• 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. SchäferH.L. Harney, J.J.M. Verbaarschot, H.A. Weidenmüller

Chaotic Scattering of Microwaves in Billiards:Induced Time-Reversal Symmetry Breaking and

Fluctuations in GOE and GUE Systems 2008

SFB 634 – C4: Quantum Chaos

2ΔxΔp h

≥⋅

The Quantum Billiard and its Simulation

SFB 634 – C4: Quantum Chaos

( ) 02 =+Δ zEk

2D microwave cavity: hz < λmin/2

( ) 02 =Ψ+Δ k

quantum billiard

cfk π2

=22h

mEk =

Helmholtz equation and Schrödinger equation are equivalent in 2D. The motion of the quantum particle in its potential can besimulated by electromagnetic waves inside a two-dimensional

microwave resonator.

Schrödinger ↔ Helmholtz

SFB 634 – C4: Quantum Chaos

• Microwave power is emitted into the resonator by antennaand 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

CompoundNucleusA+a B+b

C+c

D+d

rf powerin

rf powerout

SFB 634 – C4: Quantum Chaos

Microwave Resonator as a Model for the Compound Nucleus

• Scattering matrix for both scattering processes

• RMT description: replace Ĥ by a matrix for systems

Microwave billiardCompound-nucleusreactions

resonator Hamiltonian

coupling of resonatorstates to antenna statesand to the walls

nuclear Hamiltonian

coupling of quasi-boundstates to channel states

← Ĥ →

← Ŵ →

Ŝ(E) = - 2πi ŴT (E - Ĥ + iπ ŴŴT)-1 Ŵ

SFB 634 – C4: Quantum Chaos

Scattering Matrix Description

GOE T-invGUE T-noninv

overlapping resonances for Γ/D>1 Ericson fluctuations

isolated resonancesfor Γ/D<<1

atomic nucleus

ρ ~ exp(E1/2)

Mmicrowave cavity

ρ ~ f

M

SFB 634 – C4: Quantum Chaos

Excitation Spectra

Search for Time-Reversal Symmetry Breaking in Nuclei

SFB 634 – C4: Quantum Chaos

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

Sab

Sba

ab

• Principle of detailed balance:

• Principle of reciprocity:

a b• •

SFB 634 – C4: Quantum Chaos

Isolated Resonances - Setup

SFB 634 – C4: Quantum Chaos

Isolated Resonances - Singlets

0.002

0.004|S|

0

-π2.80 2.85 2.90

Frequency (GHz)

Arg(S)

• Reciprocity holds TRSB cannot be detected this way

• Sab• Sba

SFB 634 – C4: Quantum Chaos

Isolated Doublets of Resonances

• Violation of reciprocity due to interference of two resonances

0.005

0.010

0.015

|S|

+π0

-π2.650 2.675 2.700

Frequency (GHz)

Arg(S)

• Sab• Sba

SFB 634 – C4: Quantum Chaos

• Scattering matrix element

• Decomposition of effective Hamiltonian

• Ansatz for TRSB incorporating the FMR and its selective coupling to the microwaves

Scattering Matrix and TRSB

bWHWaiS effabab

ˆ)ˆ(ˆ2)( 1−+ −−= ωπδω

saeff HHH ˆˆˆ +=

⎟⎟⎠

⎞⎜⎜⎝

⎛

− 00

12

12a

a

HH

SFB 634 – C4: Quantum Chaos

•

TRSB Matrix Element

TiBTBiBH Ma

/)(2)(

0

2

12 −−⋅⋅⋅⋅=

ωωωλπ

couplingstrength

externalfield

spinrelaxation

time

magneticsusceptibility

• Fit parameters: andλ ω

SFB 634 – C4: Quantum Chaos

T-Violating Matrix Element

10

20

Abs

(Ha 12

) (M

Hz)

2π

π

00 20 40 60 80

Magnetic field (mT)

Arg

(Ha 12

)

• T-violating matrix element shows resonance like structure• Successful description of dependence on magnetic field

SFB 634 – C4: Quantum Chaos

Relative Strength of T-Violation

SFB 634 – C4: Quantum Chaos

• Compare: TRSB matrix element to the energy difference of two eigenvalues of the T-invariant system

aH12

ss

a

EEH

21

122−

=ξ

• Regime of isolated resonances

• Г/D small

• Resonances: eigenvalues

• Overlapping resonances

• Г/D ~ 1

• Fluctuations: Гcoh

Correlation function: )()()()()( εεε +−+= ∗∗ fSfSfSfSC

SFB 634 – C4: Quantum Chaos

Spectra and Autocorrelation Function

• Ericson fluctuations (1960):

22

22)(

εε

+ΓΓ

∝coh

cohC

• 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

• Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 for arbitrary Г/D :

• VWZ-integral:

• Rigorous test of VWZ: isolated resonances, i.e. Г << D

• Our goal: test VWZ in the intermediate regime, i.e. Г/D 1

C = C(Ti, D ; ε)

Transmission coefficients Average level distance

SFB 634 – C4: Quantum Chaos

Exact RMT Result for GOE systems

≈

• 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: S11, S22, S12

SFB 634 – C4: Quantum Chaos

Experimental Realisation in a Fully Chaotic Cavity

Example: 8-9 GHz

SFB 634 – C4: Quantum Chaos

Excitation Functions of S-Matrix Elements

Frequency (GHz)

|S|

S12

S11

S22

SFB 634 – C4: Quantum Chaos

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

~

Time domain Frequency domain

S12

S11

S22

SFB 634 – C4: Quantum Chaos

Comparison: Experiment - VWZ

Example 8-9 GHz

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(ε)

• Behaviour can be modelled through ⎟⎟⎠

⎞⎜⎜⎝

⎛= GOE

GOE

HH

H2

1

00ˆ

~

SFB 634 – C4: Quantum Chaos

TRSB in the Region of Overlapping Resonances

• 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

12

F

Violation of Detailed Balance for Overlapping Resonances

S12

SFB 634 – C4: Quantum Chaos

S21

Quantification of Reciprocity Violation

• The violation of reciprocity reflects degree of TRSB

• Definition of a contrast function

• Quantification of reciprocity violation via Δ

|||| baab

baab

SSSS

+−

=Δ

SFB 634 – C4: Quantum Chaos

Magnitude and Phase of Δ Fluctuate

SFB 634 – C4: Quantum Chaos

B 200 mT

B 0 mT:no TRSB

Crosscorrelation between S12 and S21 at ε = 0

SFB 634 – C4: Quantum Chaos

• C(S12, S21*) =

• Data: TRSB is incomplete

{1 for GOE0 for GUE

S-Matrix Fluctuations and RMT

SFB 634 – C4: Quantum Chaos

• Pure GOE VWZ description 1984

• Pure GUE V description 2007

• Partial TRSB no analytical model

• RMT as HiHH ˆˆˆ α+=

10

==

αα GOE

GUE

Test of VWZ and V Models

SFB 634 – C4: Quantum Chaos

VWZ VWZ VWZ VWZV

First approach towards RMT-description of experimental results

SFB 634 – C4: Quantum Chaos

as HiHH ˆˆˆ α+=10

==

αα GOE

GUERMT

maximal T-breaking

?

Summary

SFB 634 – C4: Quantum Chaos

• 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

T inv

T non-inv

Γ/D << 1 Γ/D 1 Γ/D > 2

non exp decay

reciprocity det balance

violated

non exp decay exp decay

non exp decayreciprocity

det balanceviolated