Chaotic Scattering in Microwave Billiards: Isolated and Overlapping Resonances

1
Chaotic Scattering in Microwave
BilliardsIsolated and Overlapping Resonances

• Time-reversal invariant (GOE) systems
• Time-reversal partially non-invariant (GOE/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.
Verbaarschot, H. A. Weidenmüller
2
The Quantum Billiard and its Simulation
Shape of the billiard implies chaotic dynamics
3
Schrödinger ? Helmholtz
2D microwave cavity hz lt ?min/2
quantum billiard
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
h

• 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
Excitation Spectra
atomic nucleus
microwave cavity

• overlapping resonances
• for G/Dgt1
• Ericson fluctuations

isolated resonances for G/Dltlt1
? exp(E1/2)
? f

• Universal description of spectra and
  fluctuations Verbaarschot, Weidenmüller
  Zirnbauer (1984)

6
Transmitted Power
7
Typical Transmission Spectrum

• Transmission measurements relative power from
  antenna a ? b

8
Scattering Matrix Description

• Scattering matrix for both scattering processes

Compound-nucleus reactions
Microwave billiard
? H ?
nuclear Hamiltonian coupling of
quasi-bound states to channel states
resonator Hamiltonian coupling of resonator
states to antenna states and to the walls
? W ?

• Experiment
  complex S-matrix elements

GOE
T-inv

• RMT description replace H by a matrix
  for systems

GUE
T-noninv
9
Resonance Parameters

• Use eigenrepresentation of
• and obtain for a scattering system with isolated
  resonances
• a ? resonator ? b
• Here
  of eigenvalues of
• Partial widths fluctuate and
  total widths also

10
Typical Part of the Spectrum

• Determination of partial and total widths in
  reflection measurements
• Depth of resonance ? partial width width of
  resonance ? total width

11
Frequency Dependence of Widths

• Gµs ? RMT PT distributed

12
Distribution of Widths

• ²-distribution with ? degrees of freedom

with

• Porter-Thomas distribution

13
Width Distribution
14
Porter-Thomas Distribution of Partial Widths
15
Spectra and Correlation of S-Matrix Elements

• Regime of isolated resonances
• ?/D small
• Resonances eigenvalues

• Overlapping resonances
• ?/D 1
• Fluctuations ?corr

Correlation function
16
Exact RMT Result for GOE Systems

• Verbaarschot, Weidenmüller and Zirnbauer (VWZ)
  1984 for arbitrary ?/D
• VWZ-integral

C C(Ti, D ?)
Transmission coefficients
Average level distance

• Rigorous test of VWZ isolated resonances, i.e. ?
  ltlt D
• First test of VWZ in the intermediate regime,
  i.e. ?/D 1, with high statistical
  significance only achievable with microwave
  billiards
• Note nuclear cross section fluctuation
  experiments yield only S2

17
Autocorrelation and DecayIsolated Resonances
Alt et al., PRL 74, 62 (1995)
18
Ericsons Prediction for G gt D

• Ericson fluctuations (1960)

• Correlation function is Lorentzian
• Measured 1964 for overlapping
• compound nuclear resonances
• Now observed in lots of different
• systems molecules, quantum dots,
• laser cavities, microwave cavities

• Different theoretical approaches for G/D 1
• - Ericson ? energy and time domain
• - VWZ ? RMT
• - Blümel and Smilansky ? semiclassical

P. v. Brentano et al., Phys. Lett. 9, 48 (1964)
19
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
• B. Dietz et al., PRE 78, 055204(R) (2008).

20
Spectra of S-Matrix Elements
Example 8-9 GHz, G/D ? 0.2
S12 ?
S
S11 ?
S22 ?
Frequency (GHz)
21
Distribution of S11-Matrix Elementand Comparison
with RMT (FSS)
8-9 GHz, G/D ? 0.2
22
Distribution of S11-Matrix Elementand Comparison
with RMT (FSS)
22-23 GHz, G/D ? 0.9

• Distributions far away from Gaussian and
  uniformly distributed phases

23
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

24
Fourier Transform vs. Autocorrelation Function
Example 8-9 GHz
? S12 ?
? S11 ?
? S22 ?
Frequency domain
Time domain
25
Distribution of Fourier Coefficients

• Distributions are Gaussian with the same
  variances (Remember Measured S-matrix
  elements are non-Gaussian distributed) ?
  Expected distribution of
  

• Fit of VWZ to data determines expectation value
  ?k

26
Distribution of Coefficients

• The Fourier transformis not stationary
• Assumption of Gaussian distributed Fourier
  coefficients is correct

• GOF test minimize

27
Corollary Hauser-Feshbach Formula

• For GgtgtD

• Distribution of S-matrix elements yields

• Over the whole measured frequency range 1 lt f lt
  10 GHz we find 3.5 gt W gt 2 in accordance with
  VWZ
• Note for isolated resonances W 3

28
Summary

• Investigated a chaotic T-invariant microwave
  resonator (i.e. a GOE system) in the regime of
  isolated and weakly overlapping resonances (G ?
  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

29
Detailed Balance in Nuclear Reactions

• Search for Time-Reversal Symmetry Breaking (TRSB)
  in nuclear reactions

30
Detailed Balance in Nuclear Reactions

• Search for Time-Reversal Symmetry Breaking (TRSB)
  innuclear reactions ? upper limits

31
Induced Time-Reversal Symmetry Breaking (TRSB) in
Billiards

• T-symmetry breaking caused by a magnetized
  ferrite
• Ferrite features Ferromagnetic Resonance (FMR)
• Coupling of microwaves to the FMR depends on the
  direction a b

F
Sab
a
b
Sba

• Principle of detailed balance

• Principle of reciprocity

32
Scattering Matrix Description
Remember Scattering matrix formalism
T-invariantT-broken

• Investigation of systems
• ?
  replace H by a matrix
• Isolated resonances singlets and doublets ? H
  is 1D or 2D
• Overlapping resonances (GgtD)

GOEGUE
33
Isolated Resonances - Setup
34
Isolated Resonances - Singlets

• Reciprocity holds ? TRSB cannot be detected this
  way

35
Isolated Doublets of Resonances

• Violation of reciprocity due to interference of
  two resonances

36
Scattering Matrix and TRSB

• Scattering matrix element ( )
• Decomposition of effective Hamiltonian
• Ansatz for TRSB incorporating the FMR and its
  selective coupling to the microwaves

37
TRSB Matrix Element

spinrelaxationtime
magneticsusceptibility
couplingstrength
externalfield

• Fit parameters and

38
T-Violating Matrix Element

• T-violating matrix element shows resonance like
  structure

• Successful description of dependence on magnetic
  field

Dietz et al., PRL 98, 074103 (2007)
39
Relative Strength of T-Violation

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

40
Search for TRSB in NucleiEricson Regime
41
Search for TRSB in NucleiEricson Regime
42
Billiard for the Study of Induced TRSB in the
Regime G ? D

• 2D tilted stadium billiard ? chaotic dynamics
• Magnetized ferrite ? T-breaking

43
TRSB in the Region of Overlapping Resonances (G ?
D)
2
1
F

• Antenna 1 and 2 in a 2D tilted stadium billiard
• Magnetized ferrite F in the stadium
• Place an additional 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 behavior
  becomes possible

44
Violation of Reciprocity
S12 S21

• Clear violation of reciprocity in the regime of
  G/D ? 1

45
Quantification of Reciprocity Violation

• The violation of reciprocity reflects degree of
  TRSB
• Definition of a contrast function
• Quantification of reciprocity violation via ?

46
Magnitude and Phase of ? Fluctuate
? B ? 200 mT
B ? 0 mTno TRSB
?
47
S-Matrix Fluctuations and RMT

• Pure GOE ? VWZ 1984
• Pure GUE ? FSS (Fyodorov, Savin Sommers)
  2005 V (Verbaarschot) 2007
• Partial TRSB ? analytical model has been
  developed (based on Pluhar, Weidenmüller, Zuk
  and Wegner, 1995)
• RMT ?
• Full T symmetry breaking sets in experimentally
  already forwith being the variance of the
  off-diagonal matrix elements of and (i.e.
  at )

GOE 0GUE 1
48
Analysis of Fluctuations with Crosscorrelation
Function
Crosscorrelation function

• Determination of T-breaking strength from the
  data
• Special interest in first coefficient (e 0)

49
Experimental Crosscorrelation coefficients
for GOE

• Data TRSB is incomplete ? mixed GOE/GUE system

for GUE
50
Exact RMT Result for Partial T Breaking

• RMT analysis based on Pluhar, Weidenmüller, Zuk,
  Lewenkopf and Wegner, 1995

T-symmetry breaking parameter
51
Exact RMT Result for Partial T Breaking

• RMT analysis based on Pluhar, Weidenmüller, Zuk,
  Lewenkopf and Wegner, 1995

for GOE

• RMT ?

for GUE
52
Determination of T-Breaking Strength

• B. Dietz et al., PRL, submitted.

53
Test of Model for Autocorrelation with
Intermediate TRSB
GOE GUE GOE/GUE ? 0.25

• Autocorrelation displays only a slight dependence
  on ?
• Large spreading of data points ? Ensemble
  measurement and GOF test

54
An Exemplary Fit Resultf 16-17 GHz and B 190
mT
GOE GUE GOE/GUE ? 0.25

• Excellent description of the data with GOE/GUE
  model

55
Elastic Enhancement Factor

• Data (red) matches RMT predicition (blue)
• Values W lt 2 seen ? clear indication of T-breaking

56
Summary

• 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 ? Sba)
• Data show, however, that TRSB is incomplete ?
  mixed GOE / GUE system
• Analytical model for partial TRSB has been
  developed
• 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
• Elastic enhancement factor determined for the
  first time in the presence of TRSB
• Next Investigation of TRSB at EPs