SCUOLA INTERNAZIONALE DI FISICA fermi Varenna sul lago

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 • 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 Tnoninvariant (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 – C 4: Quantum Chaos

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, 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 SFB 634 – C 4: Quantum Chaos

The Quantum Billiard and its Simulation Shape of the billiard implies chaotic dynamics SFB 634 – C 4: Quantum Chaos

Schrödinger Helmholtz quantum billiard 2 D microwave cavity: hz < min/2 Helmholtz equation and Schrödinger equation are equivalent in 2 D. The motion of the quantum particle in its potential can be simulated by electromagnetic waves inside a two-dimensional microwave resonator. SFB 634 – C 4: Quantum Chaos

Microwave Resonator as a Model for the Compound Nucleus C+ c rf power in out D+ d A+a 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 – C 4: Quantum Chaos B+b

Scattering Matrix Description • Scattering matrix for both scattering processes Ŝ(E) = - 2 pi ŴT (E - Ĥ + ip ŴŴT)-1 Ŵ Compound-nucleus reactions Ĥ nuclear Hamiltonian coupling of quasi-bound states to channel states • Experiment: Microwave billiard Ŵ resonator Hamiltonian coupling of resonator states to antenna states and to the walls complex S-matrix elements • RMT description: replace Ĥ by a GOE matrix for T-inv systems GUE T-noninv SFB 634 – C 4: Quantum Chaos

Excitation Spectra atomic nucleus microwave cavity overlapping resonances for G/D>1 Ericson fluctuations isolated resonances for G/D<<1 ρ ~ exp(E 1/2) ρ~f • Universal description of spectra and fluctuations: Verbaarschot, Weidenmüller + Zirnbauer (1984) SFB 634 – C 4: Quantum Chaos

Spectra and Correlation of S-Matrix Elements • Regime of isolated resonances • Overlapping resonances • Г/D small • Г/D ~ 1 • Resonances: eigenvalues • Fluctuations: Гcoh Correlation function: SFB 634 – C 4: Quantum Chaos

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 – C 4: Quantum Chaos

Fluctuations in a Fully Chaotic Cavity with T-Invariance • Tilted stadium (Primack + Smilansky, 1994) • Height of cavity 15 mm • Becomes 3 D at 10. 1 GHz • GOE behaviour checked • Measure full complex S-matrix for two antennas: S 11, S 22, S 12 SFB 634 – C 4: Quantum Chaos

Spectra of S-Matrix Elements Example: 8 -9 GHz S 11 |S| S 12 S 22 Frequency (GHz) SFB 634 – C 4: 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 SFB 634 – C 4: 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 SFB 634 – C 4: Quantum Chaos

Fourier Transform vs. Autocorrelation Function Time domain Frequency domain Example 8 -9 GHz S 12 S 11 S 22 SFB 634 – C 4: Quantum Chaos

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 SFB 634 – C 4: Quantum Chaos

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 – C 4: Quantum Chaos

What Happens in the Region of 3 D 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 – C 4: 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 SFB 634 – C 4: Quantum Chaos

Induced Time-Reversal Symmetry Breaking (TRSB) in Billiards F • T-symmetry breaking caused by a magnetized ferrite • • a b • Coupling of microwaves to the ferrite depends on the direction a Sab b a Sba • Principle of detailed balance: • Principle of reciprocity: SFB 634 – C 4: Quantum Chaos b

Search for Time-Reversal Symmetry Breaking in Nuclei SFB 634 – C 4: Quantum Chaos

TRSB in the Region of Overlapping Resonances (Γ D) 1 2 F • Antenna 1 and 2 in a 2 D 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 – C 4: Quantum Chaos

Violation of Reciprocity S 12 • Clear violation of reciprocity in the regime of Γ/D 1 SFB 634 – C 4: Quantum Chaos

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 – C 4: Quantum Chaos

Magnitude and Phase of Δ Fluctuate B 200 m. T SFB 634 – C 4: Quantum Chaos B 0 m. T: no TRSB

S-Matrix Fluctuations and RMT • Pure GOE VWZ 1984 • Pure GUE FSS (Fyodorov, Savin + Sommers) 2005 V (Verbaarschot) 2007 • Partial TRSB • RMT analytical model under development (based on Pluhař, Weidenmüller, Zuk + Wegner, 1995) GOE GUE • Full T symmetry breaking sets in experimentally already for λ α/D 1 SFB 634 – C 4: Quantum Chaos

Crosscorrelation between S 12 and S*21 at = 0 { 1 for GOE 0 for GUE • Data: TRSB is incomplete mixed GOE / GUE system *)= • C(S 12, S 21 SFB 634 – C 4: Quantum Chaos

Test of VWZ and FSS / V Models 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 – C 4: Quantum Chaos

First Approach towards the TRSB Matrix Element based on RMT maximal observed T-symmetry breaking GOE • RMT GUE • Full T-breaking already sets in for α D SFB 634 – C 4: Quantum Chaos

α (MHz) Determination of the rms value of T-breaking matrix element SFB 634 – C 4: 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 theory yet uses this large number of data points and a goodness-of-fit test SFB 634 – C 4: 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 (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 – C 4: Quantum Chaos