Chaotic Scattering of Microwaves in Billiards: Induced ...ccm08/Abstract/richter.pdf · • Induced...
Transcript of Chaotic Scattering of Microwaves in Billiards: Induced ...ccm08/Abstract/richter.pdf · • Induced...
• 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
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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
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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
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Excitation Spectra
Search for Time-Reversal Symmetry Breaking in Nuclei
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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• •
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Isolated Resonances - Setup
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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
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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
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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
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• 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)
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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
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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
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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
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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
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Comparison: Experiment - VWZ
Example 8-9 GHz
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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
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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
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Magnitude and Phase of Δ Fluctuate
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B 200 mT
B 0 mT:no TRSB
Crosscorrelation between S12 and S21 at ε = 0
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• C(S12, S21*) =
• Data: TRSB is incomplete
{1 for GOE0 for GUE
S-Matrix Fluctuations and RMT
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• 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
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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