chaotic scattering in open microwave billiards with and without t violation

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


Typical Transmission Spectrum

2

baain,bout, SP/P • Transmission measurements: relative power from antenna a → b


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


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


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)


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


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


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


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


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


Search for TRSB in Nuclei:Ericson Regime

• T. Ericson, Nuclear enhancement of T violation effects,Phys. Lett. 23, 97 (1966)


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


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


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


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


Exact RMT Result for Partial T Breaking

• RMT analysis based on Pluhař, Weidenmüller, Zuk, Lewenkopf and Wegner (1995)

for GOE

for GUE• RMT →


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


• 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


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


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)


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


Elastic Enhancement Factor

• Above ~ 5GHz data (red) match RMT prediction (blue)

• Values w < 2 seen → clear indication of T violation


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


Spectra of S-Matrix Elementsin the Ericson Regime


Distributions of S-Matrix Elementsin the Ericson Regime

• Experiment confirms assumption of Gaussian distributed S-matrix elements with random phases


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:


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:


Autocorrelation Function and Fourier Coefficients in the Ericson Regime

Time domainFrequency domain

• |C(ε)|2 is of Lorentzian shape → exponential decay


Spectra of S-Matrix Elementsin the Regime Γ/d 1≲

Example: 8-9 GHz

Frequency (GHz)

|S11

| |S

12|

|S22

|


Fourier Transform vs.Autocorrelation Function

Time domain Frequency domain

← S12 →

← S11 →

← S22 →

Example 8-9 GHz


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


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.


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


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


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


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


chaotic scattering in open microwave billiards with and without t violation

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achim richter

darmstadt sfb

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