Download Chaotic Scattering of Microwaves in Billiards: Induced Time

SCUOLA
INTERNAZIONALE
Varenna
sul
lago
DI
di
FISICA
como
-
1965
“fermi"
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 – C4: 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, 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
SFB 634 – C4: Quantum Chaos
The Quantum Billiard and its Simulation
Shape of the billiard implies
chaotic dynamics
Δp  Δx 

2
SFB 634 – C4: Quantum Chaos
Schrödinger  Helmholtz
quantum billiard
2D microwave cavity: hz < min/2
  k    0
  k 2 Ez  0
2mE
k
2
2f
k
c
2
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.
SFB 634 – C4: Quantum Chaos
Microwave Resonator as a Model for the Compound Nucleus
rf power rf power
in
out


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 – C4: Quantum Chaos
B+b
Scattering Matrix Description
• Scattering matrix for both scattering processes
Ŝ(E) =  - 2i ŴT (E - Ĥ + i ŴŴT)-1 Ŵ
Compound-nucleus
reactions
nuclear Hamiltonian
coupling of quasi-bound
states to channel states
Microwave billiard
Ĥ
resonator Hamiltonian
Ŵ
coupling of resonator
states to antenna states
and to the walls
complex S-matrix elements
• Experiment:
• RMT description: replace Ĥ by a
GOE
T-inv
matrix for
systems
GUE
T-noninv
SFB 634 – C4: Quantum Chaos
Excitation Spectra
atomic nucleus
microwave cavity


overlapping resonances
for G/D>1
Ericson fluctuations
isolated resonances
for G/D<<1
ρ ~ exp(E1/2)
ρ~f
• Universal description of spectra and fluctuations:
Verbaarschot, Weidenmüller + Zirnbauer (1984)
SFB 634 – C4: 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: C ( )  S ( f ) S  ( f   )  S ( f ) S  ( f   )
SFB 634 – C4: Quantum Chaos
Ericson’s Prediction for Γ > D
• Ericson fluctuations (1960):
2
Gcoh
C ( )  2
Gcoh   2
2
• 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 – C4: Quantum Chaos
Fluctuations in a Fully Chaotic Cavity with T-Invariance
• 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: S11, S22, S12
SFB 634 – C4: Quantum Chaos
Spectra of S-Matrix Elements
Example: 8-9 GHz
S11 
|S|
S12 
S22 
Frequency (GHz)
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
SFB 634 – C4: Quantum Chaos
Fourier Transform vs. Autocorrelation Function
Time domain
Frequency domain
Example 8-9 GHz
 S12 
 S11 
 S22 
SFB 634 – C4: 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 – C4: Quantum Chaos
Corollary: Hauser-Feshbach Formula

• For Γ>>D: Sab ( f ) Sab
( f )
iG TaTb
(1   ab )
  iG  Tc

c
• Distribution of S-matrix elements yields
W   S

fl 2
11
S
fl 2
22
1/ 2



fl 2
12
/S
2
• Over the whole measured frequency range 1 < f < 10 GHz we
find 3.5 > W > 2 in accordance with VWZ
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()
GOE

H
• Behaviour can be modelled through Hˆ   1
 0
SFB 634 – C4: Quantum Chaos
0 

GOE 
H2 
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 – C4: 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 – C4: Quantum Chaos
b
Search for Time-Reversal Symmetry Breaking in Nuclei
SFB 634 – C4: Quantum Chaos
TRSB in the Region of Overlapping Resonances (ΓD)
1
2
F
• 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
SFB 634 – C4: Quantum Chaos
Violation of Reciprocity
S12
S12
• Clear violation of reciprocity in the regime of Γ/D  1
SFB 634 – C4: Quantum Chaos
Quantification of Reciprocity Violation
• The violation of reciprocity reflects degree of TRSB
• Definition of a contrast function

Sab  Sba
| Sab |  | Sba |
• Quantification of reciprocity violation via Δ
SFB 634 – C4: Quantum Chaos
Magnitude and Phase of Δ Fluctuate
 B  200 mT

SFB 634 – C4: Quantum Chaos
B  0 mT:
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)
Hˆ  Hˆ s  i Hˆ a
  0 GOE
  1 GUE
• Full T symmetry breaking sets in experimentally already
for λ  α/D 1
SFB 634 – C4: Quantum Chaos
Crosscorrelation between S12 and S*21 at  = 0
{
1 for GOE
0 for GUE
• Data: TRSB is incomplete  mixed GOE / GUE system
*)=
• C(S12, S21
SFB 634 – C4: Quantum Chaos
Test of VWZ and FSS / V Models
VWZ
VWZ
FSS/V VWZ VWZ
• Autocorrelation functions of S-matrix fluctuations can be described
by VWZ for weak TRSB and by FSS / V for strong TRSB
SFB 634 – C4: Quantum Chaos
First Approach towards the TRSB Matrix Element based on RMT
maximal observed T-symmetry
breaking
• RMT 
Hˆ  Hˆ s  i Hˆ a
  0 GOE
  1 GUE
• Full T-breaking already sets in for α  D
SFB 634 – C4: Quantum Chaos
α (MHz)
Determination of the rms value of T-breaking matrix element
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
SFB 634 – C4: Quantum Chaos