ ω − i ∆Γ)2∆ ω + i (2Γ n + ∆Γ)(S12)As the denominator of Eq. (S10) is given by Eq. (S9),straightforward calculations finally yield the real andimaginary parts of τ :Re[ τ ] = 4Γ n ω + ∆Γ ω − ∆Γ(2Γ n + ∆Γ)4∆ ω + (2Γ n + ∆Γ) (S13)andIm[ τ ] = − n ω + ∆Γ ω (Γ n + ∆Γ)4∆ ω + (2Γ n + ∆Γ) (S14)Interestingly, at the resonance with the mode (∆ ω = 0)a phase transition appears on the real part of τ whichis positive for Γ n > Γ a , diverges for Γ n = Γ a and neg-ative for Γ n < Γ a . This theoretical prediction is fullyconfirmed by the agreement with our model in Fig. (2)of the main text and Fig. S2. E. Negative delay time
One suprizing result is that the time-delay can be neg-ative. We illustrate this effect by computing the outgoingintensity in the time domain for an incident pulse witha very small bandwidth. For the sake of simplicity, werealize a single channel-simulation ( N = 1) using the ef-fective Hamiltonian approach. We obtain the spectrumof the scattering matrix in the vicinity of a CPA con-dition for Γ a < Γ n and then for Γ a > Γ n . In the firstcase, the delay time is positive (see Fig. S3a) while it isnegative in the second case (see Fig. S3c). We obtainthe outgoing intensity in the time domain using an in-verse Fourier transform of the reflection coefficient. Asexpected, we observe that the peak of the pulse is shiftedpositively (negatively) for positive (negative) time delays(see Fig. S3b,d). We highlight, however, that the outgo-ing intensity has been normalized to visualize this effect.As the incoming wave is strongly absorbed within thesystem, the normalization factors of the outgoing waverelative to the incoming wave are 326 and 399 in thesetwo cases.Obviously, this effect does by no means break causality.In the case of strongly absorbed signals, negative time de-lays for Γ a > Γ n arise due to the distortion of the incident -0.008-0.006-0.004-0.0020 R e a l [ τ ] -400 -200 0 200 400 time (a.u.) I n t e n s i t y Frequency (a.u.) R e a l [ τ ] I n t e n s i t y Incoming waveOutgoing wave a bc d x326x399
FIG. S3. Simulation result of the time delay (a,c) and the in-coming and outgoing intensity in the time domain (b,d) in thevicinity of a CPA for (a,b) a positive time delay and (c,d) anegative time delay. The temporal intensities are found froman inverse Fourier transform of the outgoing field ψ out ( ω ) foran incoming Gaussian pulse. The width of this incident pulsein the frequency domain is 0.2% of the central frequency. Asthe incoming wave is strongly absorbed, the outgoing inten-sities are normalized by factors of 326 in b and 399 in d. pulse in resonant absorbing systems [4–8]. The signal atlong times is indeed more strongly absorbed than thesignal at early times. As a consequence, the maximumof the outgoing pulse is shifted to shorter times and thepulse appears to be delayed negatively. For Γ a < Γ n , theeffect is inverted so that the peak is delayed to longertimes. This effect hence relies on temporal variations ofthe absorption of the signal. F. Connection with the generalized
Wigner-Smithoperator
In this section, we prove that the incoming wavefront ψ CPA is also the eigenstate with maximal eigenvalue ofthe generalized
Wigner-Smith operator Q (cid:48) α for a pertur-bation α defined as Q (cid:48) α = − iS − ∂ α S [9]. Note that herethat we distinguish between Q (cid:48) α and Q α = − iS † ∂ α S .These two operators coincide for a unitary scattering ma-trix which is not the case here due to absorption.First, we estimate the pseudo-inverse of the scatteringmatrix using a singular value decomposition of S : S =Σ Nn =1 u n λ n v † n . This leads to S − = Σ Nn =1 v n (cid:18) λ n (cid:19) u † n . (S15) u n and v n are the left and right singular vectors of S associated the singular values λ n . In the vicinity of aCPA condition, one singular value of the the scatteringmatrix is vanishing, R ( ω ) = λ N →
0, for an incomingwavefront v n = ψ CPA . The singular vector u n is thenormalized outgoing field ψ out / (cid:107) ψ out (cid:107) given by Eq. (S8)with a global phase shift corresponding to the phase of3 λ -80-60-40-200 R e fl e c t i o n c o e ffi c i e n t CPA wavefrontUnoptimized wavefrontModel
FIG. S4. Variations of the reflection coefficient on a loga-rithmic scale at the CPA condition (blue crosses) and for arandom incoming wavefront (black circles), as a function ofthe perturbation λ in the effective Hamitonian model. Thered line gives the best fit of R with Eq. (S9) in which ∆ ω isreplaced by 2 πKλ , where K is a constant. the term (cid:104) − i Γ n ∆ ω + i (Γ n +∆Γ / (cid:105) .The pseudo inverse S − in the vicinity of the CPA con-dition can then be approximated by S − = v N (1 /λ n ) u † n .Using that Sψ CPA = ψ out with (cid:107) ψ out (cid:107) = R , the pseudo-inverse can also be expressed as S − ( α ) = ψ CPA ψ † out R ( α ) . (S16)As S − is a matrix of unit rank, the generalized Wigner-Smith operator Q α = − iS − ∂ α S is hence also of unitrank with a left eigenvector which is v n = ψ CPA . Wehave shown in Fig. 3a of the main text that the field-field correlation is extremely sensitive to the rotation ofthe perturber at the CPA condition. It comes thereforeas no surprise that the wavefront yielding the optimalsensitivity to the perturbation is also the CPA wavefront.
G. Simulations of a perturbation
In the main text, we find that the increase of the reflec-tion coefficient at the CPA condition for a perturbationof p meta-atoms is well predicted by Eq. (1) of the maintext in which ∆ ω is replaced by 2 πKp ( K is a constant).Here, we verify this scaling in random-matrix simulationsbased on the effective Hamiltonian. The perturbation ismodelled by a variation of the initial Hamiltonian H as H = cos( λ ) H + sin( λ )∆ H (S17)In this model, ∆ H is a random Hamiltonian which isstatistically independent of H . The parameter λ de-termines the perturbation strength and the average den-sity of states is preserved [10, 11]. The scattering ma-trix is computed from Eq. (S1). The reflection coefficient R is finally obtained in simulations for the CPA wave-front ψ in = ψ CPA and for a random incoming wavefront, ψ in = ψ rand and R is averaged over 200 CPA realizationsin each case. The linewidth detuning ∆Γ is chosen sothat 10log10( R ) = −
80 dB for λ = 0.We observe in Fig. (S4) a very good agreement be-tween the simulation results and Eq. (S9) in which ∆ ω is replaced by 2 πKλ , R ( ω ) = 4(2 πKλ ) + (∆Γ) πKλ ) + (2Γ n + ∆Γ) (S18)Here K is a constant which is adjusted to provide the bestfit of R . Eq. (S9) is however valid only for small shifts ofthe central frequency ω n . A deviation between simulationresults and the model is hence found for λ > . II. EXPERIMENTAL DETAILS
In this section, we provide further experimental de-tails on our “on-demand” realization of CPA in a pro-grammable complex scattering enclosure. We begin byproviding additional details regarding working principleand key characteristics of our programmable meta-atoms.Then, we provide supplementary descriptions of our ex-perimental setup, we characterize our chaotic cavity, anddetail our data processing. Finally, we provide an addi-tional analysis of the optimized metasurface configura-tions.
A. Programmable Metasurfaces
A programmable metasurface is an ultra-thin arrayof meta-atoms with programmable electromagnetic re-sponse. Programmable metasurfaces are emerging aspowerful young member of the metamaterial family ow-ing to their ability to dynamically shape electromag-netic fields as well as to the ease of fabricating theprogrammable metasurfaces. Initially, programmablemetasurfaces were conceived for free-space control ofwaves [12, 13] but more recently their powerful useful-ness to tune the scattering properties of complex mediahas been discovered [14]. This has led to a wide range ofapplications in indoor wireless communication [15, 16],sensing [17] and even analog computing [18].In our experiments, we use a 1-bit programmable meta-atom as unit cell of which the programmable metasur-faces are composed. Each meta-atom has two digitalizedstates, “0” and “1”, and can be configured individuallyto be in either of these states. The fundamental workingprinciple is that introduced in Ref. [19]. The meta-atomconsists of two resonators that hybridize, and the res-onance frequency of one of the two can be altered bycontrolling the bias voltage of a PIN diode. This resultsin a phase change of roughly π for the reflected wave.4Unlike the meta-atoms in Ref. [19], our prototype of-fers independent control over the two orthogonal fieldpolarizations. In short, this is achieved by fusing twosingle-polarization meta-atoms, one rotated by 90 ◦ withrespect to the other, into a single one. As seen in theinset of Fig. S5a, the two meta-atoms share the samefixed resonator but one PIN diode controls the tunableresonator for each of the two polarizations.Thoroughly characterizing the exact response of themeta-atoms is challenging since the characterization pro-cedure inevitably impacts the results. Here, we illumi-nate the metasurface with a plane wave polarized alongone of the two polarizations with a horn antenna. Wesynchronize the states of all meta-atoms for this measure-ment and measure the return loss of the horn antenna inclose proximity to the metasurface for the two possiblestates, “0” and “1”. The resulting curves are displayedin Fig. S5. Recall that the absolute values of the re-turn loss are modulated by the horn antenna’s transferfunction. The absolute value R = | S | is thereforenot an indicator of the amount of energy absorbed bythe metasurface. The phase difference of S for the twostates, however, confirms that around the working fre-quency the metasurface does indeed offer a phase shiftof roughly π . In other words, every meta-atom can beconfigured to mimic a Dirichlet or a Neumann bound-ary condition. Similar results are obtained for the otherpolarization. B. Experimental Setup
A global overview of our experimental setup is pro-vided in Fig. S6. The complex scattering enclosure is ametallic cuboid of dimensions 50 × ×
30 cm with twohemisphere deformities on two mutually perpendicularwalls. Two arrays of programmable meta-atoms with thecharacteristics detailed in the previous section are placedon two mutually perpendicular walls. The system is con-nected to the outside via eight channels. Specifically,the eight ports of our vector network analyzer (VNA,Keysight PXI) are connected via coaxial cables to eightidentical waveguide-to-coax transitions. The latter arelocated in pairs of four on two sides of the cavity, butin a chaotic system the spatial location of the scatteringchannels is irrelevant. These ports are well adapted in therange between 4 GHz and 7 GHz, as evidenced by a free-space measurement of one port’s reflection coefficient inFig. S7. A computer is connected to the VNA to initiatemeasurements of the 8 × -40-30-20-100 R ( d B ) State 1State 2
Frequency (GHz)024 P h a s e D i ff e r e n c e ab π phase shiftCPA frequency FIG. S5. (a) Reflection coefficient R = | S | measured witha horn antenna placed in front of the metasurface, for the twodistinct states “0” and “1”. (b) Phase difference of the S parameter between the two states. The frequency chosen toobtain CPA by programming the metasurfaces correspond tothat at which the metasurface responses can create a phaseshift of π .FIG. S6. The schematic drawing illustrates how the complexscattering enclosure’s eight ports are connected to a vectornetwork analyzer (VNA) as well as how a computer programsthe metasurfaces, controls the VNA and commands the step-per motor. Frequency (GHz) -20-15-10-50 R e f l e c t i on c oe ff i c i en t FIG. S7. Reflection coefficient | S | [dB] measured for a port(coax-to-waveguide transition) in free space. C. High-Precision Measurements
Given the extreme sensitivity of the CPA conditionto any sort of detuning, we take a number of measuresto enhance the precision of our measurements. Obviousmeasures include a careful choice of settings of the vectornetwork analyzer: an intermediate-frequency bandwidthof 1 kHz and an emitted power of 5 dBm. A more subtleadditional technique that we employ is to measure S ( ν )with an extremely small frequency step of 1 kHz in thevicinity of the targeted CPA frequency ν . We then fitthis data with a linear function in the complex Arganddiagram [20], as shown in Fig. S8. We thereby further re-duce the sensitivity to noise in our experiment and hencethe stability of our optimization protocol. real[S ij ] -0.0358-0.0356-0.0354-0.0352-0.035-0.0348 i m a g [ S i j ] FIG. S8. Example of a scattering coefficient in the vicinity of ν fitted in the complex Argand diagram by a linear function. D. Convergence of the minimal eigenvalue
Fig. S9 provides an illustration based on experimen-tal data of how the smallest eigenvalue of the scatteringmatrix approaches zero, both at the targeted operationfrequency as a function of iterations, and as a function offrequency for an optimal and unoptimized configuration. -0.05 0 0.05-0.0500.05
CPAunoptimized -0.005 0 real[ λ ] -0.00500.005 i m a g [ λ ] real[ λ ] i m a g [ λ ] ba FIG. S9. Minimal eigenvalue λ of the scattering matrix shownin the complex plane as a function of the iterations for ν = ν CPA (a) and as a function of frequency for an optimized andan unoptimized configuration (b). In both cases, λ → time (ns) -8 -6 -4 -2 I n t e n s i t y [ a . u . ] raw datastirred fieldfit Number of antennas γ ( M H z ) ab FIG. S10. (a) Variations of the reflected intensity in thetime domain averaged over 300 random configurations of themetasurfaces. The blue (black) curve is found from an in-verse Fourier Transform of the scattering matrix elementswith (without) removing its average value. The two curvesconverge at late times to the same exponential decay, e − γt (red dashed line). (b) By fitting the exponential decay for anumber of channels to the cavity varying from 3 to 8, we ex-tract gamma as a function of N (blue crosses). The red lineis a linear regression of the data and the black dashed linegives the extracted absorption strength γ a using Eq. (S19). E. Characterization of the Chaotic Cavity
Here, we estimate the linewidth associated with the an-tennas in the chaotic process within the cavity. We firstmeasure S for 300 random configurations of the metasur-faces. In our experiment, the metasurfaces only partiallycover the cavity walls and a substantial unstirred fieldcomponents remains that is not impacted by the meta-surface configuration. The ratio of unstirred to stirred6field components |(cid:104) S ba ( f ) (cid:105)| / (cid:104)| S ba ( f ) | (cid:105) is equal to 0.65at f = 5 .
147 GHz. Therefore, we remove the unstirredcomponents (cid:104) S (cid:105) before estimating the average linewidthassociated with the ports in the chaotic process. We thenperform an inverse Fourier transform and find the inten-sity in the time domain I ( t ). The average signal is pre-sented in Fig. S10a. We compare this signal with theintensity found in absence of the unstirred components,i.e. found from the field to which its average has beensubtracted, S = S − (cid:104) S (cid:105) . In both case, the intensityis seen to decrease exponentially, I ( t ) ∼ exp( − Γ t ) for t >
25 ns. We repeat this procedure for different num-bers of antennas N ant connected to the cavity, varying N ant from 3 to 8. In each case, we extract γ = Γ / π which is shown in Fig. S10b. This makes it possible toseparate the contribution of the antennas γ n = N ant γ c ( γ c is the average linewidth for a single antenna) and thecontribution of internal losses within the cavity γ a as γ can be decomposed as γ = N ant γ c + γ a . (S19)From a linear fit of γ , we find that N (cid:104) γ n (cid:105) = 0 .
33 MHz( N = 8), and (cid:104) γ a (cid:105) = 11 . Q = f /γ = 446 at f = 5 .
147 GHz. Finally, we now evalu-ate the modal overlap, d = γ/ ∆, of the cavity. The meanlevel spacing of successive resonances can be estimatedfrom Weyl’s law [21, 22] as ∆ = c / (8 πV f ) ∼ . d = 12 .
8, showing that many resonances con- tribute to the scattering matrix at any given frequency.
CPA realizations P e r c e n t a g e o f - s t a t e s FIG. S11. Percentage of meta-atoms in state “1” for 18 inde-pendent realizations of CPA.
F. Analysis of Optimal Metasurface Configurations
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