Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz
EPs in Microwave Billiards:Eigenvectors and the Full Hamiltonian for T-invariant and T-noninvariant Systems
Dresden 2011
• Precision experiment with microwave billiard→ extraction of full EP Hamiltonian from scattering matrix
• Properties of eigenvalues and eigenvectors at and close to an EP
• EPs in systems with violated T invariance
• Encircling the EP: geometric phases and amplitudes
• PT symmetry of the EP Hamiltonian
Supported by DFG within SFB 634
S. Bittner, B. D., M. Miski-Oglu, A. Richter, F. SchäferH. L. Harney, O. N. Kirillov, U. Günther
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 2
Microwave Resonator for the Observation of Exceptional Points
• Divide a circular microwave billiard into two approximately equal parts
• The coalescence at the EP is accomplished by the variation of two
parameters
• The opening s controls the coupling of the
eigenmodes of the two billiard parts
• The position δ of the Teflon disk mainly effects
the resonance frequencies of the left part
• Insert a ferrite F and magnetize it with an exterior
magnetic field B to induce T violation
F
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 3
Experimental setup
variation of δ
variation of s
variation of B
(B. Dietz et al., Phys. Rev. Lett. 106, 150403 (2011))
• Parameter plane (s,δ) is scanned on a very fine grid
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 4
Resonance Spectra Close to an EP (B=0)
• Scattering matrix: W)HE(W2IS 1eff
−−−= Tiπ
• Ĥeff : two-state Hamiltonian including dissipation and coupling to the exterior
• Ŵ : coupling of the resonator modes to the antenna states
Frequency (GHz) Frequency (GHz)
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 5
Two-State Matrix Model for the T-invariant Case
• Determine the non-Hermitian complex symmetric 2× 2 matrix Ĥeff and its
eigenvalues and eigenvectors for each (s,δ) from the measured Ŝ matrix
• Eigenvalues:
=
2S12
S121
effEH
HE)δ,(sH
• EPs:
S12
212S12
212,1
2H
EEZ;1ZH
2
EEe
−=+=ℜ
ℜ±
+=
EPEP ss,δδZ:0 ==↔±==ℜ i
• All entries are functions of δ and s
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 6
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
λλ
2
EE 210
+=λwith
( ){ }1b2a2b1a2b2a1b1a20
12
b20
2ab10
1aabab
VVVVVVVV)(
H
V)(
1VV
)(
1VS
++−−
−
−−
−−=
if
ffS
λ
λλδ
→ at the EP the resonance shape is not described by a Breit-Wigner form
• Note: this lineshape leads to t2-behavior (→ first talk)
• Sab has two poles of 1st order and one pole of 2nd order
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 7
Localization of an EP (B=0)
• Change of the real and the imaginary part of the eigenvalues e1,2=f1,2+i Γ1,2. They cross at s=sEP=1.68 mm and δ= δEP=41.19 mm.
j
jj,2
j,1j r
r Φ== ieνν
δ (mm)
→
=
1r
rr
2,j
1,j
j
i• At (sEP,δEP)
• Change of modulus and phase of the ratio
of the components rj,1, rj,2 of the eigenvector |rj⟩
s=1.68 mm
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 8
Eigenvalues and Ratios of Eigenvector Components in the Parameter Plane (B=0)
• S matrix is measured for each point of a grid with ∆s = ∆δ = 0.01 mm• Note the dark line, where e1-e2 is either real or purely imaginary.
There, Ĥeff → PT-symmetric Ĥ • Parameterize the contour by the variable t with t=0 at start point, t=t1 after
one loop, t=t2 after second one
t=0,t1,t2
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 9
Encircling the EP in the Parameter Plane (T-invariant Case)
• The biorthonormalized eigenvectors ⟨lj(t)| and |rj(t)⟩ with ⟨lj(t)| rj(t)⟩ =1 are
defined up to a geometric factor e± iγ j(t)
• The geometric phases γ j(t) are fixed by the condition of parallel transport
0)t(rdt
d)t( )t(γ
j)t(γ
jjj =− ii eel
• With ZZ1)t(tan 2 −+=θ the eigenvectors are
−=
=
)t(cos
)t(sin)t(r,
)t(sin
)t(cos)t(r 21 θ
θθθ
• Encircling the EP once:
−→
↔⇒+→
1
2
2
1
21r
r
r
r,ee
2
πθθ
0 ≡(t)γ=(t)γ 21⇒
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 10
Change of the Eigenvalues along the Contour (B=0)
• The real, respectively, the imaginary parts of the eigenvalues cross once during each encircling at different t
• The eigenvalues are interchanged
• Note: Im(e1) + Im(e2) ≈ const. → dissipation depends weakly on (s,δ)
→
1
2
2
1
e
e
e
e
t1 t1t2 t2
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 11
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 π
−
−→
−→
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
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 12
Summary for T-invariant case
• Full EP Hamiltonian extracted from measured scattering matrix→ direct determination of its eigenvalues and eigenvectors possible
• Behavior of eigenvalues and eigenvectors for T-invariant case as expected→ confirms validity of the procedure used for data analysis
• Next step: Investigation of the T-noninvariant case following the same procedure
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 13
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
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 14
Induced Violation of T Invariance with a Ferrite
• Spins of magnetized ferrite precess collectively with their Larmor frequency
about the external magnetic field (→ first talk) • Coupling of rf magnetic field to the ferromagnetic resonance 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
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 15
Test of Reciprocity
• Clear violation of the principle of reciprocity for nonzero magnetic field
B=0 B=53mT
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 16
Two-State Matrix Model for Broken T invariance
• Ĥeff : non-Hermitian and non-symmetric complex 2× 2 matrix
• Eigenvalues:
−+
=
0H
H0
EH
HE)δ,(sH
12
12
2S12
S121
eff A
A
i
• EPs:
2
12
2
12
2122
12
2
12
212,1
HH2
EEZ;1ZHH
2
EEe
AS
AS
+
−=++=ℜ
ℜ±
+=
EPEP ss,δδZ:0 ==↔±==ℜ i
• :H12A T-breaking matrix element
(→ first talk)
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 17
T-Violation Parameter τ
W)HE(W2IS 1eff
−−−= Tiπ
• For each set of parameters (s,δ) 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( iAS
AS
ei
i =−+
with τ∈[0,π[ real
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 18
Localization of an EP (B=53mT)
• Change of the real and the imaginary part of the eigenvalues e1,2=f1,2+i Γ1,2. They cross at s=sEP=1.66 mm and δ= δEP=41.25 mm.
• Change of modulus and phase of the ratio of
the components rj,1, rj,2 of the eigenvector |rj⟩
→
=
1
e
r
rr
τ
2j,
1,j
j
ii
j
j2,j
1,jj r
rν Φ== ieν
• At (sEP,δEP)
τ
δ (mm)
s=1.66 mm
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 19
T-Violation Parameter τ at the EP
• Φ and also the T-violating matrix element shows resonance like structure
r0
2r
12 T/)B(TB
2(B)H
iMA
−−⋅⋅⋅⋅=
ωωωλπ
couplingstrength
spinrelaxation
timemagnetic
susceptibility
• ΦEP=π/2+τ
(→ first talk)
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 20
Eigenvalues and Ratios of Eigenvector Components in Parameter Plane (B=53mT)
• S matrix is measured for each point of a grid with ∆s = ∆δ = 0.01 mm• Note the dark line, where e1-e2 is either real or purely imaginary.
There, Ĥeff → PT-symmetric Ĥ
• Parameterize the contour by the variable t with t=0 at starting point, t=t1 after one loop, t=t2 after second one
t=0,t1,t2
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 21
Encircling the EP in the Parameter Space (with T violation)
• Encircling EP once:
−→
↔
1
2
2
1
21 ,eer
r
r
r
• Eigenvectors along contour:
• Biorthonormality is defined up to a geometric factor
)(tγjj
)t(γ-jj
jj )t(r)t(R,)t()t(L ii eel ==
• Additional geometric factor: (t))Im(γ(t))γRe((t)γ jjj −= eee ii
(t)iγ± je
0dt
dτ0)(tγ
dt
dγ
dt
dγ1,2
21 ≠≠−= and for
• Condition of parallel transport yields0)t(Rdt
d(t)L jj =
→ as in T-invariant case
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 22
T-violation Parameter τ along Contour (B=53mT)
• T-violation parameter τ varies along the contour even though B is fixed
because the electromagnetic field at the ferrite changes
• τ increases (decreases) with increasing (decreasing) parameters s and δ
• τ returns after each loop around the EP to its initial value
t1 t2
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 23
Change of the Eigenvalues along the Contour (B=53mT)
• The real and the imaginary parts of the eigenvalues cross once during each
encircling at different values of t
→ the eigenvalues are interchanged
→
1
2
2
1
e
e
e
e
t1 t1t2 t2
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 24
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
eγ
γ
γ
γ
t1t1 t2t2
r1,1
r2,1r1,1
r2,1
t1 t1
• No general rule exists for the transformation scheme of the γj
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 25
Geometric Phase Re (γ j(t)) Gathered along Two Loops
t = 0: Re(γ 1(0)) = Re(γ 2(0)) = 0t = t1: Re(γ 1(t1)) = − Re(γ 2(t1)) = 0.31778
t = t2: Re(γ 1(t2)) = − Re(γ 2(t2)) = 0.09311
t = 0: Re(γ 1(0)) = Re(γ 2(0)) = 0t = t1: Re(γ 1(t1)) = − Re(γ 2(t1)) = 0.22468
t = t2: Re(γ 1(t2)) = − Re(γ 2(t2)) = −3.3⋅ 10-7
• Clearly visible that Re(γ 2(t)) = − Re(γ 1(t))t2
Twice the same loop
t1tt1 t2
Two different loops
t
• Same behavior observed for the imaginary part of γj(t)
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 26
Complex Phase γ 1(t) Gathered along Two Loops
Δ : start point
◊ : γ 1(t1) ≠ γ 1(0)
∇ : γ 1(t2) ≠ γ 1(0) if EP is encircled along different loops → | e iγ 1(t) | ≠ 1
possible → γ 1(t) depends on choice of path ⇒ geometrical phase
Two different loops Twice the same loop
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 27
Complex Phases γ j(t) Gathered when Encircling EP Twice along Double Loop
Δ : start point ∇ : γ 1(nt2) ≠ γ 1(0), n=1,2,…
• Encircle the EP 4 times along the contour with two different loops
• In the complex plane γ 1,2(t) drift away from the origin
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 28
Difference of Eigenvalues in the Parameter Plane
• Dark line: |f1-f2|=0 for s<sEP, |Γ1-Γ2|=0 for s>sEP
→ (e1-e2) = (f1-f2)+i(Γ1-Γ2) is purely imaginary for s<sEP / purely real for s>sEP
( ) Ι−= ˆHTr2
1HH effeffDL• ± (e1-e2) are the eigenvalues of
B=0 B=38mT B=61mT
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 29
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
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 30
PT symmetry of ĤDL along Dark Line
• For Vri=0 ĤDL can be brought to the form of Ĥ with the unitary transformation
zy στ/2σφU ii ee=
→ Talk by Uwe Günther
• At the EP its eigenvalues change from purely real to purely imaginary
→ exact PT symmetry is spontaneously broken
• General form of Ĥ, which fulfills [Ĥ, PT]=0:
B,A,AB
BAH
−
=i
ireal
• P : parity operator xσ=P , T : time-reversal operator T=K
• exact PT symmetry: the eigenvalues of Ĥ are real
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 31
• High precision experiments were performed in microwave billiards with and without T violation at and in the vicinity of an EP
• The behavior of the complex eigenvalues and ratios of the eigenvector components of the associated two-state Hamiltonian were investigated
• Encircling an EP:
• Eigenvalues: Eigenvectors:
• T-invariant case: γ 1(t) ≡ 0
• Violated T invariance: γ 1,2(t1) ≠ γ1,2(0), different loops: γ 1,2(t2) ≠ γ1,2(0)
• The size of T violation at the EP is determined from the phase of the ratio of the eigenvector components
• Exact PT symmetry is observed along a line in the parameter plane• Exact PT symmetry is spontaneously broken at the EP
Summary
→
1
2
2
1
e
e
e
e(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
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 32
Localization of EP for B=0 and B=53mT
B = 0: sEP=1.68 mm δEP=41.19 mm B = 53 mT: sEP=1.66 mm δEP=41.25 mm
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 33
Geometric Amplitude e−Im(γ 1(t)) Gathered Along Two Different/Equal Loops
t1t1
t = 0: Im(γ 1(0)) = Im(γ 2(0)) = 0
t = t1: Im(γ 1(t1)) = − Im(γ 2(t1)) = -0.00148
t = t2: Im(γ 1(t2)) = − Im(γ 2(t2)) = 0.01522
t = 0: Im(γ 1(0)) = Im(γ 2(0)) = 0
t = t1: Im(γ 1(t1)) = − Im(γ 2(t1)) = −0.01669
t = t2: Im(γ 1(t2)) = − Im(γ 2(t2)) = −3.5⋅ 10-7
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 34
Difference of Eigenvalues in the parameter plane
• Dark line: |f1-f2|=0 for s<sEP, |Γ1-Γ2|=0 for s>sEP
→ (e1-e2) = (f1-f2)+i(Γ1-Γ2) is real for s>sEP and purely imaginary for s<sEP
−−+
−−
=Ι+−=
2
EEHH
HH2
EE
ˆ2
EEHH
2112
S12
12S12
21
21effPT
A
A
i
i• ± (e1-e2) are the eigenvalues of
B=0 B=38mT B=61mT
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 35
Eigenvalues of HPT along Dark Line
,V2VVHε ri2
i2r
S12 i+−±=±• Eigenvalues of ĤPT:
( ) ( ) [ ]( )( )( ) ( ) [ ]( )( )4/EEImHImHImVH
4/EEReHReHReVH
221
2A12
2S12
2i
2S12
221
2A12
2S12
2r
2S12
−++=
−++=
• Dark Line: ( ) ( )( )4/EEImEEReHImHReHImHReV0 2121A12
A12
S12
S12ri −−++∝=
• Vr2 and Vi
2 cross at the EP
Dresden 2011 | Institut für Kernphysik, Darmstadt | SFB 634 | Barbara Dietz | 36
PT symmetry of HPT along Dark Line
• Unitary transformation:
• For Vri=0 ĤPT can be transformed into a PT-symmetric Hamiltonian
−=−
2φsin
HIm
2φcos
HRe
2φcos
HRe
2φsin
HIm
cosτ
1UHU
S12
S12
S12
S12
1PT
i
i
zy στ/2σφU ii ee= with ( )21
S12
EEIm
ImH
τcos
2φ2tan
−=
→ Talk by Uwe Günther
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