scuola internazionale di fisica “fermi" varenna sul lago di como - 1965
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SCUOLA INTERNAZIONALE DI FISICA “fermi"Varenna sul lago di como - 1965
Dr. h.c. Oriol BohigasTU Darmstadt 2001
• 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 (ΓD) 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. SchäferH.L. Harney, J.J.M. Verbaarschot, H.A. Weidenmüller
Chaotic Scattering in Microwave BilliardsOrsay 2008
SFB 634 – C4: Quantum Chaos
Conjecture of Bohigas, Giannoni + Schmit (1984)
SFB 634 – C4: Quantum Chaos
• 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
2ΔxΔp
The Quantum Billiard and its Simulation
SFB 634 – C4: Quantum Chaos
Shape of the billiard implies chaotic dynamics
02 zEk
2D microwave cavity: hz < min/2
02 k
quantum billiard
cf
k2
2
2
mE
k
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.
Schrödinger Helmholtz
SFB 634 – C4: Quantum Chaos
• 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
Compound NucleusA+a B+b
C+cD+d
rf power in
rf power out
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-nucleus reactions
resonator Hamiltonian
coupling of resonator states to antenna states and to the walls
nuclear Hamiltonian
coupling of quasi-boundstates to channel states
Ĥ
Ŵ
Ŝ(E) = - 2i ŴT (E - Ĥ + iŴŴT)-1 Ŵ
SFB 634 – C4: Quantum Chaos
Scattering Matrix Description
GOE T-invGUE T-noninv
• Experiment: complex S-matrix elements
overlapping resonancesfor/D>1 Ericson fluctuations
isolated resonancesfor /D<<1
atomic nucleus
ρ ~ exp(E1/2)
microwave cavity
ρ ~ f
SFB 634 – C4: Quantum Chaos
Excitation Spectra
• Universal description of spectra and fluctuations: Verbaarschot, Weidenmüller + Zirnbauer (1984)
• Regime of isolated resonances
• Г/D small
• Resonances: eigenvalues
• Overlapping resonances
• Г/D ~ 1
• Fluctuations: Гcoh
Correlation function: )()()()()( fSfSfSfSC
SFB 634 – C4: Quantum Chaos
Spectra and Correlation of S-Matrix Elements
• Ericson fluctuations (1960):
22
22
)(
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 for Γ > D
• 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
Fluctuations in a Fully Chaotic Cavity with T-Invariance
Example: 8-9 GHz
SFB 634 – C4: Quantum Chaos
Spectra of S-Matrix Elements
Frequency (GHz)
|S|
S12
S11
S22
SFB 634 – C4: 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 – C4: 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
~
Time domain Frequency domain
S12
S11
S22
SFB 634 – C4: Quantum Chaos
Fourier Transform vs. Autocorrelation Function
Example 8-9 GHz
• Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 for arbitrary Г/D :
• VWZ-integral:
• 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
C = C(Ti, D ; )
Transmission coefficients Average level distance
SFB 634 – C4: Quantum Chaos
Exact RMT Result for GOE systems
SFB 634 – C4: 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
cc
baababab T
TT
i
ifSfS
)1()()(
2/2
12
2/12
22
2
11
flflfl SSSW
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()
• Behaviour can be modelled through
GOE
GOE
H
HH
2
1
0
0ˆ
~
SFB 634 – C4: 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
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:
SFB 634 – C4: Quantum Chaos
• •
a b
F
Search for Time-Reversal Symmetry Breaking in Nuclei
SFB 634 – C4: Quantum Chaos
SFB 634 – C4: Quantum Chaos
TRSB in the Region of Overlapping Resonances (ΓD)
• 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
12
F
Violation of Reciprocity
• Clear violation of reciprocity in the regime of Γ/D 1
S12
S12
SFB 634 – C4: 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
|||| baab
baab
SS
SS
SFB 634 – C4: Quantum Chaos
Magnitude and Phase of Δ Fluctuate
SFB 634 – C4: Quantum Chaos
B 200 mT
B 0 mT:no TRSB
S-Matrix Fluctuations and RMT
SFB 634 – C4: Quantum Chaos
• Pure GOE VWZ 1984
• Pure GUE FSS (Fyodorov, Savin + Sommers) 2005V (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
as HiHH ˆˆˆ
1
0
GOE
GUE
Crosscorrelation between S12 and S21 at = 0
SFB 634 – C4: Quantum Chaos
• C(S12, S21) =
• Data: TRSB is incomplete mixed GOE / GUE system
{1 for GOE0 for GUE
*
*
Test of VWZ and FSS / V Models
SFB 634 – C4: Quantum Chaos
VWZ VWZ VWZ VWZFSS/V
• Autocorrelation functions of S-matrix fluctuations can be described by VWZ for weak TRSB and by FSS / V for strong TRSB
First Approach towards the TRSB Matrix Element based on RMT
SFB 634 – C4: Quantum Chaos
as HiHH ˆˆˆ 1
0
GOE
GUE• RMT
maximal observed T-symmetry breaking
• Full T-breaking already sets in for α D
Determination of the rms value of T-breaking matrix element
SFB 634 – C4: Quantum Chaos
α (M
Hz)
SFB 634 – C4: 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 the theory yet uses this large number of data points and a goodness-of-fit test
SFB 634 – C4: 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
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