2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 1
Exceptional Points in Microwave Billiards with and without Time-Reversal Symmetry
Ein Gedi 2013
• Precision experiment with microwave billiard→ extraction of the EP Hamiltonian from the scattering matrix
• EPs in systems with and without T invariance• Properties of eigenvalues and eigenvectors at and close to an EP• Encircling the EP: geometric phases and amplitudes• PT symmetry of the EP Hamiltonian
Supported by DFG within SFB 634
S. Bittner, B. Dietz, M. Miski-Oglu, A. R., F. SchäferH. L. Harney, O. N. Kirillov, U. Günther
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 2
Microwave and Quantum Billiards
microwave billiard quantum billiard
normal conducting resonators ~700 eigenfunctions superconducting resonators ~1000 eigenvalues
resonance frequencies eigenvalues electric field strengths eigenfunctions
0)y,x(,0)y,x()k(G
2
0)y,x(E,0)y,x(E)k(Gzz
2
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 3
Microwave Resonator asa Scattering System
• 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
rf power in rf power out
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 4
Transmission Spectrum
• 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
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 5
How to Detect an Exceptional Point?(C. Dembowski et al., Phys.Rev.Lett. 86,787 (2001))
• At an exceptional point (EP) of a dissipative system two (or more) complex eigenvalues and the associated eigenvectors coalesce• Circular microwave billiard with two approximately equal parts →
almost degenerate modes• EP were detected by varying two parameters
• The opening s controls the coupling of the eigenmodes of the two billiard parts
• The position d of the Teflon disk mainly effects the resonance frequency of the mode in the left part
• Magnetized ferrite F with an exterior magnetic field B induces T violation
F
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 6
Experimental setup
variation of d
variation of s
variation of B
(B. Dietz et al., Phys. Rev. Lett. 106, 150403 (2011))
• Parameter plane (s,d) is scanned on a very fine grid
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 7
Experimental Setup to Induce 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 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 8
Violation of T Invariance Induced with a Magnetized Ferrite
• Spins of the magnetized ferrite precess collectively with their Larmor frequency about the external magnetic field
• The magnetic component of the electromagnetic field coupled into the cavity interacts with the ferromagnetic resonance and the coupling depends on the direction a b
• T-invariant system
→ principle of reciprocity Sab = Sba
→ detailed balance |Sab|2 = |Sba|2
Sab
Sba
a b
F
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 9
Test of Reciprocity
• Clear violation of the principle of reciprocity for nonzero magnetic field
B=0 B=53mT
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 10
Resonance Spectra Close to an EP for B=0
• Scattering matrix: W)HE(W2IS 1eff
Ti• Ŵ : coupling of the resonator modes to the antenna states • Ĥeff : two-state Hamiltonian including dissipation and coupling to the exterior
Frequency (GHz) Frequency (GHz)
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 11
Two-State Effective Hamiltonian
• Eigenvalues:
0H
H0
EH
HE)δ,(sH
12
12
2S12
S121
eff A
A
i
• EPs:
2212
122
12
212,1
2EEHH
2EEe
AS
EPEP ss,δδ0
• :H12A T-breaking matrix element. It vanishes for B=0.
• Ĥeff : non-Hermitian and complex non-symmetric 22 matrix
B=0B0
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 12
Resonance Shape at the EP
• At the EP the Ĥeff is given in terms of a Jordan normal form
0
0EPEPeff 0
1)δ,(sH
2EE 21
0
with
1b2a2b1a2b2a1b1a20
12
b20
2ab10
1aabab
VVVVVVVV)(
H
V)(
1VV)(
1VS
if
ffS
d
→ at the EP the resonance shape is not described by a Breit-Wigner form• Note: this lineshape leads to a temporal t2-decay behavior at the EP
• Sab has two poles of 1st order and one pole of 2nd order
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 13
Time Decay of the Resonancesnear and at the EP
• Time spectrum = |FT{Sba(f)}|2
• Near the EP the time spectrum exhibits besides the decay of the resonances Rabi oscillations with frequency Ω = (e1-e2)/2
• At the EP and vanish no oscillations
;)t(sin)texp()t(P 2
2
)texp(t)t(P 2
• 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))
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 14
Time Decay of the Resonancesnear and at the EP
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 15
T-Violation Parameter t
W)HE(W2IS 1eff
Ti
• For each set of parameters (s,d) Ĥeff is obtained from the measured Ŝ matrix
• Ĥeff and Ŵ are determined up to common real orthogonal transformations
• Choose real orthogonal transformation such that
• T violation is expressed by a real phase → usual practice in nuclear physics
τ2
1212
1212
)HH()HH( i
AS
AS
eii
with t[ /2,/2[ real
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 16
Localization of an EP (B=0)
• Change of the real and the imaginary part of
the eigenvalues e1,2=f1,2+i 1,2/2. They cross at s=sEP=1.68 mm and ddEP=41.19 mm.
jj
j ,2
j ,1j r
r ie
δ (mm)
1r
rr
2,j
1,jj
i• At (sEP,dEP)
• Change of modulus and phase of the ratio of the components rj,1, rj,2 of the eigenvector |rj
s=1.68 mm
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 17
Localization of an EP (B=53mT)
• Change of the real and the imaginary part of the eigenvalues e1,2=f1,2+i 1,2/2. They cross at s=sEP=1.66 mm and ddEP=41.25 mm.
• Change of modulus and phase of the ratio of the components rj,1, rj,2 of the eigenvector |rj
1e
r
rr
τ
2j,
1,jj
ii
jj
2,j
1,jj r
rν ie
• At (sEP,dEP)
τ
δ (mm)
s=1.66 mm
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 18
T-Violation Parameter t at the EP(B.Dietz et al. PRL 98, 074103 (2007))
• and also the T-violating matrix element shows resonance like structure
r0
2r
12 T/)B(TB
2(B)H
iMA
couplingstrength
spinrelaxation
timemagnetic
susceptibility
• EP=/2+t
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 19
Eigenvalue Differences (e1-e2) in the Parameter Plane
• S matrix is measured for each point of a grid with s = d = 0.01 mm• The darker the color, the smaller is the respective difference• Along the darkest line e1-e2 is either real or purely imaginary
• Ĥeff PT-symmetric Ĥ
d (m
m)
s (mm)d
(mm
)s (mm)
B=0 B=53mT
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 20
Differences (|1|-|2|) and (1-2) of the Eigenvector Ratios in the Parameter Plane
• The darker the color, the smaller the respective difference• Encircle the EP twice along different loops• Parameterize the contour by the variable t with t=0 at start point, t=t1 after
one loop, t=t2 after second one
B=53mT
t=0,t1,t2
d (m
m)
s (mm)
t=0,t1,t2
d m
m)
s (mm)
B=0
t=0, t1, t2
t=0, t1, t2
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 21
Encircling the EP in the Parameter Space
• Encircling EP once:
1
2
2
121 ,ee
r
r
r
r
• Biorthonormality defines eigenvectors lj(t)| rj(t)=1 along contour up to a
geometric factor :)(tγ
jj)t(γ-
jjjj )t(r)t(R,)t()t(L ii eel
• For B0 additional geometric factor: (t))γ(Im(t))γ(Re(t)γ jjj eee ii
(t)iγ± je
0d td τ0)( tγ
d td γ
d td γ
1 ,221 and for
• Condition of parallel transport yields0)t(Rdtd(t)L jj
for all t
0 ≡(t)γ=(t)γ 21 for B=0
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 22
Change of the Eigenvalues along the Contour (B=0)
• Note: Im(e1) + Im(e2) ≈ const. → dissipation depends weakly on (s,δ)• The real and the imaginary parts of the eigenvalues cross once during each
encircling at different t → the eigenvalues are interchanged• The same result is obtained for B=53mT
t1 t1t2 t2
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 23
Evolution of the Eigenvector Components along the Contour (B=0)
• Evolution of the first component rj,1 of the eigenvector |rj as function of t• After each loop the eigenvectors are interchanged and the first one picks
up a geometric phase
2
1
1
2
2
1
r
r
r
r
r
r
t1 t2
r1,1
• Phase does not depend on choice of circuit topological phase
r2,1
t1 t2
r1,1
r2,1
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 24
T-violation Parameter t along Contour (B=53mT)
• T-violation parameter τ varies along the contour even though B is fixed • Reason: the electromagnetic field at the ferrite changes• τ increases (decreases) with increasing (decreasing) parameters s and d • τ returns after each loop around the EP to its initial value
t1 t2
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 25
Evolution of the Eigenvector Components along the Contour (B=53mT)
• Measured transformation scheme:
)(
2
)(1
)(1
)(2
2
1
21
21
11
11
r
r
r
r
r
rti
ti
ti
ti
e
e
e
e
• No general rule exists for the transformation scheme of the γj
• How does the geometric phase γj evolve along two different and equal loops?
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 26
Geometric Phase Re (j(t)) Gathered along Two Loops
t 0: Re(1(0)) Re(2(0)) 0t t1: Re(1(t1)) Re(2(t1)) 0.31778t t2: Re(1(t2)) Re(2(t2)) 0.09311
t 0: Re(1(0)) Re(2(0)) 0t t1: Re(1(t1)) Re(2(t1)) 0.22468t t2: Re(1(t2)) Re(2(t2)) 3.310-7
• Clearly visible that Re(2(t)) Re(1(t))t2
Twice the same loop
t1tt1 t2
Two different loops
t
• Same behavior observed for the imaginary part of γj(t)
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 27
Complex Phase 1(t) Gathered along Two Loops
Δ : start point : 1(t1) 1(0) : 1(t2) 1(0) if EP is encircled along different loops → | e i1(t) | 1
→ 1(t) depends on choice of path geometrical phase
Two different loops Twice the same loop
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 28
Complex Phases j(t) Gathered when Encircling EP Twice along Double Loop
Δ : start point : 1(nt2) 1(0), n=1,2,…
• Encircle the EP 4 times along the contour with two different loops• In the complex plane 1(t) drifts away from 2(t) and the origin →
‘geometric instability’• The geometric instability is physically reversible
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 29
Difference of Eigenvalues in the Parameter Plane (S.Bittner et al., Phys.Rev.Lett. 108,024101(2012))
• Dark line: |f1-f2|=0 for s<sEP, |Γ1-Γ2|=0 for s>sEP
→ (e1-e2) = (f1-f2)+i(1-2)/2 is purely imaginary for s<sEP / purely real for s>sEP
ˆHTr21HH effeffDL• (e1-e2) are the eigenvalues of
B=0 B=38mT B=61mT
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 30
Eigenvalues of ĤDL along Dark Line
riirri2i
2r V,V,V,V2VVε i• Eigenvalues of ĤDL: real
• Dark Line: Vri=0 → radicand is real
• Vr2 and Vi
2 cross at the EP
• Behavior reminds on that of the eigenvalues of a PT symmetric Hamiltonian
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 31
PT symmetry of ĤDL along Dark Line
• For Vri=0 ĤDL can be brought to the form of Ĥ with the unitary transformationzy στ/2σφU ii ee
→ ĤDL is invariant under the antilinear operator
• At the EP the eigenvalues change from purely real to purely imaginary
→ PT symmetry is spontaneously broken at the EP
• General form of Ĥ, which fulfills [Ĥ, PT]=0:
D C, B,,A,BADCDCBA
H
iiii
real
• P : parity operator xσP , T : time-reversal operator T=K
• PT symmetry: the eigenvectors of Ĥ commute with PT
TePeeeT zyyz iiii tt 2/2/U
2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 32
• High precision experiments were performed in microwave billiards with and without T violation at and in the vicinity of an EP
• The size of T violation at the EP is determined from the phase of the ratio of the eigenvector components
• Encircling an EP:
• Eigenvalues: Eigenvectors:
• T-invariant case: 1(t) 0• Violated T invariance: 1,2(t1) γ1,2(0), different loops: 1,2(t2) γ1,2(0)• PT symmetry is observed along a line in the parameter plane. It is
spontaneously broken at the EP
Summary
1
2
2
1
ee
ee
(t)γ(t)γ,r
r
r
r
r
r21)(tγ
2
)(tγ1
)(tγ1
)(tγ2
2
1
21
21
11
11
i
i
i
i
e
e
e
e