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2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 1
Exceptional Points inMicrowave Billiards:Eigenvalues and Eigenfunctions
• Microwave billiards and quantum billiards
• Microwave billiards as a scattering system
• Eigenvalues and eigenfunctions of a dissipative systemnear an exceptional point
• Properties of exceptional points in the time domain
• Induced violation of time-reversal invariance→ determination of Ĥeff at the EP
Supported by DFG within SFB 634
S. Bittner, C. Dembowski, B. Dietz, J. Metz, M. Miski-Oglu, A. R., F. SchäferH. L. Harney, H. A. Weidenmüller, D. Heiss, C. A. Stafford
Dresden 2011
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 2
Cylindrical Microwave Billiards
xyz
zμμmax ey)(x,E=)r(E ⇒2dc
=f ≤f rrr
( ) 0=y)(x,E0,=y)(x,Ek+Δ G ∂μμ2μ
scalarHelmholtz equation
( )cfπ2
=k0,=)r(E×n0,=)r(Ek+Δ μμG ∂μμ
2μ
rrrrrvectorialHelmholtz equation
cylindrical resonators
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 3
Microwave and Quantum Billiards
microwave billiard quantum billiard
normal conducting resonators → ~700 eigenfunctionssuperconducting resonators → ~1000 eigenvalues
resonance frequencies ↔ eigenvalueselectric field strengths ↔ eigenfunctions
0)y,x(,0)y,x()k(G
2 =Ψ=Ψ+Δ∂
0)y,x(E,0)y,x(E)k(Gzz
2 ==+Δ∂
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 4
Microwave Resonator asa Scattering System
• Microwave power is emitted into the resonator by antenna and theoutput signal is received by antenna → Open scattering system
• The antennas act as single scattering channels
rf power in rf power out
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 5
Transmission Spectrum
• Transmission measurements: relative power transmitted from a to b
• Scattering matrix W)WWiπ+H-(EWiπ2-I=S 1-TT
2baain,bout, SP/P ∝
• Ĥ : resonator Hamiltonian • Ŵ : coupling of resonator states to antenna states and to the walls
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 6
Resonance Parameters
∑μ
μμ
bμaμbaba (i/2)Γ+f-f
ΓΓi- δ=S
• Use eigenrepresentation of
and obtain for a scattering system with isolated resonancesa → resonator → b
Teff WWiπ-H=H
fµ = real part
Γµ = imaginary part effH• Here: of eigenvalues eµ of
• Partial widths Γµ,a , Γµ,a and total width Γµ
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 7
Typical Transmission Spectrum
a) b) c)
• Transmission measurements: relative power from antenna a → b|Sba|2 = Pout,b / Pin,a
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 8
Eigenvalues and Eigenfunctions of a Dissipative System near an EP
• At an exceptional point (EP) two (or more) complex eigenvalues and thecorresponding eigenfunctions of a dissipative system coalesce
• The crossing of two eigenvalues is accomplished by the variation of twoparameters
• Sketch of the experimental setup:
• Divide a circular microwave billiard into twoapproximately equal parts
• The opening s controls the coupling of theeigenmodes of the two billiard parts
• The position δ of the Teflon disc determines theresonance frequencies of the left part
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 9
Two-state Matrix Model
• Isolated EP→ in its vicinity the dynamics is determined by the two eigenstates→ model system with a 2d non-Hermitian symmetric matrix
• Eigenvalues:
• EPs:
• All entries are functions of δ and s
(C. Dembowski et al., Phys. Rev. E 69, 056216 (2004))
⎟⎟⎠
⎞⎜⎜⎝
⎛=
2S12
S121
eff EH
HE)δ,(sH
S12
212S12
212,1
2HEEZ;1ZH
2EEe
−=+=ℜ
ℜ±⎟⎠⎞
⎜⎝⎛ +
=
EPEP ss,δδZ:0 ==↔±==ℜ i
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 10
Encircling the EP in theParameter Space
• Encircling the EP located at the parameter values sEP and δ EP
• The eigenvalues are interchanged and one of the eigenfunctions in addition picks up a topological phase π
⎟⎟⎠
⎞⎜⎜⎝
⎛
−→⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛→⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
−
−
+
+
−
−
+
ψ
ψ
ψ
ψ,
EE
EE Ψ1
Ψ2
Ψ2
Ψ1⎟⎟⎠
⎞⎜⎜⎝
⎛→⎟⎟
⎠
⎞⎜⎜⎝
⎛
1
2
2
1
ee
ee
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 11
Experimental Setup(C. Dembowski et al., Phys. Rev. Lett. 86, 787 (2001))
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 12
Frequencies and Widths
s < sEP s > sEP
• Change of the real (resonance frequency) and the imaginary (resonancewidth) part of the eigenvalue e1,2 =f1,2+i.Γ1,2 for varying δ and fixed s
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 13
Measuring Field Distributions
• Use dielectric perturbation body, e.g. magnetic rubber
• Maier-Slater theorem:
• Reconstruct the eigenfunctions from the pattern of nodal lines, where E(x,y)=0 → extraction of Ψ(x, y)
)y,x(Ec)y,x(f 21 ⋅=δ
perturbation body
guiding magnet
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 14
Evolution of Wave Functions
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 15
Chirality of Eigenfunctions
• Eigenfunctions are interchanged and one eigenfunction picks up atopological phase of π
• Surrounding the EP four times restores the original situation
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 16
What happens at the EP?
• At the exceptional point the two eigenmodes coalesce with a phasedifference π/2
• Can this phase difference be measured?• Choose two eigenmodes which are each localized in one of the
semicircular parts of the cavity for s < sEP and δ = δ EP (i.e. f1=f2)→ these can be excited separately
|Ψ2(s = 0)⟩
|ΨEP⟩ =|Ψ1(s=0)⟩ + i|Ψ2(s=0)⟩
(C. Dembowski et al., Phys. Rev. Lett. 90, 034101 (2003))
at s = sEP and δ = δ EP
|Ψ1(s = 0)⟩
s = 0
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 17
Measured Phase Difference
• Modes Ψ1 and Ψ2 are excited separately by antennas 1 and 2• Measurement of the phase difference φ0 between the oscillating fields
|Ψ1⟩ and |Ψ2⟩ at the positions of the antennas
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 18
Time Decay of the Resonancesnear and at the EP
• In the vicinity of an EP the two eigenmodes can be described as a pair of coupled damped oscillators
• Near the EP the time spectrum exhibits besides the decay of theresonances oscillations → Rabi oscillations with frequency Ω = 2πf
• At the EP ℜ and Ω vanish → no oscillations
ℜ∝ΩΩ
ΩΓ−∝ ;)t(sin)texp()t(P 2
2
)texp(t)t(P 2 Γ−∝
• In distinction, an isolated resonance decays simply exponentially→ line-shape at EP is not of a Breit-Wigner form
(Dietz et al., Phys. Rev. E 75, 027201 (2007))
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 19
2.79 2.83f (GHz)
P(t)
= |F
TS
21|2
(dB
)
|S12
|2 (d
B)
Time Decay of the Resonances"far" from the EP
• Г1 = 1.584 ± 0.005 MHz• Г2 = 1.022 ± 0.002 MHz• Region of interaction at about 2 µs → Rabi oscillations
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 20
Time Decay of the Resonancesnear and at the EP
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 21
• Search for Time-reversal (T-) Invariance Violation (TIV) in nuclear reactions
Detailed Balance in Nuclear Reactions
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 22
Detailed Balance in Nuclear Reactions
• Search for TIV innuclear reactions → upper limits
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 23
Induced T-Invariance Violation in Microwave Billiards
• T-invariance violation caused by a magnetized ferrite
• Ferrite features Ferromagnetic Resonance (FMR)
• Coupling of microwaves to the ferrite dependson the direction a b
Sab
Sba
ab
• Principle of detailed balance: |Sab|2 = |Sba|2
• Principle of reciprocity: Sab = Sba
a b• •
(B. Dietz et al., Phys. Rev. Lett. 98, 074103 (2007))
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 24
Scattering Matrix Description
Remember: Scattering matrix formalism
• Investigation of systems
→ replace Ĥ by a matrix
• Note: Ĥeff = Ĥ – iπŴŴT is non-Hermitian in both cases
• Isolated resonances: singlets and doublets → Ĥ is 1D or 2D
• Overlapping resonances (Γ > D) Ĥ = in RMT
T-invariantT-noninvariant
real symmetricHermitian
( ) WWWiπ+H-IfWiπ2-I=(f)S 1-TT
GOEGUE
(B. Dietz et al., Phys. Rev. Lett. 103, 064101 (2009)
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 25
Isolated Resonances - Setup
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 26
Singlet with T-Violation
0.002
0.004|S|
0
-π2.80 2.85 2.90
Frequency (GHz)
Arg(S)
• Reciprocity holds → T-violation cannot be detected this way
• Sab• Sba
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 27
Doublet with T-Violation
• Violation of reciprocity due to interference of two resonances
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 28
Scattering Matrix and T-violation
• Scattering matrix element (ω = 2πf)
• Decomposition of effective Hamiltonian
• Ansatz for T-violation incorporating the ferromagnetic resonance and its selective coupling to the microwaves
• Note: Ĥs conserves T-invariance and Ĥa violates it
bW)H-(ωWaiπ2-δ=)ω(S 1-eff
+abab
saeff H+Hi=H ˆˆˆ
⎟⎟⎠
⎞⎜⎜⎝
⎛
− 00
12
12a
a
HH a
12iHa12iH-
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 29
T-Violating Matrix Element
r0
2Mra
12 i/T-ω-(B)ωω
TBλ2π
=(B)H
couplingstrength
externalfield
spinrelaxation
time
magneticsusceptibility
• Fit parameters: andλ ω
• • • •
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 30
Measured T-Violating Matrix Element
• T-violating matrix element shows resonance like structure• Successful description of dependence on magnetic field→ Ĥeff was determined and will be used to describe TIV at the EP
π/2
-π/20
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 31
Relative Strength of T-Violation
• Compare: TIV matrix element to the energy difference of two eigenvalues of the T-invariant system
a12H
s2
s1
a12
E-E2H
=ξ
2011 | Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento | Achim Richter | 32
Summary and Outlook
• Microwave billiards are precisely controlled, parameter-dependent dissipative systems
• Complex eigenvalues and eigenfunctions can be determined from the billiard scattering matrix→ Observation of EPs in microwave billiards
• Behavior of eigenvalues close to and at an EP investigated experimentally
• Also time-behavior close to and at an EP was studied→ disappearance of echoes at the EP → t2-behavior
• T-invariance violation induced in microwave billiards and observed for doublets of resonances → determination of complete Ĥeff possible at an exceptional point
• Second talk: Measurement and interpretation of T-violation
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