EPs in Microwave Billiards: Eigenvectors and the Full T ...phhqpx11/dietz.pdf · There, Ĥeff →...

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


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


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)


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


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


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


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


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⇒


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


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


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


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


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


Test of Reciprocity

• Clear violation of the principle of reciprocity for nonzero magnetic field

B=0 B=53mT


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)


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


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


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)


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


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


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


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


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


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)


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


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


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


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


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


• 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


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


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


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


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


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

EPs in Microwave Billiards: Eigenvectors and the Full T ...phhqpx11/dietz.pdf · There, Ĥeff →...

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