Cross-Section Fluctuations in Chaotic Scattering

For the theoretical prediction of cross-section fluctuations in chaotic scattering, the cross-section autocorrelation function is needed. That function is not known analytically. Using experimental data and numerical simulations, we show that an analytical approximation to the cross-section autocorrelation function can be obtained with the help of expressions first derived by Davis and Boose. Given the values of the average S-matrix elements and the mean level density of the scattering system, one can then reliably predict cross-section fluctuations.


Purpose
Quantum chaotic scattering is an ubiquitous phenomenon. It occurs, for instance, in nuclear physics [1], in electron transport through disordered mesoscopic samples [2], and in microwave billiards [3]. In all cases, the cross section displays random fluctuations versus energy or frequency. These are due to the random features of the underlying resonances. With d the average resonance spacing and Γ the average width, data on cross-section fluctuations exist for the entire range of the parameter Γ/d, from the regime of isolated resonances (Γ ≪ d) to that of strongly overlapping resonances (Γ ≫ d). The analysis of the data focuses on the value of the average cross section and on quantities that characterize the cross-section fluctuations. These are the variance of the cross section and certain correlation functions. For the analysis, one needs theoretical expressions for these quantities. These should be generic and only use a minimum of adjustable parameters.
The generic theoretical treatment of chaotic scattering employs a combination of scattering theory and randommatrix theory [1] and uses as input the values of d and of the energy-averaged elements S of the scattering matrix S. Analytical results exist for the S-matrix autocorrelation and cross-correlation functions [1] (including the value of the average cross section) and for the third and fourth moments of the S-matrix [4,5]. Because of the complexity of the problem, analytical results for higher moments of the cross section or for cross-section correlation functions cannot be expected in the foreseeable future.
The present paper aims at filling that gap. We combine the available analytical information [1,4,5], results of computer simulations, and of experimental work on microwave billiards [3,6] to study the cross-section autocorrelation function for all values of Γ/d. In particular, we address the following questions. (i) For which values of Γ/d and with which accuracy can the cross-section autocorrelation function be predicted in terms of the S-matrix autocorrelation function? (ii) Which analytical alternatives exist should that approach fail?

Framework
We consider chaotic scattering in a time-reversal invariant system described by a unitary and symmetric scattering matrix S ab (E). Here a, b = 1, . . . , Λ denote the channels and E the energy (or, in the case of microwave billiards, the frequency). The number Λ of channels may range from unity to a large number, Λ ≫ 1. Chaotic scattering is modeled by writing the S-matrix in the form [7] The real and symmetric Hamiltonian matrix H has dimension N ≫ 1 and describes the dynamics of N resonances labeled by Greek letters. These are coupled to the channels by the real matrix elements W aµ . Chaos is taken into account by choosing H as a member of the Gaussian orthogonal ensemble (GOE) of random matrices [1]. Thus, the elements of H are Gaussian random variables with zero mean values and second moments given by H µν H ρσ = (λ 2 /N )[δ µρ δ νσ + δ µσ δ νρ ]. Here and in the sequel, the overbar denotes the average over the ensemble. The parameter λ determines (or is determined by) the average level spacing d of the N resonances. It is convenient to decompose S(E) into an average and a fluctuating part, where the average S-matrix S is assumed to be diagonal. The values of the diagonal elements S aa serve as input parameters for the statistical model and are assumed to be known. That is the typical case: In nuclei, S aa is given in terms of the optical model of elastic scattering, in microwave billiards S aa is determined by the running average over a measured spectrum [3,6]. In rare cases, the average S-matrix may not be diagonal. By an orthogonal transformation in channel space, S can be reduced to diagonal form, see Refs. [5,8]. For simplicity we do not address that case. By the same transformation, the phases of the S-matrix usually appearing as factors on the right-hand side of Eq. (3), can be removed. Both for the S-matrix model Eq. (1) considered in the present work and the experimental data the average S-matrix is real and diagonal. Starting from Eq. (1), the S-matrix autocorrelation function (or "two-point function") has been calculated analytically [1] for N ≫ 1 and fixed Λ.
The resulting expression depends only on the difference ε of the two energy arguments, on the average level spacing d of the system, and on the transmission coefficients T a of all channels a defined by The transmission coefficients obey 0 ≤ T a ≤ 1. These coefficients measure the unitarity deficit of the average Smatrix and give the probability with which the resonances take part in the reaction. This is seen by using the decomposition Eq. (3) and the definition Eq. (5) to write the unitarity condition for S in the form For T a = 0 or |S aa | = 1, we have S fl ab (E) = 0 for all b, and the resonances are not reached from channel a. Conversely, b |S fl ab (E)| 2 is maximal for T a = 1 or S aa = 0 (complete absorption of the incident flux in channel a by resonance formation). The transmission coefficients T a determine the average width Γ of the resonances. An approximation for Γ is the "Weisskopf estimate" The case of strongly overlapping resonances Γ ≫ d ("Ericson regime" [9,10,11]) occurs for a T a ≫ 1: The number Λ of channels must be large and most of the individual transmission coefficients T a must not be small. Conversely, the case of nearly isolated resonances Γ ≪ d is realized when Λ is of order unity or when Λ is large but all T a are small. The theory developed in Ref. [1] and used in Refs. [4,5] applies to all values of Γ/d. Equation (7) is exact in the Ericson regime and fairly reliable elsewhere.
Under omission of kinematical factors the cross section in nuclear physics, the conductance in electron transport and the transmitted power in microwave billiards are all given by |S ab (E)| 2 or by a sum of such terms. For brevity we refer to |S ab (E)| 2 as to the cross section. The average cross section |S ab (E)| 2 = |S aa (E)| 2 δ ab + |S fl ab (E)| 2 is given in terms of S aa and of C (2) ab (0) and is, thus, known. Fluctuations of the cross section are measured in terms of the cross-section autocorrelation function That function is the object of central interest in the present paper. With the help of the decomposition Eq. (3) we write We have used that in the experiments and in the considered S-matrix model (Eq. (1)) S aa is real, that by definition S fl ab (E) = 0 and that S fl ab (E 1 )S fl ab (E 2 ) = 0 for all a, b and all E 1 , E 2 . The last relation holds because all poles of S lie in the lower half of the complex energy plane. To determine C ab (ε) we need to know the four-point function and, in the elastic case a = b, also the three-point function These functions are known analytically only for ε = 0 and in the Ericson regime (Γ ≫ d), see below.
To determine magnitude and ε-dependence of C ab (ε), we combine analytical results with numerical and experimental evidence as follows. (i) Analytical results: In Refs. [4,5] analytic expressions are given for two functions F (4) ab (ε) and F (3) ab (ε) that look similar to but actually differ from C (4) ab (ε) and C (3) ab (ε), respectively. These are defined by We note that in C (4) ab (ε) and in C     . We are going to show 2 that for ε = 0 it is possible to approximate C (4) ab (ε) in terms of F (4) ab (ε), and under certain conditions C ab (ε). For the convenience of the reader we, therefore, give in the Appendix analytic expressions for F ab (ε). We briefly show how the threefold integrals can be evaluated numerically to avoid the apparent singularities of the integrand. (ii) Numerical results: For the numerical simulations we use Eqs. (1) and (2) and fixed values for λ, for the transmission coefficients T a , and for N as initial values. Calculations were typically done for several 100 realizations to minimize statistical errors. The results agree very well with the available analytical results but go beyond them. (iii) Data: The data stem from measurements of transmission and reflection amplitudes of microwaves in a flat cylindrical resonator made of copper and coupled to two antennas, see Refs. [3,6]. Microwave power was coupled into the resonator with the help of a vector network analyzer. The range of the excitation frequency was chosen such that only one vertical electric field mode is excited. Then the microwave cavity simulates a twodimensional quantum billiard [12,13]. The resonator had the shape of a tilted stadium billiard whose classical dynamics is chaotic. Transmission and reflection amplitudes correspond to complex S-matrix elements that are theoretically modeled by Eqs. (1) and (2) [3,6].

Inelastic case (a = b)
The inelastic case is simpler than the elastic one because it involves only the function C (4) ab (ε), see Eq. (9). We begin with the Ericson regime Γ ≫ d, see Refs. [9,10,11]. In Ref. [11] it was suggested and in Ref. [15] it was shown that for Γ ≫ d the fluctuating S-matrix elements S fl ab possess a bivariate Gaussian distribution centered at zero. That fact implies that all higher moments and correlation functions can be computed from C In the Ericson regime, the two-point function has the value [9, 15] Thus, for a = b and Γ ≫ d, the cross-section autocorrelation function is known analytically. It has the shape of a Lorentzian with width Γ as given by Eq. (7). How far can we use Eq. (13) outside the Ericson regime, i.e., for smaller values of Γ/d? Figure 1 shows the ratio C   The origin of the failure of Eq. (13) for small values of Γ/d is easily understood. Qualitatively speaking, it is intuitively clear that cross-section fluctuations (measured in units of the average cross section) are much larger for isolated than for overlapping resonances. Quantitatively, the assumption that underlies Eq. (13) is that the distribution of the fluctuating S-matrix elements S fl ab (E) is Gaussian. That assumption holds only if the width γ of the distribution is sufficiently small. Indeed, the unitarity condition Eq. (6) implies that for all a and b we must have |S fl ab | 2 ≤ T a ≤ 1, and γ must be so small that the contribution of the tails of the distribution that extend beyond the values ±T  We have checked this explanation by investigating the distribution of S fl 12 . A bivariate Gaussian distribution for S fl 12 implies that the distribution of z = |S fl 12 |/|S fl 12 | has the 3 form P (z) = (π/2)z exp[−(π/4)z 2 ], and that the phase of S fl 12 is distributed uniformly in the interval {0, 2π}. In Fig. 2 and Fig. 3 and for values of Γ/d indicated above each panel, we compare in the upper row the function P (z) with experimental and numerical data, respectively. In the panels in the lower row, we show the corresponding distributions of the phase. The numerical data in the three panels of Fig. 2 are obtained by simulating absorption in the resonator in terms of a large number of fictitious channels with small transmission coefficients in each channel. Then their sum τ is the only parameter. It was determined as described in [3] from a fit of the experimental autocorrelation function to the analytic result given in [1]. The data show that agreement with the bivariate Gaussian distribution is attained for Γ/d 1. The data in Fig. 3 (generated numerically for 32 equal channels) show agreement with the bivariate Gaussian distribution only for Γ/d 3. This suggests that the limit of a bivariate Gaussian distribution for S fl ab with a = b is attained for larger values of Γ/d when all transmission coefficients are equal than when the transmission coefficients differ. These results account for the deviations from Eq. (13) displayed in Fig. 1. Concerning the ε-dependence of C    12 (0)| 2 ≈ 1, that dependence is sufficiently similar so that C (4) 12 (ε) can be reliably approximated by |C (2) 12 (ε)| 2 . (By that we mean that the full two-point function defined in Eq. (4) and not the approximate form Eq. (14) has to be used). That leaves us with the question how to approximate C An obvious possibility is offered by the function F        12 (ε)| multiplied with the same scaling factor as the experimental one. In the lowest panel we compare the numerical result for C (4) 12 (ε) with the analytic rescaled result for |F (4) 12 (ε)|. We note that the agreement is excellent for all three cases. Similarly good agreement was found also for Λ = 52 channels and several values of Γ/d. We conclude that we have reached a simple and reliable prescription for approximating C   (The formulas needed [4,5] are given in the Appendix). Rescale |F (4) 12 (ε)| so that its value at ε = 0 agrees with C

Elastic Case (a = b)
The elastic case is analogous to the inelastic one only if S aa = 0. Then all conclusions drawn in Section 3 apply. In general, that is not the case. In particular, for Γ ≪ d (isolated resonances) the last term in Eq. (1) tends to zero and S aa approaches unity. The unitarity constraint       on S implies |S aa + S fl aa | ≤ 1 and the distribution of S fl aa must then become skewed. Therefore, we have to expect that Eq. (13) applies less generally in the elastic than in the inelastic case, and that C aa (ε) in Eq. (9) plays an important role. Actually, it was pointed out in Refs. [4,5] that even in the Ericson limit, a profound difference between the elastic and the inelastic cases exists: For Γ ≫ d, |S fl aa | 3 and |S fl aa | 4 have similar values unless T a ≈ 1 or S aa ≈ 0. That shows that in order to predict C aa (ε) we need to know both C    11 (0)| 2 in the upper panel for experimental data, in the lower one for a numerical simulation with Λ = 32 equal transmission coefficients, as indicated in the figure caption. We reach the same conclusions as for a = b: When the transmission coefficients differ, the ratio is close to unity for Γ d, when they are equal for Γ 3d. As in the inelastic case, the reason for  In Fig. 8 we show that, as in the elastic case, the rescaled function |F 11 (ε) for all values of Γ/d. Such agreement was likewise found for the case of Λ = 52 channels with identical transmission coefficients. Figure 9 shows that the three-point function Re C 11 (ε) , are negligibly small as compared to C ab (ε), whereas this is not true for the experimental data with Γ/d = 1.01.

Summary
We return to the questions raised in the Introduction. We have shown that Eq. (13) holds essentially only in the Ericson regime and, for the elastic case, even there only approximately. This is because of the constraint imposed on the distribution of S fl by unitarity. Hence, in the inelastic case the cross-section autocorrelation function can be reliably predicted from Eq. (13) for Γ/d 1 for inequivalent channels and for Γ/d 3 for identical channels. Even more stringent constraints exist in the elastic case. We have demonstrated, however, that for the inelastic case for all values of Γ/d and for the elastic case under certain conditions an excellent approximation to the cross-section autocorrelation function may be obtained by replacing in Eq. (9) the three-point function C ab (ε). With these replacements, Eq. (9) takes the form The input parameters for the evaluation of Eq. (15) are the average S-matrix elements and the associated transmission coefficients for all channels and, for the dependence on ε, the average level spacing d of the scattering system. In terms of these parameters, the two-point function C (2) ab (ε) is given in Ref. [1], and C    ab (ε = 0), S fl * ab (0)|S fl ab (0)| 2 and |S fl ab (0)| 4 are given in terms of a threefold integral, c.f. Refs. [1,4], where the integration measure is given as For n = 2, i.e. for |S fl ab (0)| 2 the factor F (λ, λ 1 , λ 2 ) equals Here, r a = 1 − T a and µ a , ν a are the matrices µ a = T a λ 0 (1 + T a λ 0 ) −1 , ν a = T 2 a λ 0 (1 + λ 0 ) (1 + T a λ 0 ) −2 , where λ 0 is the 4 × 4 diagonal matrix with entries λ 1 , λ 2 , −λ and −λ.  ab (ε) and F (4) ab (ε) defined in Eqs. (12) are obtained by multiplying F (2) ab (λ, λ 1 , λ 2 ) in Eq. (18) and F The integral Eq. (16) contains several singularities. These can be handled by proceeding as in Section 5 of Ref. [14]. We define the variable Then the exponential in Eq. (22) turns into exp(−ipπε/d) .
We distinguish the two cases p ≤ 2 and p > 2. For p ≤ 2 we define two further integration variables For the inverse transformation that yields The Jacobian for this transformation is equal to 4p 2 s 3 t. The threefold integral in Eq. (16) becomes For p > 2 we define additional integration variables For the inverse transformation that yields The Jacobian equals 4s 3 t and the threefold integral in Eq. (16) becomes where the θ-function is defined by θ(x) = 0 for x ≤ 0 , (37) θ(x) = 1 for x > 0 . (38)