1. SCUOLA INTERNAZIONALE DI FISICA “fermi" Varenna sul lago di como - 1965

2. Dr. h.c. Oriol Bohigas TU Darmstadt 2001

3. BGS conjecture, quantum billiards and microwave resonators Chaotic microwave resonators as a model for the compound nucleus Fluctuation properties of the S -matrix for weakly overlapping resonances ( Γ  D ) in a T-invariant (GOE) and a T- noninvariant (GUE) system Test of model predictions based on RMT for GOE and GUE Supported by DFG within SFB 634 B. Dietz, T. Friedrich, M. Miski-Oglu, A. R., F. Schäfer H.L. Harney, J.J.M. Verbaarschot, H.A. Weidenmüller Chaotic Scattering in Microwave Billiards Orsay 2008 SFB 634 – C4: Quantum Chaos

4. Conjecture of Bohigas, Giannoni + Schmit (1984) SFB 634 – C4: Quantum Chaos For chaotic systems, the spectral fluctuation properties of eigenvalues coincide with the predictions of random-matrix theory (RMT) for matrices of the same symmetry class. Numerous tests of various spectral properties (NNSD, Σ 2, Δ 3,...) and wave functions in closed systems exist Our aim: to test this conjecture in scattering systems, i.e. in open chaotic microwave billiards in the regime of weakly overlapping resonances

5. The Quantum Billiard and its Simulation SFB 634 – C4: Quantum Chaos Shape of the billiard implies chaotic dynamics

6. 2D microwave cavity: h z < min /2quantum billiard Helmholtz equation and Schrödinger equation are equivalent in 2D. The motion of the quantum particle in its potential can be simulated by electromagnetic waves inside a two-dimensional microwave resonator. Schrödinger  Helmholtz SFB 634 – C4: Quantum Chaos

7. 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 into the walls is modelled by additive channels Compound Nucleus A+a B+b C+c D+d rf power in rf power out   SFB 634 – C4: Quantum Chaos Microwave Resonator as a Model for the Compound Nucleus

8. Scattering matrix for both scattering processes RMT description: replace Ĥ by a matrix for systems Microwave billiard Compound-nucleus reactions resonator Hamiltonian coupling of resonator states to antenna states and to the walls nuclear Hamiltonian coupling of quasi-bound states to channel states  Ĥ   Ŵ  Ŝ(E) =  - 2  i Ŵ T (E  - Ĥ + i  ŴŴ T ) -1 Ŵ SFB 634 – C4: Quantum Chaos Scattering Matrix Description GOE T-inv GUE T-noninv Experiment: complex S-matrix elements

9.  overlapping resonances  for  /D>1  Ericson fluctuations isolated resonances for  /D<<1 atomic nucleus ρ ~ exp(E 1/2 ) microwave cavity ρ ~ f SFB 634 – C4: Quantum Chaos Excitation Spectra Universal description of spectra and fluctuations: Verbaarschot, Weidenmüller + Zirnbauer (1984)

10. Regime of isolated resonances Г /D small Resonances: eigenvalues Overlapping resonances Г /D ~ 1 Fluctuations: Г coh Correlation function: SFB 634 – C4: Quantum Chaos Spectra and Correlation of S-Matrix Elements

11. Ericson fluctuations (1960): Correlation function is Lorentzian Measured 1964 for overlapping compound nuclear resonances Now observed in lots of different systems: molecules, quantum dots, laser cavities… Applicable for Г /D >> 1 and for many open channels only P. v. Brentano et al., Phys. Lett. 9, 48 (1964) SFB 634 – C4: Quantum Chaos Ericson’s Prediction for Γ > D

12. Height of cavity 15 mm Becomes 3D at 10.1 GHz Tilted stadium (Primack + Smilansky, 1994) GOE behaviour checked Measure full complex S-matrix for two antennas: S 11, S 22, S 12 SFB 634 – C4: Quantum Chaos Fluctuations in a Fully Chaotic Cavity with T-Invariance

13. Example: 8-9 GHz SFB 634 – C4: Quantum Chaos Spectra of S-Matrix Elements Frequency (GHz) |S| S 12  S 11  S 22 

14. SFB 634 – C4: Quantum Chaos Distributions of S-Matrix Elements Ericson regime: Re{S} and Im{S} should be Gaussian and phases uniformly distributed Clear deviations for Γ/D  1 but there exists no model for the distribution of S

15. SFB 634 – C4: Quantum Chaos Road to Analysis Of the Measured Fluctuations Problem: adjacent points in C(  ) are correlated Solution: FT of C(  )  uncorrelated Fourier coefficients C(t) Ericson (1965) Development: Non Gaussian fit and test procedure ~

16. Time domain Frequency domain  S 12   S 11   S 22  SFB 634 – C4: Quantum Chaos Fourier Transform vs. Autocorrelation Function Example 8-9 GHz

17. Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 for arbitrary Г /D : VWZ-integral: 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 C = C(T i, D ;  ) Transmission coefficients Average level distance SFB 634 – C4: Quantum Chaos Exact RMT Result for GOE systems

18. SFB 634 – C4: Quantum Chaos Corollary: Hauser-Feshbach Formula For Γ>>D : Distribution of S-matrix elements yields Over the whole measured frequency range 1 W > 2 in accordance with VWZ

19. SFB 634 – C4: Quantum Chaos What Happens in the Region of 3D Modes? VWZ curve in C(t) progresses through the cloud of points but it passes too high  GOF test rejects VWZ This behaviour is clearly visible in C(  ) Behaviour can be modelled through ~

20. SFB 634 – C4: Quantum Chaos Distribution of Fourier Coefficients Distributions are Gaussian with the same variances Remember: Measured S-matrix elements were non-Gaussian This still remains to be understood

21. Induced Time-Reversal Symmetry Breaking (TRSB) in Billiards T-symmetry breaking caused by a magnetized ferrite Coupling of microwaves to the ferrite depends on the direction a b S ab S ba ab Principle of detailed balance: Principle of reciprocity: SFB 634 – C4: Quantum Chaos ab F

22. Search for Time-Reversal Symmetry Breaking in Nuclei SFB 634 – C4: Quantum Chaos

23. TRSB in the Region of Overlapping Resonances ( Γ  D ) Antenna 1 and 2 in a 2D tilted stadium billiard Magnetized ferrite F in the stadium Place an additional Fe - scatterer into the stadium and move it up to 12 different positions in order to improve the statistical significance of the data sample  distinction between GOE and GUE behaviour becomes possible 1 2 F

24. Violation of Reciprocity Clear violation of reciprocity in the regime of Γ/D  1 S 12 SFB 634 – C4: Quantum Chaos

25. Quantification of Reciprocity Violation The violation of reciprocity reflects degree of TRSB Definition of a contrast function Quantification of reciprocity violation via Δ SFB 634 – C4: Quantum Chaos

26. Magnitude and Phase of Δ Fluctuate SFB 634 – C4: Quantum Chaos  B  200 mT  B  0 mT: no TRSB

27. S-Matrix Fluctuations and RMT SFB 634 – C4: Quantum Chaos Pure GOE  VWZ 1984 Pure GUE  FSS (Fyodorov, Savin + Sommers) 2005 V (Verbaarschot) 2007 Partial TRSB  analytical model under development (based on Pluhař, Weidenmüller, Zuk + Wegner, 1995) RMT  Full T symmetry breaking sets in experimentally already for λ  α/D  1 GOE GUE

28. Crosscorrelation between S 12 and S 21 at  = 0 SFB 634 – C4: Quantum Chaos C(S 12, S 21 ) = Data: TRSB is incomplete  mixed GOE / GUE system { 1 for GOE 0 for GUE * *

29. Test of VWZ and FSS / V Models SFB 634 – C4: Quantum Chaos VWZ FSS/V Autocorrelation functions of S-matrix fluctuations can be described by VWZ for weak TRSB and by FSS / V for strong TRSB

30. First Approach towards the TRSB Matrix Element based on RMT SFB 634 – C4: Quantum Chaos GOE GUE RMT  maximal observed T-symmetry breaking Full T-breaking already sets in for α  D

31. Determination of the rms value of T-breaking matrix element SFB 634 – C4: Quantum Chaos α (MHz)

32. SFB 634 – C4: Quantum Chaos Summary Investigated a chaotic T-invariant microwave resonator (i.e. a GOE system) in the regime of weakly overlapping resonances ( Γ  D ) Distributions of S-matrix elements are not Gaussian However, distribution of the 2400 uncorrelated Fourier coefficients of the scattering matrix is Gaussian Data are limited by rather small FRD errors, not by noise Data were used to test VWZ theory of chaotic scattering and the predicted non-exponential decay in time of resonator modes and the frequency dependence of the elastic enhancement factor are confirmed The most stringend test of the theory yet uses this large number of data points and a goodness-of-fit test

33. SFB 634 – C4: Quantum Chaos Summary ctd. Investigated furthermore a chaotic T-noninvariant microwave resonator (i.e. a GUE system) in the regime of weakly overlapping resonances Principle of reciprocity is strongly violated ( S ab ≠ S ab ) Data show, however, that TRSB is incomplete  mixed GOE / GUE system Data were subjected to tests of VWZ theory (GOE) and FFS / V theory (GUE) of chaotic scattering S-matrix fluctuations are described in spectral regions of weak TRSB by VWZ and for strong TRSB by FSS / V Analytical model for partial TRSB is under development First approach using RMT shows that full TRSB sets already in when the symmetry breaking matrix element is of the order of the mean level spacing of the overlapping resonances