Chaotic Scattering without and with Time Reversal Violation

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 | 1

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

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

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

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

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

Resonance Parameters • Use eigenrepresentation of and obtain for a scattering system with isolated resonances a → resonator → b • Here: real part of eigenvalues of imaginary part • Partial widths fluctuate and total widths 2017 | SFB 1245 | Achim Richter | 7 also

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 nonexponential decay 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 | 8

Excitation Spectra atomic nucleus overlapping resonances for Γ/d > 1 Ericson fluctuations microwave cavity isolated resonances for Γ/d << 1 ρ ~ exp • Universal description of spectra and fluctuations: Verbaarschot, Weidenmüller and Zirnbauer (1984) 2017 | SFB 1245 | Achim Richter | 9 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 2017 | SFB 1245 | Achim Richter | 10

Spectra and Correlations of S-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 | 11

Experimental Distribution of S 11 and Comparison with RMT Predictions of Fyodorov, Savin and Sommers (2005) Exp: RMT: • Distributions of modulus z are not bivariate Gaussians • Distributions of phases are not uniform 2017 | SFB 1245 | Achim Richter | 12

Experimental Distribution of S 12 and Comparison with RMT Simulations Exp: RMT: • For Γ/d =1. 02 the distribution of S 12 is close to Gaussian and the phases become uniformly distributed → Ericson regime • Recent analytical results agree with data 2017 | SFB 1245 | Achim Richter | 13

Spectra of S-Matrix Elements in the Ericson Regime 2017 | SFB 1245 | Achim Richter | 14

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

Experimental Distribution of S 12 and Comparison with Predictions of Guhr, Kumar, Nock and Sommers (2013) • Frequency range between 24 and 25 GHz (Γ/d =1. 21) 2017 | SFB 1245 | Achim Richter | 16

Spectra of S-Matrix Elements in the Regime Γ/d ≲ 1 |S 22| |S 11| |S 12| Example: 8 -9 GHz • How does the system decay? Frequency (GHz) 2017 | SFB 1245 | Achim Richter | 17

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 2017 | SFB 1245 | Achim Richter | 18

Fourier Transform vs. Autocorrelation Function Time domain Example 8 -9 GHz ← S 12 → ← S 11 → ← S 22 → 2017 | SFB 1245 | Achim Richter | 19 Frequency domain

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 | 20

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 2017 | SFB 1245 | Achim Richter | 21

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 • T-noninvariant system 2017 | SFB 1245 | Achim Richter | 22 Sab = Sba |Sab|2 = |Sba|2

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

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 | 24

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

S-Matrix Correlation Functions and RMT • Pure GOE • Pure GUE • Partial T violation VWZ 1984 FSS (Fyodorov, Savin, Sommers) 2005 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 2017 | SFB 1245 | Achim Richter | 26

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 | 27

Exact RMT Result for Partial T Breaking • RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner (1995) • RMT → 2017 | SFB 1245 | Achim Richter | 28 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) 2017 | SFB 1245 | Achim Richter | 29

Summary • Scattering processes from highly complex systems show characteristics fluctuation phenomena. • Their average features are of generic nature independently of the details of the underlying interactions and show sensitivity to the energy or frequency. • Such fluctuations are characteristic for quantum chaotic scattering occuring in nuclei, molecules, in the conductance of mesoscopic semiconductor devices and disordered open systems, in electronic transport in ballistic open quantum dots, and in microwave cavities (see also T. E. O. Ericson, B. Dietz and A. R. , PRE 94, 042207 (2016)). • Investigated experimentally and theoretically chaotic scattering in open systems without and with induced T violation in the regime of weakly overlapping resonances (Г ≤ d). • Data show that T violation is incomplete no pure GUE system but one of GOE/GUE • Analytical model for partial T violation which interpolates between GOE and GUE has been developed 2017 | SFB 1245 | Achim Richter | 30

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