Slide 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. Schfer, H. L. Harney, J.J.M Verbaarschot, H. A. Weidenmller Porquerolles 2013 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 1 Slide 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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 2 Slide 3 Typical Transmission Spectrum Transmission measurements: relative power from antenna a b 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 3 Slide 4 Scattering Matrix Description Scattering matrix for both scattering processes as function of excitation frequency f : RMT description: replace 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 GOE T-inv GUE T-noninv Experiment: complex S -matrix elements ( f ) = I - 2 i T ( f I - + i T ) -1 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 4 Slide 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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 5 Slide 6 Excitation Spectra overlapping resonances for /d > 1 Ericson fluctuations isolated resonances for /d < 1 atomic nucleus ~ exp(E 1/2 ) microwave cavity ~ f Universal description of spectra and fluctuations: Verbaarschot, Weidenmller and Zirnbauer (1984) 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 6 Slide 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 21 1 2 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 7 Slide 8 Spectra and Correlations of S -Matrix Elements /d Enhancement Factor for Systems with Partial T-Violation Hauser-Feshbach for >>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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 22 Slide 23 Elastic Enhancement Factor Above ~ 5GHz data (red) match RMT prediction (blue) Values w < 2 seen clear indication of T violation 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 23 Slide 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 12 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 24 Slide 25 Spectra of S -Matrix Elements in the Ericson Regime 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 25 Slide 26 Distributions of S -Matrix Elements in the Ericson Regime Experiment confirms assumption of Gaussian distributed S -matrix elements with random phases 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 26 Slide 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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 27 Slide 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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 28 Slide 29 Autocorrelation Function and Fourier Coefficients in the Ericson Regime Time domainFrequency domain |C()| 2 is of Lorentzian shape exponential decay 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 29 Slide 30 Spectra of S -Matrix Elements in the Regime /d 1 Example: 8-9 GHz Frequency (GHz) |S 11 | |S 12 | |S 22 | 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 30 Slide 31 Fourier Transform vs. Autocorrelation Function Time domainFrequency domain S 12 S 11 S 22 Example 8-9 GHz 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 31 Slide 32 Exact RMT Result for GOE Systems C = C(T i, d ; ) Transmission coefficients Average level distance Verbaarschot, Weidenmller and Zirnbauer (VWZ) 1984 for arbitrary /d VWZ-Integral Rigorous test of VWZ: isolated resonances, i.e. d First test of VWZ in the intermediate regime, i.e. /d 1, with high statistical significance only achievable with microwave billiards Note: nuclear cross section fluctuation experiments yield only | S | 2 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 32 Slide 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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 33 Slide 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 Boos 1988) Ericson regime: 2013 |Institut fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 34 Slide 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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 35 Slide 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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 36 Slide 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 fr Kernphysik, Darmstadt | SFB 634 | Achim Richter | 37