2013 | institute of nuclear physics darmstadt/sfb 634 | achim richter | 1 exceptional points in...

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


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


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


Transmission Spectrum

• Relative power transmitted from a to b

• Scattering matrix W)WWiπ+H-(EWiπ2-I=S 1-TT

2

baain,bout, SP/P

• Ĥ : resonator Hamiltonian

• Ŵ : coupling of resonator states to antenna states and to the walls


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


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


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


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


Test of Reciprocity

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

B=0 B=53mT


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)


Two-State Effective Hamiltonian

• Eigenvalues:

0H

H0

EH

HE)δ,(sH

12

12

2S12

S121

eff A

A

i

• EPs:

2

212

12

2

12

212,1

2

EEHH

2

EEe

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


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

2

EE 210

with

1b2a2b1a2b2a1b1a20

12

b20

2ab10

1aabab

VVVVVVVV)(

H

V)(

1VV

)(

1VS

if

ffS

→ 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


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(P2

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))


Time Decay of the Resonancesnear and at the EP


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( iAS

AS

ei

i

with t[ /2,/2[ real


Localization of an EP (B=0)

• Change of the real and the imaginary part of

the eigenvalues e1,2=f1,2+i G1,2/2. They cross at s=sEP=1.68 mm and d=dEP=41.19 mm.

j

jj ,2

j ,1j r

r ie

δ (mm)

1r

rr

2,j

1,j

j

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


Localization of an EP (B=53mT)

• Change of the real and the imaginary part of the eigenvalues e1,2=f1,2+i G1,2/2. They cross at s=sEP=1.66 mm and d=dEP=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 ie

• At (sEP,dEP)

τ

δ (mm)

s=1.66 mm


T-Violation Parameter t at the EP(B.Dietz et al. PRL 98, 074103 (2007))

• F 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


Eigenvalue Differences (e1-e2) in the Parameter Plane

• S matrix is measured for each point of a grid with s = = 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 (

mm

)

s (mm)d

(m

m)

s (mm)

B=0 B=53mT


Differences (|n1|-|n2|) and (F1-F2) 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 (

mm

)

s (mm)

t=0,t1,t2

d m

m)

s (mm)

B=0

t=0, t1, t2

t=0, t1, t2


Encircling the EP in the Parameter Space

• Encircling EP once:

1

2

2

1

21 ,eer

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 t

d τ0)( tγ

d t

d γ

d t

d γ1 ,2

21 and for

• Condition of parallel transport yields0)t(Rdt

d(t)L jj

for all t

0 ≡(t)γ=(t)γ 21 for B=0


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


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


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


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?


Geometric Phase Re (gj(t)) Gathered along Two Loops

t 0: Re(1(0)) Re(2(0)) 0t t1: Re(1(t1)) Re(2(t1)) 0.31778

t 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)) 0.22468

t 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)


Complex Phase g1(t) Gathered along Two Loops

Δ : start point

: g1(t1) g 1(0)

: g1(t2) g 1(0) if EP is encircled along different loops → | e ig1(t) | 1 → g1(t) depends on choice of path geometrical phase

Two different loops Twice the same loop


Complex Phases gj(t) Gathered when Encircling EP Twice along Double Loop

Δ : start point : g1(nt2) g 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


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

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

• Behavior reminds on that of the eigenvalues of a PT symmetric Hamiltonian


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

→ 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,BADC

DCBAH

ii

iireal

• 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


• 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

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

2013 | institute of nuclear physics darmstadt/sfb 634 | achim richter | 1 exceptional points in...

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