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Universal Properties of 2-Port Scattering, Impedance and Admittance Matrices of Wave Chaotic Systems

Statistical fluctuations in the eigenvalues of the scattering, impedance and admittance matrices of 2-Port wave-chaotic systems are studied experimentally using a chaotic microwave cavity. These fluctuations are universal in that their properties are dependent only upon the degree of loss in the cavity. We remove the direct processes introduced by the non-ideally coupled driving ports through a matrix-normalization process that involves the radiation-impedance matrix of the two driving ports. We find good agreement between the experimentally obtained marginal probability density functions (PDFs) of the eigenvalues of the normalized impedance, admittance and scattering matrix and those from Random Matrix Theory (RMT). We also experimentally study the evolution of the joint PDF of the eigenphases of the normalized scattering matrix as a function of loss. Experimental agreement with the theory by Brouwer and Beenakker for the joint PDF of the magnitude of the eigenvalues of the normalized scattering matrix is also shown.


I A: Introduction:
The scattering of short-wavelength waves inside enclosures manifests itself in several fields of physics and engineering such as quantum dots [1], atomic nuclei [2], acoustic resonators [3,4], electromagnetic compatibility [5], etc. Of particular interest is the case when the ray trajectories within the enclosure show chaotic dynamics in the classical limit. This interest has spawned the field of "wave chaos" (or "quantum chaos"), and has attracted much theoretical and experimental work [6, 7] to understand its nature. On account of the small wavelength of the scattered waves, as compared to the characteristic length-scale of the enclosure, the response of these systems exhibit extreme sensitivity to small changes in configuration, driving frequency, nature of driving ports, ambient conditions such as temperature, etc. Thus, an intimate knowledge of the response of any such system for a given well-defined stimulus or system configuration will not provide any foresight in predicting the response of a similar system when the stimulus or system configuration is slightly altered. This calls for a statistical approach to quantify the nature of such wave-chaotic systems.
In this regard, Random Matrix Theory (RMT) [8] has proved to be an integral tool in predicting universal statistical aspects of wave chaotic systems. It has been conjectured that in the short-wavelength regime, RMT can be used to model wave-chaotic systems. In particular, the statistics of systems that show Time-Reversal Symmetry are conjectured to be described by the Gaussian Orthogonal Ensemble (GOE) of random matrices, while the statistics of systems showing Broken Time-Reversal Symmetry are conjectured to be described by the Gaussian Unitary Ensemble (GUE) of random matrices. There is also a third random matrix ensemble corresponding to certain systems with spininteractions (Gaussian Symplectic Ensemble). RMT provides a potential framework for uncovering universal statistical properties of short-wavelength wave chaotic systems (e.g. Ericson fluctuations in nuclear scattering [7,9] and universal conductance fluctuations (UCF) in quantum-transport systems [10] ).
Since the applicability of RMT and the concomitant universal statistics is conjectural rather than rigorous, and since this conjectured applicability is said to be asymptotic in the limit of wavelength small compared to the system size, it is important to test the RMT conjecture against results obtained for specific real situations.
Experimentally, however, validating the applicability of RMT has always proved challenging. One of the most common problems encountered by experimentalists is the presence of non-universal, system-specific artifacts introduced into the measured data by the experimental apparatus. These are generally referred to as the "direct processes", as opposed to the "equilibrated processes" which describe the chaotic scattering within the system [11]. A typical example presents itself while measuring the statistical fluctuations in the scattering of microwaves through cavities with chaotic ray dynamics. These fluctuations are studied by exciting the cavity through coupled ports and observing the response (reflection and transmission) for a given excitation. Generally, it is not possible to perfectly couple (ideally match) the ports to the cavity at all frequencies. Here by "perfect coupling" we refer to the situation in which there is no prompt reflection of a wave incident from an incoming channel on a cavity port. Thus, such a wave would be entirely transmitted into the cavity, and any reflection coefficient measured from that port is the result of waves that have entered the cavity, bounced around, and subsequently been reflected back toward the port. (Subsequently in this section and section I (b), we give a more precise definition of perfect coupling in terms of the port radiation impedance and the characteristic impedance of the incoming channel.) We refer to the deviation from perfect coupling as "mismatch". This mismatch, which is strongly determined by the geometry of the port, manifests itself as systematic fluctuations in the measured data. The result is that the measured data depends on the non-universal, direct processes of the ports, as well as the underlying universal, equilibrated processes of the chaotic scattering system. Several approaches have been formulated to account for these direct processes [12, 13, 14] of which the "Poisson Kernel" approach introduced by Mello, Pereyra and Seligman is of special mention. Based on an information-theoretic model, the "Poisson Kernel" characterizes the direct processes between the ports and the cavity by the ensemble-averaged 1 such a surrogate for for a specific system used system configuration averaging. We denote this surrogate for as . Averaging over configurations, however, may suffer from excessive statistical error if the number of configurations averaged over is insufficiently large. Thus, to improve the estimate of the scattering coefficient statistics, Refs. [15,16], which treat one port (scalar ) scatterers, make use of an ergodic hypothesis [17,18] to include an additional running average over frequency ranges that include many resonances but are sufficiently small that the scattering coefficient statistics can be assumed to be nearly constant (i.e., a frequency range where the port coupling strengths are nearly constant). Using this approach, Refs. [15,16] have investigated the universal fluctuations in the reflection coefficient of 1-port wave-chaotic microwave cavities. This was shown to produce favorable results for 1port systems when compared with RMT predictions. We note, however, that the analysis is highly dependent on the accuracy of the experimentally-obtained , which is prone to statistical errors.
The situation can become even more complicated when dealing with ports. In the recent 2-port paper, Ref. [19], the authors circumvent such problems by taking careful steps to ensure that the driving ports are nearly perfectlycoupled to the cavity in the frequency range where the data is analyzed. In doing so, Ref. [19] achieves good agreement between the experimental results for the fluctuations in the transmission coefficient, and the RMT predictions for timereversal-symmetric and for broken-time-reversal-symmetric cavities. We note, however, that Ref. [19] is for the case of perfectly coupled ports and that it is desirable to also deal with arbitrary port couplings.

N
In Ref. [20,21] a novel method to characterize the direct processes between the cavity and the driving ports was introduced. This method, which is motivated by electromagnetic-wave propagation inside complex enclosures, makes use of impedances to characterize the direct-processes rather than the ensemble-averaged scattering matrix as in Ref. [12]. For an -port scattering system, the Scattering Matrix is the real, diagonal matrix whose elements are the characteristic impedances of the waveguide (or transmission line) input channels at the driving ports. Like

NxN
is also a well-established physical quantity in quantum mechanics. Just as the elements of represent the transition probabilities from one state to the other in a quantum scattering system, S t Z t is an electromagnetic analog to Wigner's Reaction Matrix [22], which linearly relates the wave function to its normal derivative at the boundary separating the scattering region from the outside world.
References [20,21] have shown that the direct processes can be quantified by the "radiation impedance" of the driving-ports. For a cavity driven by a single port, the radiation impedance of the port is that impedance observed at the reference plane of the port which retains its coupling geometry but has the distant walls of the cavity moved out to infinity and an outward radiation condition imposed. Experimentally, this can be realized by lining the walls that are distant from the port with microwave absorber. The oneport radiation impedance, denoted , is thus a frequencydependant, complex scalar quantity which depends only on the local structure of the port and is not influenced by the shape of the distant cavity boundaries; is the "radiation resistance" which quantifies the energy dissipated in the far-field of the radiating port, and is the "radiation reactance" which arises from energy stored in the near-field of the radiating port. thus presents a nonstatistical experimentally viable way to quantify the direct processes in a wave-chaotic system for any given port geometry; without resorting to averaging.
The normalized scattering matrix is , where is the identity matrix.
and by then using Eqs. (2) and (3). The matrix o Z t is a real diagonal matrix whose elements are the characteristic impedances of the transmission lines connected to the driving ports.
The normalized quantities z t and represent the impedance and scattering matrix when the ports are perfectly coupled to the cavity, i.e., when is a smoothly varying function of frequency and of the coupling-port structure, Eqs. (2) and (3) yield the perfectly-coupled (ideally matched) impedance and scattering matrix over any arbitrarily large range of frequency and for any port geometry.
Reference [21] predicts that the PDFs of the eigenvalues of z t which are contained in the diagonal matrix the wavenumber for the incoming frequency and is the mean-spacing of the adjacent eigenvalues of the Helmholtz operator, , as predicted by the Weyl Formula [30] for the closed system. The use of the Weyl Formula here is conventionally accepted for lack of a more complete treatment which is applicable to open systems or to systems with high absorption (as discussed in this paper). The quantity Q represents the loaded quality factor of the cavity. The quantity thus represents the ratio of the frequency width of the cavity resonances due to distributed losses to the average spacing between resonant frequencies. Reference [20] also predicts that the variance( ) of the PDFs of This relation has been verified experimentally in Ref.
[23] for a one-port cavity and will be assumed to hold true for the 2port results discussed in this paper for data-sets with . For data-sets with , the following procedure is employed. First, we numerically generate marginal PDFs of the real and imaginary parts of the normalized impedance eigenvalues using random-matrix Monte-Carlo simulations with square matrices of size to obtain a unique estimate of corresponding to that experimental data-set.
This paper is organized into the following sections. Section II explains the experimental setup and the data acquisition process. By using the measured frequencydependent radiation impedance matrix, we carry out the normalization of the cavity impedance to uncover the universally fluctuating z t ; which in turn yields s t and the normalized admittance matrix . Section III is divided into four sub-sections and presents our experimental results on the universal fluctuations in the eigenvalues of , and and also compares these experimentally obtained PDFs with those from RMT. A technical issue encountered in these 2-port experiments is the presence of non-zero, off-diagonal terms in the measured y t rad Z t . These terms account for the direct-path processes ("cross-talk") between the two ports and come about because of the finite physical separation between the two-ports in the experiment during the radiation measurement. The role of these non-zero, off-diagonal rad Z t terms in determining the universal PDFs of z t is explained in Section IV. Section V concludes this paper with a summary of our experimental findings.

II: Experimental Setup, Data Acquisition and
Construction of normalized z t , s t and y t : Microwave-cavity resonators with irregular shapes (where the classical ray trajectories are chaotic) have proved to be a favored test-bed to validate statistical predictions on chaotic scattering [7]. In this paper, we present findings on an air-filled, quarter bow-tie shaped billiard cavity [ Fig.1  To set up the investigation, we introduce two driving ports [ Fig.1(b)] which are placed roughly 20 cm apart, and are labeled Port-1 and Port-2 . The ports are located sufficiently far away from the side-walls of the cavity so that the near-field structure of each port is not altered by the walls. Both ports are sections of coaxial transmission lines, where the exposed center-conductor extends from the top plate of the cavity and makes contact with the bottom plate, injecting current into the bottom plate [ Fig.1(c)]. The ports are non-identical; the diameter of the inner conductor is 2a=1.27 mm for Port-1 and 2a=0.635 mm for Port-2. As in our previous studies [23,24], the normalization of the measured data is a two-step procedure. The first step, what we refer to as the "Cavity Case" involves measuring a large ensemble of the full-2x2 scattering matrix using an Agilent E8364B Vector Network Analyzer. To realize this large ensemble, two metallic perturbers (shown as gray solids in Fig.1.(a) ), each of typical dimensions 6.5 cm x 4 cm x 0.78 cm are used. The perturbers are roughly the order of a wavelength in size at 5 GHz. The edges of the perturbers are intentionally serrated to further randomize the wave scattering within the cavity by preventing the formation of standing waves between the straight wall segments of the cavity and the edges of the perturbations. The perturbers are systematically translated and rotated through one hundred different locations within the volume of the cavity. Hence each orientation of the two perturbers results in a different internal field structure within the cavity. Thus we measure one hundred cavity configurations all having the same volume, coupling geometry for the driving ports, and almost exactly the same cavity cavity scattering matrices is collected. Special care is taken not to bring the perturbers too close to the ports so as not to alter the near-field structure of the ports.

S t
The dominant loss mechanism in the empty cavity is due to ohmic losses in the broad top and bottom plates of the cavity. The fluctuations in loss from mode-to-mode are small and come from differences in field configurations around the side walls. The degree of loss can be increased in a controlled manner by partially lining the inner side-walls with 2 cm-long strips of microwave absorber [ Fig.1(a)] having uniform spacing. Three lossy Cavity Cases -labeled "Loss Case 0" : with no absorbing strips, "Loss Case 1" : with 16 absorbing strips and "Loss Case 2" : with 32 absorbing strips, are measured. Along with frequency, the three "loss cases" lead to an experimental control over the value of from 0.9 to 28.
represents the case when the ports are perfectly matched (mismatched) to the cavity. The inset in Fig.2 shows the PDF of the measured (i.e. ) for a Loss Case 0 cavity from 3 to 18 GHz. The PDF is fairly widely spread over the range 0 to 1 with a mean value of ~0.7, and with a standard deviation of ~0.
3. An analysis of the coupling and loss for the scattering matrix in similar microwave cavities is presented in Ref. [ 37].
The degree to which the two perturbations produce a change in the internal field structure of the cavity can be qualitatively inferred by looking at the frequency correlations in the measured S t data. In Fig. 2   . We have previously identified the fact that short ray orbits inside the cavity will produce systematic deviations of the finite ensemble average from a true ensemble average [20]. We therefore invoke ergodicity and also employ frequency averaging of the data. Since the frequency averaging ranges that we use below are very much larger than Weyl f Δ (typically by a factor of ~20), this confirms that our frequency (in addition to perturber configuration averaging) is an effective means of approximating a true ensemble average. The second step of our normalization procedure is what we refer to as the "Radiation Case" [ Fig.1(b)]. In this step, the side-walls of the cavity are completely lined with ~ 2 mm thick microwave absorber (ARC-Tech DD 10017) which gives about 20-25 dB reflection loss between 3 and 18 GHz for normal incidence. The perturbers are removed so as not to produce any reflections back to the ports. Port-1 and Port-2 are left untouched-so that they retain the same coupling geometry as in the "Cavity Case". The radiation measurement now involves measuring the resultant 2x2-scattering matrix, which t the same 16000 frequency steps as in the "Cavity Case". The microwave absorber serves to severely suppress any reflections from the side-walls. This effectively simulates the situation of the side-walls of the cavity being moved out to infinity (radiation-boundary condition). The off-diagonal terms in rad S t correspond to direct-path processes between the two ports. The contribution of these terms has been taken into account in the analysis and results that follow (Section III (A,B,C,D)). The hazards associated with ignoring these terms in the normalization process deserves special mention and are discussed in Section IV.
Having measured the ensemble of cavity S t and the corresponding radiation rad S t , we convert these quantities into the corresponding cavity impedance Z t  = z y t t ). These normalized quantities represent the corresponding electromagnetic response of the chaotic-cavity in the limit of perfect coupling between the driving ports and the cavity over the entire frequency range of the experiment from 3 to 18 GHz.

III: Experimental Results:
In this section, we give our experimental results on  < λ . Reference [24] has shown that for a 1-port system, the magnitudes and phases of the normalized 1-port scattering coefficient are statistically independent. The independence was shown to be extremely robust and is unaffected by the presence of loss. For a two-port setup, as in the experiments presented in this paper, this would imply statistical independence of the magnitude and phases of the eigenvalues of s s t .
To test this hypothesis, the two complex eigenvalues of the s t ensemble are grouped into one list, which we shall refer to as " ". We observe that grouping the two eigenvalues together as opposed to randomly choosing one of the two eigenvalues does not change the statistical properties of the results that follow. Figure 3 (a) shows a plot in the complex plane of the eigenvalue density for a representative set of measured s t λˆs t ranging between 7.6 to 8.1 GHz where the loss-parameter is roughly constant. The gray-scale level at any point in Fig.3(a) Fig. 3(b). It can be observed that the PDF approximations are essentially identical and independent of the angular-slice.  Fig. 3(b) which f the typical statistical binning error of the experimental histograms show that the data agrees well with the numerical RMT PDF.
In Fig.3(c), the histogram approximations of the phase of the points lying within two-annular rings defined by  In order to make comparisons of the data with numerical computations from RMT, we transform the eigenphases 1 φ and 2 φ to 1 κ and 2 κ , as follows, 2 2 where is the Heaviside step function ( ). This transformation of variables has the effect of making 0 > x 1 κ and 2 κ statistically independent, with all the correlation information between 1 φ and 2 φ being contained in 1 κ ; and 2 κ being uniformly distributed (as shown in Fig.3(c)). In the lossless case, it can be easily deduced from , that t nger confined to move along the unit-circle; but rathe are distributed inside the unit circle in a manner dependent pon the loss in the system (as was shown in Fig.3(a)). The sub-unitary modulus of the eigenvalues thus presents an extra degree of freedom for eigenvalue avoidance, hence we expect a reduced anticorrelation of the eigenphases as the losses increase. To our knowledge, there exists no analytic formula for the evolution of the joint PDF of the eigenphases of s t as a function of loss. In the following paragraphs, we thus compare our experimental results for the joint PDF of the eigenphases of Hermitian, its eigenvalues are purely real. The matrix s s t t † is of significant interest in the quantum-transport community as it determines the conductance fluctuations of ballistic quantum-dots in the presence of dephasing/loss. Owing to the analogy between the time-independent Schrödinger equation and the two-dimensional Helmholtz equation, the microwave billiard experiment presents itself as an ideal platform to test statistical theories for these quantum fluctuations without the complicating effects of thermal smearing [40] and Coulomb interactions, as discussed in Ref. [41].
Models have been introduced to quantify the loss of quantum phase coherence (dephasing) of transport electrons in quantum dots [42,43,44,45]. These models generally utilize a fictitious lead attached to the dot that has a number of channels each with transparency . Electrons that enter one of the channels of this lead are re-injected into the dot with a phase that is uncorrelated with their initial phase, and there is no net current through the fictitious lead. An alternative model of electron transport employs a uniform imaginary term in the electron potential [46,47], leading to loss of probability density with time, similar to the loss of microwave energy in a cavity due to uniformly distributed losses in the walls and lids. As far as the conductance is concerned, it was shown that these two models are equivalent in the limit when the number of channels in the dephasing lead [31,43,48]. In this case, the dephasing parameter γ is equivalent to a loss parameter describing the strength of uniformly distributed losses in the system. Other models have been proposed that consider parasitic channels [43,49] or an "absorbing patch" or "absorbing mirror" [50] to describe losses in a microwave cavity. Here we examine the predictions of Brouwer and Beenakker using the dephasing lead model in the limit mentioned above. In this case the dephasing parameter γ is treated as a loss parameter describing fairly uniformly distributed losses in our microwave cavity, and is found to be proportional to the loss approaching 1 (Fig. 5(b)). We also observe a strong anti-correlation in for . This anti-correlation is manifested in all the data measured at varying degrees of loss from =0.9 to 28.   Fig. 5(b), respectively. Using these values of 〉 〈T γ , we plot the analytic contour curves defined by Eq. (7) for the two loss cases, shown as the solid black lines in Fig. 5. The theoretical curves reflect the same number of contour levels shown in the data. We observe relatively good agreement between the theoretical prediction of Ref.
[31] and the experimental data. This agreement between the experimental data and the theoretical prediction is also observed to extend over other loss-cases and frequency ranges. References [20,21] show that attaching an arbitrary lossless two-port network at the interface between the plane of measurement, and the cavity does not alter the statistics of z t . If we now assume that this lossless two-port is a transmission line with an electrical-length equal to one-quarter wavelength at the driving frequency, then the lossless two-port acts as an "impedance inverter" [25] thereby presenting a cavity admittance at the plane of measurement. This similarity in the statistical description of z t and y t is predicted to be extremely robust and independent of loss in the cavity, coupling, driving frequency, etc. For our experimental test of this prediction, we consider the three loss cases, Loss Case 0, 1 and 2, in the frequency range 10.5-12GHz. By an eigenvalue decomposition, each z t and matrix yields two complex eigenvalues, which we group together to form and respectively. We observe that grouping the two eigenvalues together as opposed to randomly considering one of the two eigenvalues separately does not alter the statistical results that follow. Histograms of the real and imaginary parts of and y t z t λˆy t λẑ t λˆ y t λˆ are plotted in Fig. 6. The hollow stars, circles and triangles in Fig.6(a) (Fig.6(b)) correspond to the histogram approximations of the PDF of ( ) for Loss case 0, 1 and 2 respectively. The evolution of these PDFs for and with increasing loss, are in qualitative agreement with the description given in Ref. [21]. As losses increase, we observe that the PDFs of shifts from being peaked at ~ 0.6 (Loss Case 0) to developing a Gaussian-type distribution that peaks near ~1 (Loss Case 2). While in Fig. 6(b), as losses increase, the PDFs lose their long tails and become sharper. The solid stars, circles and triangles in Fig. 6(a) (Fig. 6(b)