Chaotic Scattering in Open Microwave Billiards with and without T Violation

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 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

2. rf power in rf power out C+c D+d  Compound Nucleus  B+b A+a Microwave Resonator as a Model for the Compound Nucleus 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 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 2

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 ( fI - 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  • Experiment: complex S-matrix elements GOE T-inv • RMT description: replace Ĥ by a matrix for systems T-noninv GUE 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 4

5. real part imaginary part 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 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 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 7

8. Spectra and Correlations ofS-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. N S N S 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 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 9

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 a b Sba • T-invariant system → reciprocity • → detailed balance Sab= Sba |Sab|2= |Sba|2 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 10

11. Detailed Balance in Microwave Billiard S12 S21 • 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

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. if T-invariance holds if T-invariance is violated Analysis of T violation with Cross-Correlation Function • Cross-correlation function: • Special interest in cross-correlation coefficient Ccross(ε = 0) 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 13

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) in1995 • RMT  GOE: 0GUE: 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

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  Ŵis real • Ohmic losses are mimicked by additional fictitious channels • Ŵ is approximately constant in 1GHz frequency intervals 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 15

16. RMT Result for Partial T Violation(Phys. Rev. Lett. 103, 064101 (2009)) • 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 transmission coefficients  from autocorrelation function 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 16

17. Exact RMT Result for Partial T Breaking • RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner (1995) for GOE • RMT → for GUE 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 17

18. cross-correlation coefficient T-Violation Parameter ξ • Largest value of T-violation parameter achieved is ξ≃ 0.3 • Published in Phys. Rev. Lett. 103, 064101 (2009) 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 18

19. log(FT(C12(ε))) t (ns) Autocorrelation Function and its Fourier Transform 6 4 |C12(ε)|2 x (10-4) 2 0 • Adjacent points in Cab() are correlated • FT of Cab() • For a fixed t are Gaussian distributed about their • time-dependent mean 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 19

20. Distribution of Fourier Coefficients • Distribution ofmodulus divided by its local mean is bivariate Gaussian • Distribution of phases is uniform 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 20

21. 3-4 GHz log(FT(C12(ε))) 9-10 GHz t (ns) • Fit determines expectation value of , tabs and improved values • for T1, T2,  • GOF test relies on Gaussian distributed Determination of τabs from FT of Autocorrelation Function • Decay isnon exponential → autocorrelation function is not Lorentzian 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 21

22. Enhancement Factor for Systems with Partial T-Violation • Hauser-Feshbach for Γ>>d: • Elastic enhancement factor w 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

23. Elastic Enhancement Factor • 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

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 Elementsin the Ericson Regime 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 25

26. Distributions of S-Matrix Elementsin 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. Exp: RMT: Experimental Distribution of S11and Comparison with Fyodorov, Savin and Sommers Prediction (2005) • 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

28. Exp: RMT: Experimental Distribution of S12 and Comparison with RMT Simulations • 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

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 Elementsin 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 Frequency domain Example 8-9 GHz ← 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; ) Average level distance Transmission coefficients • 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. Ericson regime: Three- and Four-Point Correlation Functions • Autocorrelation function of cross-sections s =|Sab|2 is a four-point correlation • function of Sab • Express Cab(e) in terms of 2 -, 3 -, and 4 - point correlation functions of Sfl • 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

35. Ratio of Cross-Section- and Squared S-Matrix-Autocorrelation Coefficients:2-Point Correlation Functions exp: o, o red: a = b black: a ≠ b • 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

36. Experimental and Analytical 3- and 4-Point Correlation Functions • |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

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 andxfrom 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