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Chaotic Scattering in Open Microwave Billiards with and without T Violation Microwave resonator as a model for the compound nucleus Induced violation of.

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Presentation on theme: "Chaotic Scattering in Open Microwave Billiards with and without T Violation Microwave resonator as a model for the compound nucleus Induced violation of."— Presentation transcript:

1 Chaotic Scattering in Open Microwave Billiards with and without T Violation 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 Porquerolles |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 1

2 Microwave Resonator as a Model for the Compound Nucleus 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 Compound Nucleus A+a B+b C+c D+d rf power in rf power out   h 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 : RMT description: replace Ĥ by a matrix for systems Microwave billiard Compound-nucleus reactions resonator Hamiltonian coupling of resonator states to antenna states and to absorptive channels nuclear Hamiltonian coupling of quasi-bound states to channel states  Ĥ  Ĥ   Ŵ  Ŵ  GOE T-inv GUE T-noninv Experiment: complex S -matrix elements Ŝ( f ) = I - 2  i Ŵ T ( f I - Ĥ + i  ŴŴ T ) -1 Ŵ 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 4

5 Resonance Parameters Use eigenrepresentation of and obtain for a scattering system with isolated resonances a → resonator → b Here:of eigenvalues of Partial widthsfluctuate and total widthsalso real part imaginary part 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 5

6 Excitation Spectra overlapping resonances for Γ/d > 1 Ericson fluctuations isolated resonances for Γ/d < 1 atomic nucleus ρ ~ exp(E 1/2 ) microwave cavity 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 TM 0 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: S 11, S 22, S 12, S |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 7

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 NSNS NSNS 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 S ab S ba a b 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 T-invariant system → reciprocity → detailed balance F S ab = S ba |S ab | 2 = |S ba | |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 10

11 Detailed Balance in Microwave Billiard Clear violation of principle of detailed balance for nonzero magnetic field B → How can we determine the strength of T violation? S 12 S |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 Analysis of T violation with Cross-Correlation Function Cross-correlation function: Special interest in cross-correlation coefficient C cross (ε = 0) if T-invariance holds if T-invariance is violated 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) in 1995 RMT  GOE: 0 GUE: 1 Complete T-invariance violation sets in already for  ≃ 1 Based on Efetov´s supersymmetry method we derived analytical expressions for C (2) (ε) and C cross (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  Ŵ is 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, (2009)) RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner, 1995 parameter for strength of T violation  from cross-correlation coefficient transmission coefficients  from autocorrelation function mean resonance spacing → from Weyl‘s formula (2) d 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 for GUE RMT → 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 17

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

19 Adjacent points in C ab (  ) are correlated FT of C ab (  )  For a fixed t are Gaussian distributed about their time-dependent mean Autocorrelation Function and its Fourier Transform t (ns) log(FT(C 12 (ε))) |C 12 (ε)| 2 x (10 -4 )  2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 19

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

21 Determination of τ abs from FT of Autocorrelation Function Fit determines expectation value of,  abs and improved values for T 1, T 2,  GOF test relies on Gaussian distributed Decay is non exponential → autocorrelation function is not Lorentzian 3-4 GHz 9-10 GHz log(FT(C 12 (ε))) t (ns) 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 21

22 Enhancement Factor for Systems with Partial T-Violation Hauser-Feshbach for Γ>>d : Isolated resonances: w = 3 for GOE, w = 2 for GUE Strongly overlapping resonances: w = 2 for GOE, w = 1 for GUE Elastic enhancement factor w 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: S 11, S 22, S |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 24

25 Spectra of S -Matrix Elements in the Ericson Regime 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 25

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 S 11 and Comparison with Fyodorov, Savin and Sommers Prediction (2005) Distributions of modulus z is not a bivariate Gaussian Distributions of phases  are not uniform Exp: RMT: 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 27

28 Experimental Distribution of S 12 and Comparison with RMT Simulations No analytical expression for distribution of off-diagonal S -matrix elements For  / d =1.02 the distribution of S 12 is close to Gaussian → Ericson regime? Recent analytical results agree with data → Thomas Guhr Exp: RMT: 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 28

29 Autocorrelation Function and Fourier Coefficients in the Ericson Regime Time domainFrequency 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 Frequency (GHz) |S 11 | |S 12 | |S 22 | 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 30

31 Fourier Transform vs. Autocorrelation Function Time domainFrequency domain ← S 12 → ← S 11 → ← S 22 → Example 8-9 GHz 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 31

32 Exact RMT Result for GOE Systems C = C(T i, d ;  ) Transmission coefficients Average level distance Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 for arbitrary Г/d VWZ-Integral 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 | |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 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 33

34 Three- and Four-Point Correlation Functions Autocorrelation function of cross-sections  =| S ab | 2 is a four-point correlation function of S ab Express C ab (  ) in terms of 2 -, 3 -, and 4 - point correlation functions of S fl C (3) (0), C (4) (0) and therefore C ab (0) are known analytically (Davis and Boosé 1988) Ericson regime:  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 Excellent agreement between experimental and analytical result Dietz et al., Phys. Lett. B 685, 263 (2010) exp: o, o red: a = b black: a ≠ b 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 35

36 Experimental and Analytical 3- and 4-Point Correlation Functions | F (3) (  )| and F (4) (  ) have similar values → Ericson theory is inconsistent as it retains C (4) (  ) but discards C (3) (  )  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 (  ≤ 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 T 1, T 2,  abs and  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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