1. Chaotic Scattering in Open Microwave Billiards with and without T Violation
Porquerolles
2013
Microwave resonator as a model for the compound nucleus
Induced violation of T invariance in microwave billiards
RMT description for chaotic scattering systems with T violation
Experimental test of RMT results on fluctuation properties of
S-matrix elements
I will speak about old and new results obtained based on random matrix theory and the supersymmetry method for S-matrix correlation functions and the distributions of S-matrix elements for systems with and without time reversal invariance violation. Although widely used to discover signatures of T-violation in compound nucleus reactions the random matrix theory results were tested if at all only in very few experiments. We tested these both with RMT simulations and with experimental data. The experiments were performed with chaotic microwave billiards. I will first shortly outline how the experimental data may be used to test analytic results obtained for the S-matrix correlations in compound nucleus reactions.
Supported by DFG within SFB 634
B. Dietz, M. Miski-Oglu, A. Richter
F. Schäfer, H. L. Harney, J.J.M Verbaarschot, H. A. Weidenmüller
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 1
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2. Microwave Resonator as a Model for the Compound Nucleus
rf power in
out  
C+c
D+d
Compound
Nucleus
A+a
B+b h
Microwave power is emitted into the resonator by antenna 
and the output signal is received by antenna   Open scattering system
The antennas act as single scattering channels
Absorption at the walls is modeled by additive channels
For the measurement of a spectrum the resonator is coupled to the exterior through antennas. An input signal is emitted into the resonator by antenna 1, thereby exciting an electric field mode inside the resonator and the output signal is received by antenna 2. Hence, the resonator is an open scattering system, where the antennas act as single scattering channels. This scattering process is equivalent to that of a compound nucleus reaction. There a target nucleus capital A is bombarded by a particle small a, leading to a compound nucleus, which eventually decays into a particle small b and a residual nucleus capital B. Attaching more antennas to the resonator or dissipation of microwave power in the walls of the resonator corresponds to more open channels in the compound nucleus.
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 2
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3. Typical Transmission Spectrum
Transmission measurements: relative power from antenna a → b
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 3

4. Scattering Matrix Description
Scattering matrix for both scattering processes as function of excitation
frequency f :
Ŝ( f ) = I - 2pi ŴT ( f I - Ĥ + ip ŴŴT)-1 Ŵ
Compound-nucleus
reactions
Microwave billiard
 Ĥ 
nuclear Hamiltonian
coupling of quasi-bound
states to channel states
resonator Hamiltonian
coupling of resonator
states to antenna states
and to absorptive channels
 Ŵ 
Both scattering processes are described by a scattering matrix, which was derived by Mahaux and Weidenmüller in the context of compound-nucleus reaction theory. Here, f is the excitation frequency and 1I is the unit matrix. For compound nucleus reactions H is the nuclear Hamiltonian, for a microwave resonator it is the Hamiltonian of the resonator. Likewise for compound nucleus reactions the matrix W couples the quasi-bound states of the compound nucleus to the open channels, and in the microwave billiard the resonator modes to the antenna channels. While in nuclear physics one can only measure cross sections we have access to the full scattering matrix in microwave experiments. For a statistical description of properties of the S-matrix the matrix H is replaced by a random matrix from a suitable ensemble. In case of time-reversal invariance this ensemble is the GOE, in case of complete T violation it is the GUE..
Experiment: complex S-matrix elements
GOE
T-inv
RMT description: replace Ĥ by a matrix for systems
GUE
T-noninv
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 4
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5. Resonance Parameters Use eigenrepresentation of
and obtain for a scattering system with isolated resonances a → resonator → b
Here: of eigenvalues of
Partial widths fluctuate and total widths also
real part
imaginary part
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 5

6. Excitation Spectra atomic nucleus microwave cavity
overlapping resonances
for Γ/d > 1
Ericson fluctuations
isolated resonances
for Γ/d < 1
ρ ~ exp(E1/2)
ρ ~ f
Universal description of spectra and fluctuations: Verbaarschot, Weidenmüller and Zirnbauer (1984)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 6

7. Fully Chaotic Microwave Billiard
Tilted stadium (Primack+Smilansky, 1994)
Only vertical TM0 mode is excited in resonator → simulates a quantum
billiard with dissipation
Additional scatterer → improves statistical significance of the data sample
Measure complex S-matrix for two antennas 1 and 2: S11, S22, S12, S21 2 1
We use a microwave cavity made of Copper coupled to two antennas and measure the response to an external field as a function of radiofrequency f. The microwave cavity has the shape of a tilted stadium billiard. The dynamics of the corresponding classical billiard is
chaotic. The tilted shape was used in order to avoid bouncing--ball orbits between parallel walls. The experiment is performed at room temperature, with Ohmic losses at the walls of the cavity.
The range of the excitation frequency is chosen such, that only a single vertical mode in the microwave cavity is excited. Then, the cavity simulates a two-dimensional chaotic quantum system and is a microwave billiard. An additional scatterer was inserted in the resonances and spectra were measured for six different positions of it to improve the statistical significance of the data. A vector network analyzer couples microwave power in and out of the resonator via either one or both antennas and yields the complex elements Sab(f) of the scattering matrix, where a, b = 1, 2. The range 1GHz f25GHz was covered in steps of D=100kHz in reflection and transmission measurements (yielding S12(f), S21(f)).
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 7
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8. Spectra and Correlations of S-Matrix Elements
Г/d <<1: isolated resonances → eigenvalues, partial and total widths
Г/d ≲ 1: weakly overlapping resonances → investigate S-matrix fluctuation properties with the autocorrelation function and its Fourier transform
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 8

9. Microwave Billiard for the Study of Induced T Violation
A cylindrical ferrite is placed in the resonator
An external magnetic field is applied perpendicular to the billiard plane
The strength of the magnetic field is varied by changing the distance
between the magnets
In the experiment a cylindrical ferrite is placed in the resonator and an external magnetic field is applied perpendicular to the billiard plane. The magnetic field is generated by two permanent magnets which are placed above the top and below the bottom plate of the resonator. The magnetic field is varied by changing the distance between the magnets.
Field strengths up to 390 mT could be attained. Here we focus on the results for B=190 mT as there the effects are most clearly visible.
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 9
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10. Induced Violation of T Invariance with Ferrite
Spins of magnetized ferrite precess collectively with their Larmor frequency about the external magnetic field
Coupling of rf magnetic field to the ferromagnetic resonance depends on the direction a b
Sab F b a
Sba
Time-reversal invariance is violated by a ferrite cylinder of 4 mm diameter and 5 mm height. When it is magnetized by an external magnetic field, the spins of the ferrite precess collectively with their Larmor frequency about the external field. The rf magnetic field component of the resonator modes are in general elliptically polarized and couple to the spins of the ferrite. At the ferromagnetic resonance microwaves whose magnetic field rotates in the same sense as the spins are absorbed, whereas waves with magnetic field rotating in the opposite direction pass it unattenuated. Thus the coupling depends on the rotational direction of the rf field. An interchange of input and output channels changes the rotational direction and thus the coupling of the resonator modes to the ferrite.
T-invariant system → reciprocity
→ detailed balance
Sab = Sba
|Sab|2 = |Sba|2
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 10
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11. Detailed Balance in Microwave Billiard
S12
S21
This figure compares measured intensities of the transmission from antenna 1 to 2 and vice versa for an experiment with and one without magnetization of the ferrite. It clearly demonstrates the violation of the principle of detailed balance for nonzero magnetic field.
Clear violation of principle of detailed balance for nonzero magnetic field B
→ How can we determine the strength of T violation?
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 11
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12. Search for TRSB in Nuclei: Ericson Regime
T. Ericson, Nuclear enhancement of T violation effects, Phys. Lett. 23, 97 (1966)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 12

13. Analysis of T violation with Cross-Correlation Function
Special interest in cross-correlation coefficient Ccross(ε = 0)
A measure which is most sensitive to the strength of T-invariance violation is the cross correlation function between the S-matrix element S12 and the complex conjugate of S21. To observe the effect of T breaking it suffices to consider the cross correlation coefficients obtained for e=0. If T-invariance holds, we have Ccross(e=0)=1 while for complete violation of T invariance S12 and S21* are uncorrelated and thus Ccross(e=0)=0.
if T-invariance holds
if T-invariance is violated
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 13
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14. S-Matrix Correlation Functions and RMT
Pure GOE  VWZ 1984
Pure GUE  FSS (Fyodorov, Savin, Sommers) 2005
Partial T violation  Pluhař, Weidenmüller, Zuk, Lewenkopf + Wegner
derived analytical expression for C(2) (ε=0) in 1995
RMT 
GOE: 0 GUE: 1
For the determination of the strength of the T-invariance violation we need a theoretical description in the transition regime of partial T-violation. As our system is classically chaotic the most direct approach to the description of universal properties of S-matrix fluctuations comes from RMT. In 1984 Verbaarschot, Weidenmüller and Zirnbauer derived a universal expression for the autocorrelation function of S-matrix elements based on Efetov‘s supersymmetry method for T-invariant chaotic scattering systems. In 2005 Fyodorov, Savion and Sommers derived an analytical expression for the autocorrelation function in the limit of complete T-violation. In 1995 Pluhar…. derived again based on Efetov‘s supersymmetry method an analytical expression for the autocorrelation coefficient Cab(e=0), which continuously connects the regimes of no and complete T-violation. The Hamiltonian matrix consists of a real symmetric part plus i times a coefficient times a real and antisymmetric part. For ξ=0 it equals a random matrix from the GOE, for …=1 from the GUE. Full T-violation is already achieved as soon as the T-breaking matrix element, which equals the coefficient times the square root of the variance of the matrix elements of H, is of the order of one mean level spacing. This is the case for ξ=1.
Complete T-invariance violation sets in already for x ≃1
Based on Efetov´s supersymmetry method we derived analytical
expressions for C (2) (ε) and Ccross(ε=0)  valid for all values of 
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 14
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15. S-Matrix Ansatz for Chaotic Systems with Partial T Violation
Insert Hamiltonian Ĥ (ξ) for partial T-violation into scattering matrix
T-violation is due to ferrite only  Ŵ is real
Ohmic losses are mimicked by additional fictitious channels
Ŵ is approximately constant in 1GHz frequency intervals
The starting point is the scattering matrix for the scattering from one antenna to another. For the analysis in the regime of partial T violation we insert the corresponding Hamiltonian into this expression. We assume that the T-invariance violation is due to the ferrite only. Then the matrix elements W for the coupling of the resonator to the antenna modes is real. We have two antenna. Additional fictitious channels c are introduced to mimic Ohmic absorption in the walls of the cavity and the ferrite. Thus in the analysis and in the random matrix simulations these matrix elements are chosen real and of dimension 2 times M+2, where M is the number of fictitious channels. Based on Efetov‘s supersymmetry method the ensemble average for the first autocorrelation coefficient was derived for arbitrary values of the T-breaking parameter ξ. Based on this work we derived analytical expressions for the auto- and the crosscorrelation function.
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 15
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16. RMT Result for Partial T Violation (Phys. Rev. Lett
RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner, 1995
mean resonance spacing
→ from Weyl‘s formula
(2) d
parameter for strength of T violation
 from cross-correlation coefficient
The result is given in terms of a threefold integral. This part coincides with the VWZ result for the case of T-invariance. An important result is, that as in the GOE case it only depends on a few parameters. It depends on the mean level spacing d, which can be computed directly from Weyl’s formula, the T-breaking parameter ξ, which we determined from the cross-correlation function, and the transmission coefficients Tc. Ta and Tb can be computed from the measured S-matrix elements using this relation. A second fit parameter, however, is τabs, which is obtained from a fit to the autocorrelation function.
transmission coefficients
 from autocorrelation function
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 16
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17. Exact RMT Result for Partial T Breaking
RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner (1995)
In the relevant frequency ranges of partial T-breaking the cross-correlation coefficient depends only weakly on the values of the transmission coefficients. We thus computed it for a typical set of values and then for each measured value of the cross-correlation coefficients read off the size of the T-breaking parameter ξ. In the intervals from 1 to 25 GHz the strength ξ varies from zero to 0.3, while the ratio of the average resonance width Γ to the average resonance spacing varies from 0.01 to 1.2.
for GOE
RMT →
for GUE
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 17
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18. T-Violation Parameter ξ
cross-correlation
coefficient
Here are shown again the experimental values of the cross-correlation coefficient and the parameter ξ for the T-violation deduced from these. The error bars indicate the r.m.s. variation of Ccross(0) over the 6 realizations.
Largest value of T-violation parameter achieved is ξ ≃ 0.3
Published in Phys. Rev. Lett. 103, (2009)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 18
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19. Autocorrelation Function and its Fourier Transform
|C12(ε)|2 x (10-4) 2 4 6
t (ns)
log(FT(C12(ε)))
Adjacent points in C ab() are correlated
FT of Cab() 
For a fixed t are Gaussian distributed about their
time-dependent mean
Next the parameter tabs was determined more precisely from a fit of the autocorrelation function resulting from the random matrix model to the experimental one. In this figure I show an experimental autocorrelation function evaluated in a 1 GHz frequency window, where G/d equals We observe that adjacent points of the correlation function in the frequency range are correlated. This is not the case for the Fourier transformed autocorrelation function. Therefore, we have converted the scattering functions Sab(f) (measured at M equidistant frequencies with step width D) into complex Fourier coefficients Šab(t) with t0. In this figure I compare the corresponding analytic result, which by the way is a twofold instead of a threefold integral, with Monte Carlo simulations. The agreement is very good. Thus the random matrix model seems to work well and we used it for the determination of the two parameters x and t where again the data taken between 1 and 25GHz were analyzed in 1GHz intervals with the help of a goodness of fit test (maximum likelihood fit). For a fixed time t the Fourier coefficients are Gaussian distributed about a time dependent mean value. This is demonstrated in the following view graph.
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 19
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20. Distribution of Fourier Coefficients
Here is shown the distribution of the measured Fourier coefficients divided by their local mean. Gaussian distributed real and imaginary parts of the Fourier coefficients divided by their local mean implies a bivariate Gaussian distribution for the modulus and a uniform distribution for the phases. That the distributions of the modulus of the measured Fourier coefficients indeed are bivariate Gaussian is demonstrated here, and the phases are uniformly distributed.
For the fitting procedure we developed a GOF test which is based on the assumption, that the Fourier coefficients show this behavior.
Distribution of modulus divided by its local mean is bivariate Gaussian
Distribution of phases is uniform
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 20
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21. Determination of τabs from FT of Autocorrelation Function
3-4 GHz
9-10 GHz
log(FT(C12(ε)))
t (ns)
Fit determines expectation value of , tabs and improved values
for T1, T2, 
The fit of the analytical expression for the autocorrelation function to the experimental one yields the expectation value and the parameter entering of the RMT theory, namely T1, T2, tabs and x.
The parameters of the fits are the mean level spacing d, the transmission coefficients Tc and x. The mean resonance spacing is computed from Weyl’s formula. A starting value for x is obtained from the experimental cross-correlation coefficient for the transmission coefficients T1 and T2, i.e. the antenna channels, from the measured values of S11 and S22. The fit to the autocorrelation functions yielded values within 5 % of these starting values. In this figure I compare the analytic autocorrelation function for the case of partial T-violation and the two limiting cases of GOE and GUE obtained for a typical set of thus determined parameters. All three curves are very close to one another. Moreover the data points obtained from one measurement show a large spreading about the average behavior. This demonstrates the necessity of the measurement with several realizations of the system under consideration and a GOF test for a reliable fit to the data..
For each of 6 realizations the spectra of Sab(f) were divided into intervals of length 1 GHz.In each interval the FT of the autocorrelation functions was computed for times shorter than 200 ns. The time resolution is 1/Df=1ns. We measured the excitation functions S11, S22, S12, and S21; that yields in total 4800 Fourier coefficients from each interval. At t~200 ns the values of the FT(C(t)) have decayed over more than three orders of magnitude, and noise limits the analysis. The spread of the data is large.
GOF test relies on Gaussian distributed
Decay is non exponential → autocorrelation function is not Lorentzian
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 21
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22. Enhancement Factor for Systems with Partial T-Violation
Hauser-Feshbach for Γ>>d:
Elastic enhancement factor w
In the Ericson regime the autocorrelation function is well approximated by the Hauser-Feshbach term. This term is the so-called elastic enhancement factor. In chaotic scattering, elastic processes are known to be systematically enhanced over inelastic ones. The effect was first found in nuclear physics. The enhancement depends on the degree of T-violation and on G/d. The values of the scattering matrix Sab(f) are seen to be correlated, with a
correlation width G  several MHz. With S^{fl}_{a b} = S_{a b} - <S_{a b}> we have also determined the ``elastic enhancement factor'' W = <|S^{fl}_{1 1}|^2|S^{fl}_{2 2}|^2>^{1/2} /<|S^{fl}_{1 2}|^2> as a function of f, both from the autocorrelation functions and from the widths of the distributions of the imaginary parts of the scattering matrix. Both results agree very well and yield a smooth decrease of W with f from W3 to W1.5. The computation of the enhancement factors based on the theoretical expression for the S-matrix autocorrelation function yields similar values.
Isolated resonances: w = 3 for GOE, w = 2 for GUE
Strongly overlapping resonances: w = 2 for GOE, w = 1 for GUE
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 22
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23. Elastic Enhancement Factor
Here we compare the analytic result of the enhancement factor with the fitted parameters as input with the experimental result. At low frequencies only few resonances contribute and thus the errors of the experimental values for W are large. Moreover, W is determined from only a single value of Cab(0) of the measured autocorrelation function! In contradistinction, the analytic result is based on a fit of the complete autocorrelation function. The figure shows that at low frequencies we have W=3. With increasing G/D W decreases slowly. It has a dip at 5-6 GHz caused by the ferromagnetic resonance. Around GHz two further minima in W are observed. At those frequencies were ξ is largest, the enhancement factor drops below the value 2.
Above ~ 5GHz data (red) match RMT prediction (blue)
Values w < 2 seen → clear indication of T violation
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 23
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24. Fluctuations in a Fully Chaotic Cavity with T-Invariance
Tilted stadium (Primack + Smilansky, 1994)
GOE behavior checked
Measure full complex S-matrix for two antennas: S11, S22, S12
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 24

25. Spectra of S-Matrix Elements in the Ericson Regime
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26. Distributions of S-Matrix Elements in the Ericson Regime
Experiment confirms assumption of Gaussian distributed S-matrix elements with random phases
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 26

27. Experimental Distribution of S11 and Comparison with Fyodorov, Savin and Sommers Prediction (2005)
RMT:
Here I compare the experimental result for the distribution of the modulus and phase of the matrix element S11 with the analytic results. As an input I used the parameters obtained from the comparison of the random matrix model for the correlation functions with the experimental ones. The agreement with the analytical expressions is very good! We observe a clear deviation from the expected Gaussian result.
Distributions of modulus z is not a bivariate Gaussian
Distributions of phases  are not uniform
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 27
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28. Experimental Distribution of S12 and Comparison with RMT Simulations
Hitherto there exists no analytic expression for the distribution of the off diagonal elements for a higher dimensional S-matrix. Here I compare the experimental results with RMT simulations and again we find a good agreement. Thus even these are well described by random matrix theory. The values for the transmission coefficients are the same as in the figures for the S11 distribution in the regime of T invariance. Note that for G/d=1.02 we already have a good agreement with the Gaussian result which is a Wigner distribution with mean value not necessarily equal to one.
No analytical expression for distribution of off-diagonal S-matrix elements
For G/d=1.02 the distribution of S12 is close to Gaussian → Ericson regime?
Recent analytical results agree with data → Thomas Guhr
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 28
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29. Autocorrelation Function and Fourier Coefficients in the Ericson Regime
Frequency domain
Time domain
|C(ε)|2 is of Lorentzian shape → exponential decay
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 29

30. Spectra of S-Matrix Elements in the Regime Γ/d ≲ 1
Example: 8-9 GHz
|S12|
|S11|
|S22|
Frequency (GHz)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 30

31. Fourier Transform vs. Autocorrelation Function
Time domain
Example 8-9 GHz
Frequency domain
← S12 →
← S11 →
← S22 →
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 31

32. Exact RMT Result for GOE Systems
Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 for arbitrary Г/d
VWZ-Integral
C = C(Ti, d ; )
Transmission coefficients
Average level distance
Rigorous test of VWZ: isolated resonances, i.e. Г ≪ d
First test of VWZ in the intermediate regime, i.e. Г/d ≈ 1, with high statistical significance only achievable with microwave billiards
Note: nuclear cross section fluctuation experiments yield only |S|2
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 32

33. Corollary
Present work: S-matrix → Fourier transform → decay time (indirectly measured)
Future work at short-pulse high-power laser facilities: Direct measurement of the decay time of an excited nucleus might become possible by exciting all nuclear resonances (or a subset of them) simultaneously by a short laser pulse.
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 33

34. Three- and Four-Point Correlation Functions
Autocorrelation function of cross-sections s =|Sab|2 is a four-point correlation
function of Sab
Ericson regime:
Express Cab(e) in terms of 2 -, 3 -, and 4 - point correlation functions of Sfl
A typical example for a four-point correlation function is the cross-section autocorrelation function which is generally evaluated if one has only access to the modulus of the S-matrix elements. In the Ericson regime cross-section autocorrelation functions equals the square of the S-matrix autocorrelation function and thus can be computed analytically. In the regime of small and moderate value of G/d there exists an exact analytic expression for this quantity for e=0 and for e>0 an approximate analytic expression in terms of two- three and four-point functions of S-matrix elements. For this the cross-section correlation function has to be expressed in terms of correlation functions of the fluctuating part of the S-matrix elements. Davis and Boose derived an analytic expression for the cross-section autocorrelation function at e=0
C(3)(0), C(4)(0) and therefore Cab(0) are known analytically
(Davis and Boosé 1988)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 34
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35. Ratio of Cross-Section- and Squared S-Matrix-Autocorrelation Coefficients: 2-Point Correlation Functions
exp: o, o
red: a = b
black: a ≠ b
In this figure I compare the experimental results for the cross-section autocorrelation function at e=0 divided by the squared autocorrelation function of the S-matrix elements obtained from an experiment with a microwave billiard with preserved T-invariance with the corresponding analytic results. We find a good agreement. The black curve corresponds to the inelastic case. For G/d=1.02 the ratio is close to one as expected for Gaussian distributed S-matrix elements with zero mean value and still far away from one for the elastic case, in accordance with the results obtained for the distribution of the S-matrix elements.
Excellent agreement between experimental and analytical result
Dietz et al., Phys. Lett. B 685, 263 (2010)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 35
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36. Experimental and Analytical 3- and 4-Point Correlation Functions
Based on random matrix theory and the compound nucleus S-matrix formalism Davis and Boose derived analytical expressions for this three-point and this four-point function. The correlation functions C(3) and C(4) are obtained from F(3) and F(4) by taking the modulus of the S-matrix elements in the square brackets. For e=0 C coincides with and for e≠0 it can be approximated by the corresponding F functions. Due to lack of time I won’t go deeper into this but would like to note, that as can be seen from the figures, F(3) and F(4) are of the same order of magnitude, even for G>>d, unless the transmission coefficients are close to one, or equivalently the expectation value of the S-matrix is vanishingly small. Therefore the Ericson theory is inconsistent as it simply neglects the terms containing the three-point function.
|F(3)(e )| and F(4)(e ) have similar values
→ Ericson theory is inconsistent as it retains C(4)(e ) but discards C(3)(e )
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 36
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37. Summary
Investigated experimentally and theoretically chaotic scattering in open systems with and without induced T violation in the regime of weakly overlapping resonances (G ≤ d)
Data show that T violation is incomplete  no pure GUE system
Analytical model for partial T violation which interpolates between GOE and GUE has been developed
A large number of data points and a GOF test are used to determine T1, T2, tabs and x from cross- and autocorrelation function
GOF test relies on Gaussian distribution of the Fourier coefficients of the S-matrix
RMT theory describes the distribution of the S-matrix elements and two -, three - and four - point correlation functions well
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 37
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