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

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SCUOLA INTERNAZIONALE DI FISICA “fermi" Varenna sul lago di como - 1965. Dr. h.c. Oriol Bohigas TU Darmstadt 2001. Chaotic Scattering in Microwave Billiards. Orsay 2008. BGS conjecture, quantum billiards and microwave resonators - PowerPoint PPT Presentation

Transcript of SCUOLA INTERNAZIONALE DI FISICA “fermi" Varenna sul lago di como - 1965

Page 1: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

SCUOLA INTERNAZIONALE DI FISICA “fermi"Varenna sul lago di como - 1965

Page 2: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

Dr. h.c. Oriol BohigasTU Darmstadt 2001

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• 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

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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

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2ΔxΔp

The Quantum Billiard and its Simulation

SFB 634 – C4: Quantum Chaos

Shape of the billiard implies chaotic dynamics

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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

Page 7: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

• 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

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• 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

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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)

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• 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

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• 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

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• 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

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Example: 8-9 GHz

SFB 634 – C4: Quantum Chaos

Spectra of S-Matrix Elements

Frequency (GHz)

|S|

S12

S11

S22

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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

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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

~

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Time domain Frequency domain

S12

S11

S22

SFB 634 – C4: Quantum Chaos

Fourier Transform vs. Autocorrelation Function

Example 8-9 GHz

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• 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

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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

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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

~

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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

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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

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Search for Time-Reversal Symmetry Breaking in Nuclei

SFB 634 – C4: Quantum Chaos

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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

Page 24: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

Violation of Reciprocity

• Clear violation of reciprocity in the regime of Γ/D 1

S12

S12

SFB 634 – C4: Quantum Chaos

Page 25: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

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

Page 26: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

Magnitude and Phase of Δ Fluctuate

SFB 634 – C4: Quantum Chaos

B 200 mT

B 0 mT:no TRSB

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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

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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

*

*

Page 29: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

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

Page 30: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

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

Page 31: SCUOLA  INTERNAZIONALE  DI  FISICA   “fermi" Varenna  sul  lago  di  como  -  1965

Determination of the rms value of T-breaking matrix element

SFB 634 – C4: Quantum Chaos

α (M

Hz)

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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

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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