Exceptional Points in Microwave Billiards with and without Time- Reversal Symmetry

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

2baain,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:

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


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


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))


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( i

AS

AS

eii

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 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


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


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


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


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


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


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 (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)


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


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


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


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 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


• 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

Exceptional Points in Microwave Billiards with and without Time- Reversal Symmetry

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