Classical -, quantum - and microwave billiards

1. ChaoticScatteringin Normal- and SuperconductingMicrowave Billiards,Induced Time- Reversal Violation and the VWZ Approach ECT* 2019 • Classical-, quantum- and microwavebilliards • Billiards as a modelforthe compound nucleus • InducedviolationofTinvariance in billiards • RMT descriptionforchaoticscatteringsystemswithTviolation • Experimental testof RMT results on fluctuationpropertiesof S-matrix elementswith normal and superconductingmicrowavebilliards Supported by DFG within SFB 1245 B. Dietz, T. Klaus, M. Miski-Oglu, A. Richter, M. Wunderle 2019 | SFB 1245 | Achim Richter |

2. Classical-, Quantum- and Microwave Billiards 2019 | SFB 1245 | Achim Richter |

3. Quantum billiard Microwave billiard d eigenvalue Ewave number Schrödinger- and Microwave Billards Analogy eigenfunction  electric field strength Ez 2019 | 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 isemittedintotheresonatorbyantenna and theoutputsignalisreceivedbyantenna Open scatteringsystem • 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 2019 | SFB 1245 | Achim Richter |

5. Typical Transmission Spectrum • Transmission measurements: relative power fromantenna a → b 2019 | 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 2019 | SFB 1245 | Achim Richter |

7. real part imaginary part Resonance Parameters • Use eigenrepresentationof and obtainfor a scatteringsystemwithisolatedresonances a → resonator → b • Here: ofeigenvaluesof • Partial widthsfluctuate and total widths also 2019 | SFB 1245 | Achim Richter |

8. Fluctuations of Total Widths and Partial Widths 2019 | SFB 1245 | Achim Richter | • 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)

9. Excitation Spectra atomic nucleus microwave cavity overlappingresonances forΓ/d > 1 Ericson fluctuations weaklyoverlappingresonancesforΓ/d ≲ 1 isolatedresonances forΓ/d << 1 ρ ~ exp ρ ~ f • Universal descriptionofspectra and fluctuations:Verbaarschot, Weidenmüller and Zirnbauer(1984) 2019 | SFB 1245 | Achim Richter |

10. Fully Chaotic Microwave Billiard • Tiltedstadium (Primack+Smilansky, 1994) • Onlyvertical TM0modeisexcited in resonator→ simulates a quantum billiardwithdissipation • Additional scatterer→improvesstatisticalsignificanceofthedata sample • Measurecomplex S-matrix fortwoantennas 1 and 2:S11, S22, S12, S21 2 1 (Dietz, Friedrich, Harney, Miski-Oglu, Richter, Schäfer andWeidenmüller, 2008) 2019 | SFB 1245 | Achim Richter |

11. Spectra and Correlations ofS-Matrix Elements • Γ/d <<1: isolatedresonances → eigenvalues, partial and total widths • Г/d ≲ 1: weaklyoverlappingresonances and stronglyoverlappingresonances(Г/d >1) → investigate S-matrix fluctuationpropertieswiththeautocorrelationfunction and its Fourier transform 2019 | SFB 1245 | Achim Richter |

12. Spectra of S-Matrix Elementsin the Regime Γ/d ≲ 1 Example: 8-9 GHz |S12| |S11| |S22| • How does the system decay? Frequency (GHz) 2019 | SFB 1245 | Achim Richter |

13. Exact RMT Result for GOE Systems • Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 forarbitraryГ/d • VWZ-Integral C = C(Ti, d; ) Average level distance Transmission coefficients • Rigoroustestof VWZ: isolatedresonances, i.e. Г≪ d • First testof VWZ in the intermediate regime, i.e. Г/d≈1, with high statisticalsignificanceonlyachievablewithmicrowavebilliards • Note: nuclearcrosssectionfluctuationexperimentsyieldonly |S|2 2019 | SFB 1245 | Achim Richter |

14. Fourier Transform vs.Autocorrelation Function Time domain Frequency domain Example 8-9 GHz ← S12→ ← S11→ ← S22→ 2019 | SFB 1245 | Achim Richter |

15. Corollary • Presentwork:S-matrix → Fourier transform → decay time (indirectlymeasured) • Future work at short-pulse high-power laserfacilities:Directmeasurementofthedecay time of an excitednucleusmightbecome possible byexciting all nuclearresonances (or a subsetofthem) simultaneouslyby a shortlaser pulse. 2019 | SFB 1245 | Achim Richter |

16. N S N S Fully Chaotic Microwave Billiard andthe Study of Induced T Violation • A cylindricalferriteisplaced in theresonator • An external magneticfieldisappliedperpendiculartothebilliard plane • The strengthofthemagneticfieldisvariedbychangingthedistance betweenthemagnets 2019 | SFB 1245 | Achim Richter |

17. Induced Violation of T Invariancewith Ferrite • SpinsofmagnetizedferriteprecesscollectivelywiththeirLarmorfrequencyaboutthe external magneticfield • Couplingofrfmagneticfieldtotheferromagneticresonancedepends on thedirection a b Sab F a b Sba • T-invariant system → reciprocity → detailedbalance • T-noninvariantsystem Sab= Sba |Sab|2= |Sba|2 2019 | SFB 1245 | Achim Richter |

18. Detailed Balance in the Microwave Billiard S12 S21 • Clear violationofprincipleofdetailedbalancefornonzeromagneticfield B → Howcanwedeterminethestrengthof T violation? 2019 | SFB 1245 | Achim Richter |

19. Search for TRSB in Nuclei:Ericson Regime 2019 | SFB 1245 | Achim Richter |

20. if T-invariance holds if T-invariance is violated Analysis of T violation with a Crosscorrelation Function • Crosscorrelationfunction • Special interest in crosscorrelationcoefficientCcross(ε = 0) 2019 | SFB 1245 | Achim Richter |

21. S-Matrix CorrelationFunctions 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 2019 | SFB 1245 | Achim Richter |

22. RMT Resultfor 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 2019 | SFB 1245 | Achim Richter |

23. Exact RMT Resultfor Partial T Breaking • RMT analysisbased on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner (1995) for GOE • RMT → for GUE 2019 | SFB 1245 | Achim Richter |

24. cross-correlation coefficient T-Violation Parameter ξ • Largestvalueof T-violation parameterachievedisξ≃ 0.3 (Dietz, Friedrich, Harney, Miski-Oglu, Richter, Schäfer, Verbaarschot andWeidenmüller, 2009) 2019 | SFB 1245 | Achim Richter |

25. Induced Time Reversal Symmetry Breaking in a Superconducting Microwave Billiard • In room- temperatureexperimentsTinvarianceviolation was inducedbyinserting a ferritintothebilliardresonator and magnetizingitwith an external B-field • Due tothe Meissner- Ochsenfeld effectthisis, however, nolonger possible in caseofsuperconductingcavitiesconstructedfrom lead-coatedcopperorbrass sincebelow a criticaltemperatureTCthefieldisexpelledfromthesuperconductor • Wethusused a billiardcavitymade out ofniobium – a type II superconductor– formagneticfieldsbetweentwocriticalvalues, Bc1 = 153 mTandBc2 = 268 mT • Betweenthesetwocriticalvaluesmagneticfluxstartstopenetratethesuperconductor via so calledvorticesorfluxtubes 2019 | SFB 1245 | Achim Richter |

26. Billiard Cavity Made out of Niobium • The top plate (left) isremovedfromthebottomplate (right) • A lead-coated brass frame (right) withtheshapeifthe African continent("Africabilliard", Berry and Robnik (1986)) was insertedintothecircularniobiumcavity • The circlesindicatethe 6 antennapositionsfor in- and outputtingrf power • The ferriteposition at thecenterismarkedby a cross 2019 | SFB 1245 | Achim Richter |

27. Principleofthe Measurement • Ratio ismeasuredforB = 0and B ≠ 0 • Quality factorQ = fμ / Γμ > 105→ all resonancesaredetected and • Fromthemeasured S-matrix theresonance strengthΓμa∙ Γμbisdetermined and comparedto RMT predictions 2019 | SFB 1245 | Achim Richter |

28. Measured Transmission Spectra • About 150 resonances (all resonances) • Spectra at B = 200 mTexhibit a broadpeakbetween 16.5 and 18 GHz whichmaybeattributedtoelectricfieldmodestrappedinsidetheferrit ("FMR") • ForB = 0 mTand also below and abovethe FMR theprincipleofreciprocityholds, i.e. Sab(f ) = Sba(f ), but itisclearlyviolatedwithin it. This isshown in the zoom fortwopairsofresonances at 15.59 and 15.62 GHz and at 17.08 and 17.11 GHz, respectively. 2019 | SFB 1245 | Achim Richter |

29. Fluctuations in the Transmission Spectra • WemodeltheHamiltonianofthe fully chaoticAfricabilliardby an ensembleofN×N-dimensional randommatriceswithentries Hsbeing a real-symmetric and Ha a real-antisymmetricmatrixwithuncorrelatedGaussian-distributedmatrixelements. The parameterξdeterminesthemagnitudeofT violation. • The strengthdistributionisderivedfromthatofthe partial widths: • Best fittinganalyticalcurveyieldsξ. withconservedTinvariance (ξ = 0): Porter-Thomas forchaoticsystems with fully violated (ξ > 1): exponential 2019 | SFB 1245 | Achim Richter |

30. Experimental and TheoreticalStrengthDistributions • Experiment: histogramm • GOE: solid line • GUE: dashedline • Best fittinganalyticalcurve (red) yielding: ξ = 0.2 2019 | SFB 1245 | Achim Richter |

31. Distribution oftheModulusoftheMeasured off-diagonal S-Matrix Elements • In ordertoobtainthetransmissioncoefficientsTa,b and τabsensemblesof 500 randomSmatricesweregenerated. Theirdistribution (reddashedlines) iscomparedtothemeasuredone (black solid lines) in 4 different frequencyintervals. • Note thatthedistributionclearlyreflectstheexistanceof T breaking (ξ ≠ 0) at thepositionifthe FMR whereTa and Tbare larger than outside of it. 2019 | SFB 1245 | Achim Richter |

32. Fluctuation Properties oftheResonanceFrequencies • Spectralproperties (NNSD P(s) and thecummulativeoneI(s), thevarianceΣ2(L) and the Dyson-MethastatisticΔ3(L)) for external magneticfieldsB = 0 mT(left) and B = 200 mT (right). • Experimental data (histograms and triangles) arecomparedto GOE predictions (solid lines) and GUE predictions (dash-dottedlines). • The red solid linesshowthecorrespondingbestfittingcurvesyieldingξ = 0.2 and thusverifythevaluederivedfromthetransitionstrengthdistribution. 2019 | SFB 1245 | Achim Richter |

33. Summary • We experimentally realized for the first time a superconducting microwave billiard with partially violated T invariance. In order to obtain longer and complete sequences of eigenvalues and to realize a stronger violation of T invariance the area of the cavity, i.e., the billiard needs to be increased and a few more ferrites must be added. Yet, care has to be taken that additional absorption leaves the resonances isolated in order to ensure completelevelsequences. 2019 | SFB 1245 | Achim Richter |

34. Perspectives and Thanks • I spokehereaboutthe last experimentpreformed in "our" billiardgroup at the Institut ofNuclear Physics at the TU Darmstadt. The group was founded in 1992 so tosayas an offspringfromourexperience in rfsuperconductivityfromdeveloping and buildingthe S-DALINAC and ourresearch in nuclearphysics. • Duringthealmost 30 yearsthebilliardgrouphasexistedwehadmanycollaboratorsfromthe outside, someofthemarehere at themeeting – Thomas Guhr, Thomas Seligman, JacVerbaarschot, Jochen Wambach and Hans Weidenmüller. This givesmepleasuretothank all ofthem. • Finally I am verypleasedthattheworkwithsuperconductingbilliardsisnowbeingcontinued at Lanzhou University in China. Mylong time collaborator at Darmstadt Barbara Dietz is the prime investigatorthere, and I wishyou, Barbara, greatsuccess in yourendeavours. 2019 | SFB 1245 | Achim Richter |