Chaotic Scattering in Open Microwave Billiards with and
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
Microwave Resonator as a Model for the Compound Nucleus C+ c rf power in out D+ d h A+a Compound Nucleus B+b • 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
Typical Transmission Spectrum • Transmission measurements: relative power from antenna a → b 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 3
Scattering Matrix Description • Scattering matrix for both scattering processes as function of excitation frequency f : Ŝ( f ) = I - 2 pi ŴT ( f I - Ĥ + ip ŴŴT)-1 Ŵ Compound-nucleus reactions Microwave billiard nuclear Hamiltonian Ĥ resonator Hamiltonian coupling of quasi-bound states to channel states Ŵ coupling of resonator states to antenna states and to absorptive channels • Experiment: • RMT description: replace Ĥ by a complex S-matrix elements GOE T-inv matrix for systems T-noninv GUE 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 4
Resonance Parameters • Use eigenrepresentation of and obtain for a scattering system with isolated resonances a → resonator → b • Here: • Partial widths real part imaginary part of eigenvalues of fluctuate and total widths 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 5 also
Excitation Spectra atomic nucleus microwave cavity overlapping resonances for Γ/d > 1 Ericson fluctuations isolated resonances for Γ/d < 1 ρ ~ exp(E 1/2) • Universal description of spectra and fluctuations: Verbaarschot, Weidenmüller and Zirnbauer (1984) 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 6 ρ~f
Fully Chaotic Microwave Billiard • Tilted stadium (Primack+Smilansky, 1994) 1 2 • 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 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 7
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
Microwave Billiard for the Study of Induced T Violation N S • 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
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 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 10 Sab = Sba |Sab|2 = |Sba|2
Detailed Balance in Microwave Billiard S 12 S 21 • 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
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
Analysis of T violation with Cross-Correlation Function • Cross-correlation function: • Special interest in cross-correlation coefficient Ccross(ε = 0) if T-invariance holds if T-invariance is violated 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 13
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 Ccross(ε=0) valid for all values of 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 14
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 1 GHz frequency intervals 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 15
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
Exact RMT Result for Partial T Breaking • RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner (1995) • RMT → 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 17 for GOE for GUE
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
|C 12(ε)|2 x (10 -4) 6 4 2 log(FT(C 12(ε))) Autocorrelation Function and its Fourier Transform 0 t (ns) • Adjacent points in C ab( ) 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
Distribution of Fourier Coefficients • 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
Determination of τabs from FT of Autocorrelation Function log(FT(C 12(ε))) 3 -4 GHz 9 -10 GHz t (ns) • Fit determines expectation value of for T 1, T 2, • GOF test relies on Gaussian distributed , tabs and improved values • Decay is non exponential → autocorrelation function is not Lorentzian 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 21
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
Elastic Enhancement Factor • Above ~ 5 GHz 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
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 für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 24
Spectra of S-Matrix Elements in the Ericson Regime 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 25
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
Experimental Distribution of S 11 and Comparison with Fyodorov, Savin and Sommers Prediction (2005) Exp: RMT: • 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
Experimental Distribution of S 12 and Comparison with RMT Simulations Exp: RMT: • No analytical expression for distribution of off-diagonal S-matrix elements • For G/d=1. 02 the distribution of S 12 is close to Gaussian → Ericson regime? • Recent analytical results agree with data → Thomas Guhr 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 28
Autocorrelation Function and Fourier Coefficients in the Ericson Regime Frequency domain • |C(ε)|2 is of Lorentzian shape 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 29 Time domain → exponential decay
Spectra of S-Matrix Elements in the Regime Γ/d ≲ 1 |S 22| |S 11| |S 12| Example: 8 -9 GHz Frequency (GHz) 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 30
Fourier Transform vs. Autocorrelation Function Time domain Example 8 -9 GHz ← S 12 → ← S 11 → ← S 22 → 2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 31 Frequency domain
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
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
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( ) 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
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
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
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 T 1, T 2, tabs 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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