Impact of non-Hemiticity on modal strength and correlation in transmission through random open cavities

The nonorthogonality of eigenfunctions over the volume of non-Hermitian systems determines the nature of waves in complex systems. Here, we show in microwave measurements of the transmission matrix that the non-Hermiticity of open random systems leads to enhanced modal excitation and strong correlation between modes. Modal transmission coefficients reach values comparable to the dimensionless conductance which may be much larger than unity. This is accompanied by strong negative correlation between modal speckle patterns ensuring that net transmission is never larger than the incident power.

A single mode may be excited when the sample is illuminated on resonance with a spectrally isolated mode.However, as the coupling to the exterior and internal dissipation increase, modes broaden and overlap spectrally.In such non-Hermitian systems, the eigenfunctions are nonorthogonal [1,5,[16][17][18][19].The non-orthogonality of eigenfunctions leads to the existence of exceptional points in systems that incorporate both gain and loss but in which parity-time symmetry is preserved [20,21] and to the enhancement of the linewidth and spontaneous emission rates in laser resonators [22][23][24].Its impact has also been explored in localized plasmonic surfaces [25], dielectric microcavities [26,27] and chaotic systems with small perturbations [28,29].However, direct demonstrations of the nonorthogonality of modes in open systems and its influence on the statistics of modal excitation have remained a challenge.
Microwave measurements in the region of moderate modal overlap have shown that modal strength in transmission may be enhanced in isolated cases and interference between modal speckle patterns tends to suppress transmission below the incoherent sum of modal contributions [3,30,31].Determining the statistics of individual modes and the degree of interference between modal speckle patterns is central to describing the propagation of waves and to controlling the flow of radiation by shaping the incident wavefront in photonic [32][33][34] and plasmonic systems [4,6,35].
Here we demonstrate the relation between the correlation of eigenfunctions over the volume of the sample and the correlation of modal components of the transmission matrix (TM) in non-Hermitian systems.This leads to a systematic linkage of enhanced modal transmission and destructive interference among correlated eigenfunctions of overlapping resonances.The degree of modal overlap in random media is varied by changing the degree of disorder or the openness of the sample boundaries.The modal overlap may be characterized by the Thouless number , which is the ratio of the average linewidth to the average level spacing = n / .δ reflects the degree of spatial localization since tightly localized modes have narrow linewidths since they couple weakly to the surroundings through the sample boundaries [10,36].The Thouless number is equal to the average of the dimensionless conductance = , which similarly falls as modes are more strongly localized and transport is suppressed.For classical waves, the dimensionless conductance corresponds to the average transmittance, = , where, = Σ | | is the sum over flux transmission coefficients between all incoming and outgoing channels, a and b, respectively [10,14,36,37].
We measure the TM of a multichannel two-dimensional random system (see Fig. 1(a)).The disordered aluminum cavity of height = mm, width = mm and length = mm supports a single transverse mode in the vertical direction.The randomly positioned scattering elements are 6-mm-diameter aluminum cylinders.The TM which is the part of the scattering matrix associated with the transmission coefficients between incoming and outgoing channels on the left and right side of the sample, respectively, is measured in the microwave range between two linear arrays of = antennas that are coaxial to waveguide adapters [31].Spectra of each transmission coefficients of the TM are successively obtained using two electro-mechanical switches and the two ports of a vector network analyzer (VNA).The openings of the system are fully controlled by the antennas [38] but strong internal reflection may appear at the interfaces due to the metallic region surrounding each coupler in comparison to a waveguide which is fully open on both ends.The resonances ̃ and the projection of eigenfunctions on the interfaces are found in an analysis of spectra of the TM as a superposition of modal TMs (MTMs) [31] = −Σ T − ̃ . ( and are the components of the vector associated with the left and right sides of the sample, respectively.The modal analysis is performed using the Harmonic Inversion technique to extract the central frequencies and linewidths from spectra of transmission coefficients [2,31].Each modal transmission coefficient giving the vectors and is then reconstructed from a fit of the corresponding transmission coefficient spectrum as a superposition of Lorentzian lines.Equation (1) shows that the MTM = − T is of unit rank and and correspond to the modal speckle patterns of the nth mode.The validity of the expansion of the TM into MTMs is confirmed within the accuracy of the modal decomposition by the finding that the ratio between the two first eigenvalues of each MTM is higher than 10 2 .The vectors and are then extracted from the singular value decomposition of each MTM. between eigenvectors over the volume [17,19,39].The diagonal elements of are equal to the Petermann factor, ≡ , which is a measure of the degree of complexness of the eigenfunctions [40][41][42][43].The Petermann factor characterizes the excess spontaneous emission for laser cavities and governs the linewidth of lasing modes [22][23][24]44].For small modal overlap, the eigenfunctions coincide closely with the real eigenfunctions of the closed system so that ∼ .
However, increases as the coupling of the sample to its surroundings increases [41] and can have values exceeding one thousand [23].
The off-diagonal elements of give the degree of correlation between eigenfunctions.Since the eigenfunctions are complete, Σ = [17], the enhancement of with increasing implies that non-vanishing and negative-on-average correlation ≠ < between overlapping eigenfunctions, in contrast to the orthogonality in Hermitian systems.
A direct probe of the overlap matrix would require a non-invasive scan of the spatial profile of the eigenfunctions inside the sample, which is almost impossible in most cases.However, the correlation of eigenfunctions is expressed in the correlation of their projections onto the coupling channels [5,45,46 (2) For = , Eq. ( 2) gives the relation between the linewidths and the coupling vectors, = ‖ ‖ 4 / [45].In principle, Eq. ( 2) makes it possible to obtain the degree of nonorthogonality of eigenfunctions from the decomposition of the scattering matrix into a sum of Lorentzian lines.Because we measure the TM rather than the scattering matrix, it is not possible to find the vectors and separately, only the MTMs = − T can be calculated.The relative phase and magnitudes of the vectors and are unknown.We compute the "transmission overlap matrix" The diagonal elements are ̃ = ̃ ̃ , ̃ = .̃ ̃ , ̃ and ̃ , ̃ are equal for extended states in the limit for which ‖ R ‖ ∼ ‖ L ‖ [46].We also compensate the impact of absorption on the operator ̃.Since the linewidths are broadened by absorption, we replace in the denominator of Eq. ( 3) ̃ − ̃ * by ̃ − ̃ * + , where is the homogeneous absorption rate, with ∼ .MHz in the low frequency range with weakly coupled antennas and ∼ MHz in the higher frequency range with strongly coupled antennas [46].
The real parts of ̃ ̃ , ̃ are shown in Fig. 2(f-h) for the three samples studied.When the antennas are weakly coupled, the overlap matrix ̃ is seen to be close to diagonal as it would be for a closed system.Uniform losses that broaden the linewidths indeed do not alter the orthogonality of eigenfunctions.For the case of strong coupling and ∼ ., the transmission overlap matrix is also mostly diagonal.However, when two resonances overlap enhanced diagonal and negative off-diagonal elements are observed, as seen, for instance, for the two resonances with central frequencies around 11.03 GHz.For the sample with the smaller number of scatterers and hence shorter mode lifetime and larger mode linewidth, ∼ ., the diagonal part for most modes increases while the off-diagonal terms become more negative.
To carry out measurements on a random ensemble, we move a mm-diameter magnet along a line within the medium in steps of / = .mm, where is the wavelength at 12 GHz.The magnet within the sample is moved by the force of a second magnet above the top plate of the cavity.We find resonances and associated modal coefficients for more than a thousand modes in 40 realizations of two ensembles: 1) weakly coupled antennas with 30 scatterers giving = ., and 2) strongly coupled antennas with 280 scatterers.Since no scatterers are positioned along the line of motion of the magnet, for this ensemble is increased to = .from = ., for the sample where there is no excluded volume for scatterers.
We first explore the degree of correlation between different modes, the off-diagonal elements of ̃ ̃ , ̃ .The average ̃ ̃ , ̃ is shown in Fig. 2 as a function of the complex shift between two resonances | ̃ − ̃ | normalized by .̃ ̃ , ̃ is seen to be negative with a magnitude which decreases with | ̃ − ̃ |.The magnitude of all elements are seen to be stronger for strongly coupled antennas as a consequence of greater nonorthogonality.
Chalker and Mehlig predicted that the eigenvector correlator of x non-Hermitian random matrices of the Ginibre complex Gaussian ensemble is given by [17,19,39] where ̃= √ ̃ − ̃ is essentially the complex spacing between resonances.This result was confirmed for non-Hermitian random matrices describing the statistical properties of resonances in open chaotic cavities [18].To compare theoretical and experimental results, the complex spacing is normalized by the level spacing and a scale factor of order of unity, | ̃| = | ̃ − ̃ |/ .A good fit of experimental results for ̃ ̃ , ̃ in Fig. 2  The distribution of the modal strengths in transmission which are the diagonal elements = ̃ is shown in Fig. 3(a).The distribution extends between 0 and unity and is peaked near = .Values as large as = are found in the tail of the distribution for > .
We now analyze the variation of the enhancement of with .In the absence of absorption, √ = ‖ ‖ so that can be expressed as the product of two terms of different origin Here, , with , is the coupling asymmetry for the nth eigenfunctions between the left and right boundaries, which reflects the spatial pattern of the eigenfunctions within the sample.
We investigate the statistics of in random media in the crossover from diffusion to localization using the tight-binding Hamiltonian (TBH) model [47,48].The Hamiltonian of the closed random system of dimension The on-site potential is independently and uniformly distributed on the interval [−/ , / ].Each lead is modeled by a 1D semi-infinite TBH so that the effective Hamiltonian is given by = − , where V is an x matrix with elements equal to unity for sites to which the leads are attached and zero elsewhere [47,48] .Hence, Eq. ( 5) gives, ∼ cosh − − .Assuming a uniform distribution of between 0 and leads to a bimodal distribution of [49].This is in agreement with the formula proposed for isolated peaks in the transmission spectrum of 1D samples using a resonator model associated with effective cavities of length = ℓ [49,50].When transmission is dominated by a single mode, ∼ and ∼ , but modes may occasionally overlap even in an ensemble in which < [51].may then be large and can significantly exceeds unity to produce a tail in .
For diffusive waves, > , the coupling to the surroundings increases and modes overlap spectrally.The eigenstates are extended throughout the sample and the coupling to the modes from the left and the right sides are typically similar so that ‖ ‖ ∼ ‖ ‖ .Hence, ∼ and the lower peak in and disappears.The probability distributions of and ∼ are then broad with peaks shifting towards values much greater than unity.Values of as large as are found.
The variation of , and with are shown in Fig. 3(d).and first increase with as the correlation between eigenfunctions increases, but then decrease once the sample is translucent, / .The eigenfunctions in this regime are only slightly perturbed from the orthonormal eigenfunctions of the empty waveguide so the degree of nonorthogonality is small.It is worthwhile to consider separately the contributions to the transmittance of the on-and offdiagonal terms.From the modal decomposition of the TM given in Eq. ( 1), the transmittance can be expressed in terms of modal components as A perturbative approach in the limit of small modal overlap [40] shows that ≡ increases as ∼ + [46].modes contribute to transmission so that the incoherent sum of modal strengths inc = Σ is increased relative to = by a term scaling as ∼ .Transmission is then reduced by destructive interference between correlated modal components with ≠ in Eq. ( 6) ensuring that transmission is bounded by unity.The contribution of offdiagonal terms to the average transmittance scales as − , as expected.Numerical results from RMT simulations shown in Fig. 4 are in excellent agreement with the perturbative approach for < . .The decomposition of the TM into its modal components provides a fresh vantage point from which to understand and control transport through and energy within disordered photonic and plasmonic media, chaotic cavities, and multimode fibers [32,33].The ability to approach perfect transmission in diffusive systems by exciting the first transmission eigenchannel and the maximal contrast in focusing through random systems via control of the TM increases with the number of resonant modes participating in transmission [34].Thus the correlation in transmission eigenvalues is linked to the correlation in the modes of the medium.This is particularly important since modes are defined over the full spectrum, while eigenchannels are defined at a single frequency.The spectral correlation of modes may therefore be used to enhance control of transmission and delay times for broadband pulses [30,52,53].
The modal overlap matrix given in Eq. ( 2) of the main text is = − ϯ / ̃ − ̃ * . 4,5The components of the vector associated with the left and right sides of the sample are and , respectively, so that = [ ].This gives However, from measurements of the transmission matrix (TM), we can only extract the modal TMs (MTMs) = − T .Hence, the vectors and cannot be found separately and the relative phase and amplitude between them cannot be determined from the TM.However, we can write ϯ When the disorder is uniform in space and when eigenfunctions extend throughout the sample, the average degree of correlation between modal components on the left is the same as for modal components on the right.The statistical properties of the vectors and are then equivalent so that ‖ ‖ ∼ ‖ ‖ .
In the limit , we can approximate This can also be expressed in terms of the trace of the MTMs as, (S6) The diagonal elements of ̃ ̃ , ̃ give the modal strengths in transmission, = ‖ ‖ ‖ ‖ / | | .We show in the main text that can be written as the product = .Here = ‖ ‖ ‖ ‖ / ‖ ‖ +‖ ‖ is the asymmetry in coupling of the nth eigenfunctions to the left and right boundaries.When ∼ , the degree of correlation between eigenfunctions is the same on the two sides so that the right hand sides of Eq. (S6) and Eq.(S3) are equal.

B. Estimation of the homogenous losses
To estimate the linewidth associated with homogeneous absorption and losses through the antennas ̃ , related by = ̃ + , we observe that the average ̃ scales linearly with the number of coupled antennas, ̃ = ̃ , where ̃ is the average linewidth associated to a single antenna coupled to the sample.By disconnecting the antennas from the switches, it is possible to decrease the number of coupled channels and thereby obtain an estimate for .In the weak coupling regime, we find ̃ ∼ .MHz and ∼ .MHz, so that the broadening of the resonances in the weak coupling regime is therefore mainly due to homogeneous absorption within the sample.In the strong coupling regime associated to a higher frequency range, ̃ ∼ MHz MHz and ∼ MHz show that losses are dominated by the coupling through the antennas for = .
The MM  internal Hamiltonian of the effective Hamiltonian = − is modeled by a real symmetric matrix drawn from the Gaussian Orthogonal Ensemble with = / .The coupling matrix V is a real random matrix with Gaussian distribution and = / .The modal overlap increases with increasing coupling strength of the channels to the system, = / .The coupling strength to the continuum is = / , where = / is the mean level spacing at the center of the band, = .The coupling strength is related to the transmission coefficient through the channels, , with = / + .

B. Distribution of asymmetry factor
We first express the asymmetry between projections of the eigenfunctions on the incoming and outgoing channel, , as Since the eigenfunctions are uniformly distributed over the volume for chaotic systems, the terms ‖ ‖ and ‖ ‖ are the sum of independent random variables.The average of depends on and the degree of complexness of the eigenfunctions defined as = Im / Re . is related to the phase rigidity of the eigenfunctions and the Petermann factor, = − , with = − / + .For a given in the limit , Eq. (S8) yields ∼ + / + + .When the modal coupling vectors and are real, which is the case of the weak coupling regime, ∼ gives = / + .However, for complex coupling vectors with statistically equivalent real and imaginary parts, ∼ , and = / + .
The term ‖ ‖ − ‖ ‖ / ‖ ‖ + ‖ ‖ is mainly the difference of two independent random variables ‖ ‖ and ‖ ‖ .In the limit , this is a Gaussian variable.− is the square of a Gaussian variable and therefore has the Porter-Thomas distribution 14 The distribution of is finally given by = ∫ , .This however requires the distribution of , which is known analytically only in the weak coupling regime defined by √Var .
Nevertheless, in the limit , fluctuations in are small so that we can make the approximation ∼ .Numerical results shown in Supplementary Fig. S2a are in good agreement with the analytical result with = for = , but deviations are observed for small values of due to fluctuations in .

C. Distribution of and
The distribution of the Petermann factor and modal strength are shown in Supplementary Figs.S2b,c.In the weak coupling regime ( ), most resonances are isolated so that and are both peaked near = .Values of smaller than unity are a consequence of the asymmetry of modes for which < .As increases, the two distributions broaden.Long tails are observed for and for = .and = .Values of the Petermann factor and modal strengths as high as 300 are found.
The sum of the diagonal and off-diagonal terms satisfies the relation with of order 1, in agreement with the Thouless relation in random media = .
The eigenfunctions are the complex right | ⟩ and left | eigenvectors of the wave equation with outgoing boundary conditions, + = .They form two complete bi-orthogonal sets satisfying the orthogonality relations | = and are associated with complex eigenvalues ̃ = − / , where is the central frequency and is the linewidth.For systems with time-reversal symmetry, the eigenvectors are related by the transpose | = | ⟩ and the matrix of eigenfunctions is normalized by = .The expansion of the scattering matrix in terms of quasi-normal modes is then = − T [ − Ω ̃]− .The matrix of vectors is the projection of eigenfunctions onto the channels of the sample and Ω ̃ is the diagonal matrix of eigenvalues ̃ .

Figure 1 (
Figure 1(b) shows the spectrum of the transmission through the antennas determined from = − | | , where is the mean reflection parameter at each antenna.We carry out measurements in three ensembles with moderate modal overlap in frequency ranges in which: (1) the antennas are weakly coupled to a sample ( ∼ .) with 30 cylinders contained within a cavity, for which ∼ .; and (2,3) the antennas are strongly coupled to a sample ( ∼ .) and the disorder is strong.For the samples with 280 and 200 scatterers, ∼ .and ∼ ., respectively.
FIG. 2: Measured correlator . The wavevector is / in the center of the band at ∼ .TBH simulations with = and = are carried out for different values of with ranging from 5.6 to 0.02.For localized waves, , the distribution shown in Fig.3(c) is bimodal with peaks at = and = .This is a consequence of the bimodal distribution of asymmetry factors of spatially localized modes.The bulk of the distribution can be explained by considering the coupling to a localized eigenstate exponentially peaked at in the sample with localization length .The strength of the eigenfunctions at the left and right interfaces is given by ‖ ‖ ∼

FIG. 3 :
FIG. 3: (a) Distribution of experimental modal strengths in the strong coupling regime with = . .(b,c) Simulations of and in the inset for samples with = .(blue line), = .(black line), = .(red line).(d) Variation of , and with the conductance .

FIG. 4 :
FIG. 4: Random matrix theory simulations of the scaling of the transmittance (blue circles), and the sum of the diagonal (black triangles) and off-diagonal (red crosses) modal contributions as a function of .The dashed lines are the fits using analytical expression given in Supplementary Material [46].
therefore compute the transmission modal overlap matrix (see Eq. (3) of the main text)