Chaotic Scattering of Microwaves in Billiards: Induced Time-Reversal Symmetry Breaking and Fluctuations in GOE and GUE Systems 2008

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1 Chaotic Scattering of Microwaves in Billiards: Induced Time-Reversal Symmetry Breaking and Fluctuations in GOE and GUE Systems 2008 Quantum billiards and microwave resonators as a model of the compound nucleus Induced time-reversal symmetry breaking in billiards - isolated resonances Fluctuation properties of S-matrix elements - overlapping resonances Test of models based on RMT for GOE and GUE systems Supported by DFG within SFB 634 S. Bittner, B. Dietz, T. Friedrich, M. Miski-Oglu, P. Oria-Iriarte, A. R., F. Schäfer H.L. Harney, J.J.M. Verbaarschot, H.A. Weidenmüller

2 The Quantum Billiard and its Simulation Δp Δx h 2

3 Schrödinger Helmholtz quantum billiard 2D microwave cavity: h z < λ min /2 ( 2 Δ + k ) Ψ = 0 ( 2 Δ + k ) E = 0 z k = 2mE h 2 k = 2πf c 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.

4 Microwave Resonator as a Model for the Compound Nucleus rf power in rf power out C+c 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 into the walls is modelled by additive channels

5 Scattering Matrix Description Scattering matrix for both scattering processes Ŝ(E) = -2πi Ŵ T (E - Ĥ + iπ ŴŴ T ) -1 Ŵ Compound-nucleus reactions nuclear Hamiltonian coupling of quasi-bound states to channel states Ĥ Ŵ Microwave billiard resonator Hamiltonian coupling of resonator states to antenna states and to the walls RMT description: replace Ĥ by a GOE GUE matrix for T-inv systems T-noninv

6 atomic nucleus M Excitation Spectra overlapping resonances for Γ/D>1 Ericson fluctuations microwave cavity M isolated resonances for Γ/D<<1 ρ ~ exp(e 1/2 ) ρ ~ f

7 Search for Time-Reversal Symmetry Breaking in Nuclei

8 Induced Time-Reversal Symmetry Breaking (TRSB) in Billiards T-symmetry breaking caused by a magnetized ferrite a b Coupling of microwaves to the ferrite depends on the direction a b S ab b a S ba Principle of detailed balance: Principle of reciprocity:

9 Isolated Resonances - Setup

10 Isolated Resonances - Singlets S S ab S ba Arg(S) 0 -π Frequency (GHz) Reciprocity holds TRSB cannot be detected this way

11 Isolated Doublets of Resonances S S ab S ba Arg(S) +π 0 -π Frequency (GHz) Violation of reciprocity due to interference of two resonances

12 Scattering Matrix and TRSB Scattering matrix element S ab i a Wˆ + ( ω) = δ 2π ( ω ab Hˆ eff ) 1 Wˆ b Decomposition of effective Hamiltonian ˆ eff a = Hˆ + H Hˆ s 0 H a H12 0 a 12 Ansatz for TRSB incorporating the FMR and its selective coupling to the microwaves

13 TRSB Matrix Element H a 12 ( B) = iπ 2 λ B T 2 ωm ω ( B) ω i 0 / T coupling strength external field spin relaxation time magnetic susceptibility Fit parameters: λ and ω

14 T-Violating Matrix Element 20 Abs(H a 12 ) (MHz) 10 Arg(H a 12 ) 2π π Magnetic field (mt) T-violating matrix element shows resonance like structure Successful description of dependence on magnetic field

15 Relative Strength of T-Violation a H 12 Compare: TRSB matrix element to the energy difference of two eigenvalues of the T-invariant system ξ = 2H E s 1 a 12 E s 2

16 Spectra and Autocorrelation Function Regime of isolated resonances Г/D small Resonances: eigenvalues Overlapping resonances Г/D ~ 1 Fluctuations: Г coh Correlation function: C( ε ) = S( f ) S ( f + ε ) S ( f ) S ( f + ε )

17 Ericson s Prediction Ericson fluctuations (1960): C( ε ) 2 Γ Γ 2 coh 2 coh + 2 ε Correlation function is Lorentzian Measured 1964 for overlapping compound nuclear resonances Now observed in lots of different systems: molecules, quantum dots, laser cavities P. v. Brentano et al., Phys. Lett. 9, 48 (1964) Applicable for Г/D >> 1 and for many open channels only

18 Exact RMT Result for GOE systems Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 for arbitrary Г/D : VWZ-integral: C = C(T i, D ; ε) Transmission coefficients Average level distance Rigorous test of VWZ: isolated resonances, i.e. Г << D Our goal: test VWZ in the intermediate regime, i.e. Г/D 1

19 Experimental Realisation in a Fully Chaotic Cavity 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: S 11, S 22, S 12

20 Excitation Functions of S-Matrix Elements Example: 8-9 GHz S 12 S 11 S S 22 Frequency (GHz)

21 Road to Analysis Problem: adjacent points in C(ε) are correlated ~ Solution: FT of C(ε) uncorrelated Fourier coefficients C(t) Ericson (1965) Development: Non Gaussian fit and test procedure

22 Comparison: Experiment - VWZ Time domain Example 8-9 GHz Frequency domain S 12 S 11 S 22

23 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(ε) Behaviour can be modelled through Hˆ H1 0 GOE 0 = GOE H 2

24 TRSB in the Region of Overlapping Resonances 1 F 2 Antenna 1 and 2 Place a magnetized ferrite F into tilted stadium billiard Place an additional Fe - scatterer into the stadium and move it into different positions in order to improve the statistical significance of the data sample distinction between GOE and GUE becomes possible

25 Violation of Detailed Balance for Overlapping Resonances S 12 S 21

26 Quantification of Reciprocity Violation The violation of reciprocity reflects degree of TRSB Definition of a contrast function Δ = Sab Sba Sab + Sba Quantification of reciprocity violation via Δ

27 Magnitude and Phase of Δ Fluctuate B 200 mt B 0 mt: no TRSB

28 Crosscorrelation between S 12 and S 21 at ε = 0 { 1 for GOE C(S 12, S 21 *) = 0 for GUE Data: TRSB is incomplete

29 S-Matrix Fluctuations and RMT Pure GOE VWZ description 1984 Pure GUE V description 2007 Partial TRSB no analytical model RMT Hˆ = Hˆ s + iα Hˆ a α = 0 α = 1 GOE GUE

30 Test of VWZ and V Models VWZ VWZ V VWZ VWZ

31 First approach towards RMT-description of experimental results maximal T-breaking RMT Hˆ = Hˆ s + iα Hˆ a α = 0 α = 1 GOE GUE

32 Summary Open microwave resonators are excellent model systems to test fluctuation properties of the compound nucleus RMT based models (VWZ, V) for GOE and GUE can be tested with high precision Γ/D << 1 Γ/D 1 Γ/D > 2 T inv non exp decay non exp decay exp decay T non-inv reciprocity det balance violated non exp decay reciprocity det balance violated?