Chaotic Scattering without and with Time Reversal Violation in Open Microwave Billiards

1. Chaotic Scattering without and with Time Reversal Violation in Open Microwave Billiards Modelling Nuclei ECT* 2017 • Classical-, quantum- and microwave billiards • 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 and SFB 1245 B. Dietz, M. Miski-Oglu, A. Richter F. Schäfer, H. L. Harney, J.J.M Verbaarschot, H. A. Weidenmüller 2017 | SFB 1245 | Achim Richter |

2. Classical-, Quantum- and Microwave Billiards 2017 | SFB 1245 | Achim Richter |

3. Quantum billiard Microwave billiard d eigenvalue Ewave number Schrödinger- and Microwave Billards Analogy eigenfunction  electric field strength Ez 2017 | SFB 1245 | Achim Richter |

4. rf power in rf power out C+c D+d  Compound Nucleus  B+b A+a Microwave Resonator as a Model fortheCompound Nucleus d • 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 • Manufactured at CERN from surplus Nb metal sheets of sc LEP cavities 2017 | SFB 1245 | Achim Richter |

5. Typical Transmission Spectrum • Transmission measurements: relative power from antenna a → b 2017 | SFB 1245 | Achim Richter |

6. Scattering Matrix Description within the“Heidelberg Approach“ Nucleus Microwave billiard E, f energy E nuclear Hamiltonian coupling of quasi-bound states to channel states frequency f resonator Hamiltonian coupling of resonator states to antenna states and to absorptive channels H  W  • Experiment measures complex S-matrix elements GOE T-inv • RMT description: replace H by a matrix for systems T-noninv GUE 2017 | SFB 1245 | Achim Richter |

7. 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 2017 | SFB 1245 | Achim Richter |

8. Fluctuations of Total Widths and Partial Widths • Both quantities fluctuate randomly about a slow secular variation of the mean. • The partial widths obey a Porter-Thomas distribution. • There is a lack of correlations among partial widths and also a lack of correlations between total widths and resonance energies. Finally, we observed a nonexponentialdecay of the system. • …“one of the strongest tests ever for the assertion that upon quantization a classically chaotic system respecting T-invariance attains GOE fluctuation properties“. (Alt, Gräf, Harney Lengeler, Richter, Schardt and Weidenmüller, 1995) 2017 | SFB 1245 | Achim Richter |

9. Excitation Spectra atomic nucleus microwave cavity overlapping resonances for Γ/d > 1 Ericson fluctuations isolated resonances for Γ/d << 1 ρ ~ exp ρ ~ f • Universal description of spectra and fluctuations:Verbaarschot, Weidenmüller and Zirnbauer (1984) 2017 | SFB 1245 | Achim Richter |

10. 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 2017 | SFB 1245 | Achim Richter |

11. Spectra and Correlations ofS-Matrix Elements • Γ/d <<1: isolated resonances → eigenvalues, partial and total widths • Г/d ≲ 1: weakly overlapping resonances and strongly overlapping resonances(Г/d >1) → investigate S-matrix fluctuation properties with the autocorrelation function and its Fourier transform 2017 | SFB 1245 | Achim Richter |

12. Exp: RMT: Experimental Distribution of S11and Comparison with RMT Predictions of Fyodorov, Savin and Sommers (2005) • Distributions of modulus z are not bivariate Gaussians • Distributions of phases  are not uniform 2017 | SFB 1245 | Achim Richter |

13. Exp: RMT: Experimental Distribution of S12 and Comparison with RMT Simulations • For Γ/d =1.02 the distribution of S12 is close to Gaussian and the phases become uniformly distributed → Ericson regime • Recent analytical results agree with data 2017 | SFB 1245 | Achim Richter |

14. Spectra of S-Matrix Elementsin the Ericson Regime 2017 | SFB 1245 | Achim Richter |

15. DistributionsofS-Matrix Elementsin the Ericson Regime • Experiment confirms assumption of Gaussian distributed S-matrix elements with random phases 2017 | SFB 1245 | Achim Richter |

16. Experimental Distribution of S12 and Comparison with Predictions of Guhr, Kumar, Nock and Sommers (2013) • Frequencyrangebetween 24 and 25 GHz (Γ/d =1.21) 2017 | SFB 1245 | Achim Richter |

17. Spectra of S-Matrix Elementsin the Regime Γ/d ≲ 1 Example: 8-9 GHz |S12| |S11| |S22| • How does the system decay? Frequency (GHz) 2017 | SFB 1245 | Achim Richter |

18. 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 2017 | SFB 1245 | Achim Richter |

19. Fourier Transform vs.Autocorrelation Function Time domain Frequency domain Example 8-9 GHz ← S12→ ← S11→ ← S22→ 2017 | SFB 1245 | Achim Richter |

20. 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. 2017 | SFB 1245 | Achim Richter |

21. 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 2017 | SFB 1245 | Achim Richter |

22. Induced Violation of T Invariancewith Ferrite • Spinsofmagnetized 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 → detailedbalance • T-noninvariantsystem Sab= Sba |Sab|2= |Sba|2 2017 | SFB 1245 | Achim Richter |

23. Detailed Balance in the 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? 2017 | SFB 1245 | Achim Richter |

24. Search for TRSB in Nuclei:Ericson Regime • T. Ericson, Nuclear enhancement of T violation effects,Phys. Lett. 23, 97 (1966) 2017 | SFB 1245 | Achim Richter |

25. if T-invariance holds if T-invariance is violated Analysis of T violation with a Crosscorrelation Function • Crosscorrelation function • Special interest in crosscorrelation coefficient Ccross(ε = 0) 2017 | SFB 1245 | Achim Richter |

26. 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 2017 | SFB 1245 | Achim Richter |

27. 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 crosscorrelation coefficient transmission coefficients  from autocorrelation function 2017 | SFB 1245 | Achim Richter |

28. Exact RMT Result for Partial T Breaking • RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner (1995) for GOE • RMT → for GUE 2017 | SFB 1245 | Achim Richter |

29. 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) 2017 | SFB 1245 | Achim Richter |

30. Summary • Scatteringprocessesfromhighlycomplexsystemsshowcharacteristicsfluctuationphenomena. • Theiraveragefeaturesareofgenericnatureindependentlyofthedetailsoftheunderlyinginteractionsandshowsensitivitytotheenergyorfrequency. • Such fluctuationsarecharacteristicforquantumchaoticscatteringoccuring in nuclei, molecules, in theconductanceofmesoscopicsemiconductordevicesanddisordered open systems, in electronic transport in ballistic open quantumdots, and in microwavecavities (see also T.E.O. Ericson, B. Dietz and A.R., PRE 94, 042207 (2016)). • Investigatedexperimentallyandtheoreticallychaoticscattering in open systemswithoutandwithinduced T violation in theregimeofweaklyoverlappingresonances(Г≤ d). • Data showthat T violationisincomplete no pure GUE system but oneof GOE/GUE • Analytical modelfor partial T violationwhichinterpolatesbetween GOE and GUE hasbeendeveloped 2017 | SFB 1245 | Achim Richter |

31. Universality of Nearest Neighbour Level Spacing Distributions 208Pb Stadium Billiard [H.-D. Gräf et al., PRL 69, 1296(1992)] • Universal (generic) behavior of the two systems of very different sizes [B. Dietz et al., PRL 118, 012501(2017)] 2017 | SFB 1245 | Achim Richter |