chaotic scattering in open microwave billiards with and without t violation
DESCRIPTION
Chaotic Scattering in Open Microwave Billiards with and without T Violation. Porquerolles 2013. Microwave resonator as a model for the compound nucleus Induced violation of T invariance in microwave billiards RMT description for chaotic scattering systems with T violation - PowerPoint PPT PresentationTRANSCRIPT
Chaotic Scattering in Open Microwave
Billiards with and without T Violation
• Microwave resonator as a model for the compound nucleus
• Induced violation of T invariance in microwave billiards
• RMT description for chaotic scattering systems with T violation
• Experimental test of RMT results on fluctuation properties of
S-matrix elements
Supported by DFG within SFB 634
B. Dietz, M. Miski-Oglu, A. Richter
F. Schäfer, H. L. Harney, J.J.M Verbaarschot, H. A. Weidenmüller
Porquerolles2013
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 1
Microwave Resonator as a Model for the 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 at the walls is modeled by additive channels
Compound NucleusA+a B+b
C+cD+d
rf power in
rf power out
h
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 2
Typical Transmission Spectrum
2
baain,bout, SP/P • Transmission measurements: relative power from antenna a → b
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 3
Scattering Matrix Description
• Scattering matrix for both scattering processes as function of excitation frequency f :
• RMT description: replace Ĥ by a matrix for systems
Microwave billiardCompound-nucleus reactions
resonator Hamiltonian
coupling of resonator states to antenna states and to absorptive channels
nuclear Hamiltonian
coupling of quasi-boundstates to channel states
Ĥ
Ŵ
GOE T-invGUE T-noninv
• Experiment: complex S-matrix elements
Ŝ( f ) = I - 2i ŴT ( f I - Ĥ + i ŴŴT)-1 Ŵ
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 4
Resonance Parameters
)2/(iff
iSba
baba
• Use eigenrepresentation of
and obtain for a scattering system with isolated resonancesa → resonator → b
• Here: of eigenvalues of
• Partial widths fluctuate and total widths also
Teff WWiHH ˆˆˆˆ
ba ,
f real part
imaginary parteffH
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 5
Excitation Spectra
overlapping resonancesfor Γ/d > 1Ericson fluctuations
isolated resonancesfor Γ/d < 1
atomic nucleus
ρ ~ exp(E1/2)
microwave cavity
ρ ~ f
• Universal description of spectra and fluctuations:Verbaarschot, Weidenmüller and Zirnbauer (1984)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 6
Fully Chaotic Microwave Billiard
• Tilted stadium (Primack+Smilansky, 1994)
• Only vertical TM0 mode is excited in resonator → simulates a quantum
billiard with dissipation• Additional scatterer → improves statistical significance of the data sample
• Measure complex S-matrix for two antennas 1 and 2: S11, S22, S12, S21
12
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 7
Spectra and Correlations ofS-Matrix Elements
• Г/d <<1: isolated resonances → eigenvalues, partial and total widths
• Г/d ≲ 1: weakly overlapping resonances → investigate S-matrix fluctuation properties with the autocorrelation function and its Fourier transform )(S)(S)(S)(S)( **)2( ffffC ababababab
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 8
Microwave Billiard for the Study of Induced T Violation
NS
NS
• A cylindrical ferrite is placed in the resonator
• An external magnetic field is applied perpendicular to the billiard plane
• The strength of the magnetic field is varied by changing the distance
between the magnets
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 9
Sab
Sba
ab
Induced Violation of T Invariance with Ferrite
• Spins of magnetized ferrite precess collectively with their Larmor frequency
about the external magnetic field • Coupling of rf magnetic field to the ferromagnetic resonance depends on the
direction a b
• T-invariant system → reciprocity
→ detailed balance
F
Sab = Sba
|Sab|2 = |Sba|2
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 10
Detailed Balance in Microwave Billiard
• Clear violation of principle of detailed balance for nonzero magnetic field B→ How can we determine the strength of T violation?
S12
S21
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 11
Search for TRSB in Nuclei:Ericson Regime
• T. Ericson, Nuclear enhancement of T violation effects,Phys. Lett. 23, 97 (1966)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 12
Analysis of T violation with Cross-Correlation Function
2
21
2
12
*2112
)ε(S)(S
)ε(S)(SRe)ε(
ff
ffCcross
• Cross-correlation function:
• Special interest in cross-correlation coefficient Ccross(ε = 0)
0
1)0ε(crossC
if T-invariance holds
if T-invariance is violated
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 13
S-Matrix Correlation Functions and RMT
• Pure GOE VWZ 1984
• Pure GUE FSS (Fyodorov, Savin, Sommers) 2005
• Partial T violation Pluhař, Weidenmüller, Zuk, Lewenkopf + Wegner
derived analytical expression for C(2) (ε=0) in 1995
• RMT
as HNiHH ˆ/ξˆ)ξ(ˆ
GOE: 0GUE: 1
• Complete T-invariance violation sets in already for ≃1• Based on Efetov´s supersymmetry method we derived analytical
expressions for C (2) (ε) and Ccross(ε=0) valid for all values of
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 14
S-Matrix Ansatz for Chaotic Systems with Partial T Violation
• Insert Hamiltonian Ĥ (ξ) for partial T-violation into scattering matrix
WWWiHIfWiIf TT ˆˆˆ)ξ(ˆˆ2)ξ;(S1
• T-violation is due to ferrite only Ŵ is real
• Ohmic losses are mimicked by additional fictitious channels
• Ŵ is approximately constant in 1GHz frequency intervals
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 15
RMT Result for Partial T Violation(Phys. Rev. Lett. 103, 064101 (2009))
channelsabsorptive
abs
2
T
1T
a
ccc S
• RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner, 1995
parameter for strength of T violation from cross-correlation coefficient
transmission coefficients from autocorrelation function
2t
mean resonance spacing→ from Weyl‘s formula
(2)
d )ξ,ε,d;,T,T( abs21)2( abC
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 16
Exact RMT Result for Partial T Breaking
• RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner (1995)
for GOE
for GUE• RMT →
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 17
1
0ξ,ˆ/ξˆ)ξ(ˆ as HNiHH
T-Violation Parameter ξ
• Largest value of T-violation parameter achieved is ξ ≃ 0.3• Published in Phys. Rev. Lett. 103, 064101 (2009)
cros
s-co
rrel
atio
nco
effi
cien
t
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 18
• Adjacent points in C ab() are correlated
• FT of Cab()
• For a fixed t are Gaussian distributed about their
time-dependent mean
Autocorrelation Function and its Fourier Transform
)S~
Im(),S~
Re( abab
2)(S
~)(
~ttC abab
t (ns)lo
g(F
T(C
12(ε
)))
|C12
(ε)|2 x
(10
-4)
0
2
4
6
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 19
Distribution of Fourier Coefficients
• Distribution of phases is uniform
2
4
2|S
~|/|S
~|
z
abab ezzP
• Distribution of modulus divided by its local mean is bivariate Gaussian
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 20
Determination of τabs from FT of Autocorrelation Function
2
)(S~
kabk tx • Fit determines expectation value of , abs and improved values for T1, T2, • GOF test relies on Gaussian distributed )S
~Im(),S
~Re( abab
• Decay is non exponential → autocorrelation function is not Lorentzian
3-4 GHz
9-10 GHz
log(
FT
(C12
(ε))
)
t (ns)
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 21
Enhancement Factor for Systems with Partial T-Violation
• Hauser-Feshbach for Γ>>d:
Isolated resonances: w = 3 for GOE, w = 2 for GUE
Strongly overlapping resonances: w = 2 for GOE, w = 1 for GUE
)0(/)0()0(
S/SS
)2(12
)2(22
)2(11
2fl12
2fl22
2fl11
CCC
w
cc
baabab f
T
TT)1()(S
2fl
• Elastic enhancement factor w
ababab SSSfl
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 22
Elastic Enhancement Factor
• Above ~ 5GHz data (red) match RMT prediction (blue)
• Values w < 2 seen → clear indication of T violation
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 23
Fluctuations in a Fully Chaotic Cavity with T-Invariance
• Tilted stadium (Primack + Smilansky, 1994)
• GOE behavior checked
• Measure full complex S-matrix for two antennas: S11, S22, S12
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 24
Spectra of S-Matrix Elementsin the Ericson Regime
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 25
Distributions of S-Matrix Elementsin the Ericson Regime
• Experiment confirms assumption of Gaussian distributed S-matrix elements with random phases
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 26
Experimental Distribution of S11 and Comparison with Fyodorov, Savin and Sommers Prediction (2005)
• Distributions of modulus z is not a bivariate Gaussian
• Distributions of phases are not uniform
Exp:
RMT:
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 27
Experimental Distribution of S12 and Comparison with RMT Simulations
• No analytical expression for distribution of off-diagonal S-matrix elements
• For /d=1.02 the distribution of S12 is close to Gaussian → Ericson regime?
• Recent analytical results agree with data → Thomas Guhr
Exp:
RMT:
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 28
Autocorrelation Function and Fourier Coefficients in the Ericson Regime
Time domainFrequency domain
• |C(ε)|2 is of Lorentzian shape → exponential decay
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 29
Spectra of S-Matrix Elementsin the Regime Γ/d 1≲
Example: 8-9 GHz
Frequency (GHz)
|S11
| |S
12|
|S22
|
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 30
Fourier Transform vs.Autocorrelation Function
Time domain Frequency domain
← S12 →
← S11 →
← S22 →
Example 8-9 GHz
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 31
Exact RMT Result for GOE Systems
C = C(Ti, d ; )
Transmission coefficients Average level distance
• 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
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 32
Corollary
• Present work:S-matrix → Fourier transform → decay time (indirectly measured)
• Future work at short-pulse high-power laser facilities:Direct measurement of the decay time of an excited nucleus might become possible by exciting all nuclear resonances (or a subset of them) simultaneously by a short laser pulse.
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 33
Three- and Four-Point Correlation Functions
2222
)(S)ε(S)(S)ε(
fffC abababab
• Autocorrelation function of cross-sections =|Sab|2 is a four-point correlation
function of Sab
• Express Cab() in terms of 2 -, 3 -, and 4 - point correlation functions of Sfl
)ε()ε()ε(Re2)ε()ε( )3()3()2(2
)4( abababababababab CCSCSCC
• C(3)(0), C(4)(0) and therefore Cab(0) are known analytically (Davis and Boosé 1988)
• Ericson regime: 2)2( )ε()ε( abab CC
,)(S)ε(S)ε(2flfl)3(
abababab ffC 2
2fl2fl2fl)4( )(S)ε(S)(S)ε(
fffC abababab
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 34
Ratio of Cross-Section- and Squared S-Matrix-Autocorrelation Coefficients:2-Point Correlation Functions
• Excellent agreement between experimental and analytical result• Dietz et al., Phys. Lett. B 685, 263 (2010)
exp: o, o red: a = bblack: a ≠ b
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 35
Experimental and Analytical 3- and 4-Point Correlation Functions
• |F(3)( )| and F(4)( ) have similar values→ Ericson theory is inconsistent as it retains C(4)() but discards C(3)()
abababab ffF 2flfl)3( )(S)ε(S)ε( 2fl2fl)4( )ε(S)(S)ε( ffF ababab
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 36
Summary
• Investigated experimentally and theoretically chaotic scattering in open systems with and without induced T violation in the regime of weakly overlapping resonances (≤ d)
• Data show that T violation is incomplete no pure GUE system
• Analytical model for partial T violation which interpolates between GOE and GUE has been developed
• A large number of data points and a GOF test are used to determine T1, T2, abs and from cross- and autocorrelation function
• GOF test relies on Gaussian distribution of the Fourier coefficients of the S-matrix
• RMT theory describes the distribution of the S-matrix elements and two -, three - and four - point correlation functions well
2013 |Institut für Kernphysik, Darmstadt | SFB 634 | Achim Richter | 37