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

SCUOLA INTERNAZIONALE DI FISICA “fermi" Varenna sul lago di como - 1965. Dr. h.c. Oriol Bohigas TU Darmstadt 2001. Chaotic Scattering in Microwave Billiards. Orsay 2008. BGS conjecture, quantum billiards and microwave resonators

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

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

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

  3. Chaotic Scattering in Microwave Billiards Orsay 2008 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 SFB 634 – C4: Quantum Chaos

  4. Conjecture of Bohigas, Giannoni + Schmit (1984) 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, Σ2, Δ3,...) 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 SFB 634 – C4: Quantum Chaos

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

  6. Schrödinger Helmholtz quantum billiard 2D microwave cavity: hz < min/2 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. SFB 634 – C4: Quantum Chaos

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

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

  9. Excitation Spectra atomic nucleus microwave cavity overlapping resonances • for G/D>1 • Ericson fluctuations isolated resonances for G/D<<1 ρ ~ exp(E1/2) ρ ~ f Universal description of spectra and fluctuations: Verbaarschot, Weidenmüller + Zirnbauer (1984) SFB 634 – C4: Quantum Chaos

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

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

  12. Fluctuations in a Fully Chaotic Cavity with T-Invariance Tilted stadium (Primack + Smilansky, 1994) • Height of cavity 15 mm • Becomes 3D at 10.1 GHz GOE behaviour checked Measure full complex S-matrix for two antennas: S11, S22, S12 SFB 634 – C4: Quantum Chaos

  13. Spectra of S-Matrix Elements Example: 8-9 GHz S12 |S| S11 S22 Frequency (GHz) SFB 634 – C4: Quantum Chaos

  14. 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 SFB 634 – C4: Quantum Chaos

  15. 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 ~ SFB 634 – C4: Quantum Chaos

  16. Fourier Transform vs. Autocorrelation Function Time domain Frequency domain Example 8-9 GHz  S12  S11  S22 SFB 634 – C4: Quantum Chaos

  17. Exact RMT Result for GOE systems Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 for arbitraryГ/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 SFB 634 – C4: Quantum Chaos

  18. Corollary: Hauser-Feshbach Formula For Γ>>D: Distribution of S-matrix elements yields Over the whole measured frequency range 1 < f < 10 GHz we find 3.5 > W > 2 in accordance with VWZ SFB 634 – C4: Quantum Chaos

  19. 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 SFB 634 – C4: Quantum Chaos

  20. 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 SFB 634 – C4: Quantum Chaos

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

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

  23. TRSB in the Region of Overlapping Resonances (ΓD) 2 1 F • 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 SFB 634 – C4: Quantum Chaos

  24. Violation of Reciprocity S12 S12 Clear violation of reciprocity in the regime of Γ/D  1 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  B  200 mT B  0 mT:no TRSB  SFB 634 – C4: Quantum Chaos

  27. GOE GUE S-Matrix Fluctuations and RMT 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 SFB 634 – C4: Quantum Chaos

  28. * Crosscorrelation between S12 and S21 at = 0 { 1 for GOE0 for GUE * C(S12, S21) = Data: TRSB is incomplete  mixed GOE / GUE system SFB 634 – C4: Quantum Chaos

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

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

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

  32. 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 SFB 634 – C4: Quantum Chaos

  33. 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 (Sab ≠ Sab) 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 SFB 634 – C4: Quantum Chaos

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