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

1. Exceptional Points in MicrowaveBilliardswithandwithout Time-ReversalSymmetry EinGedi 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 Supportedby DFG within SFB 634 S. Bittner, B. Dietz, M. Miski-Oglu, A. R., F. Schäfer H. L. Harney, O. N. Kirillov, U. Günther

2. Microwave and Quantum Billiards microwave billiard quantum billiard resonance frequencies  eigenvalues electric field strengths eigenfunctions normal conductingresonators~700 eigenfunctions superconductingresonators~1000eigenvalues

3. rf power in rf power out Microwave Resonator asa Scattering System   • Microwave power isemittedintotheresonatorbyantennaandtheoutputsignalisreceivedbyantenna Open scatteringsystem • The antennas act as single scattering channels

4. Transmission Spectrum • Relative power transmittedfrom a to b • Scatteringmatrix • Ĥ : resonator Hamiltonian • Ŵ : coupling of resonator states to antenna states and to the walls

5. HowtoDetect an ExceptionalPoint?(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

6. Experimental setup (B. Dietz et al., Phys. Rev. Lett. 106, 150403 (2011)) variation of d variation ofB variation ofs • Parameter plane (s,d) is scanned on a very fine grid

7. N S N S Experimental Setup to Induce T Violation • 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

8. F Sab a b Sba Violation of T Invariance Induced with a Magnetized Ferrite • SpinsofthemagnetizedferriteprecesscollectivelywiththeirLarmorfrequencyabouttheexternalmagneticfield • The magneticcomponentoftheelectromagneticfieldcoupledintothecavityinteractswiththeferromagneticresonanceandthecouplingdependson thedirection a b • T-invariant system→ principle of reciprocitySab= Sba →detailed balance|Sab|2= |Sba|2

9. Test of Reciprocity B=0 B=53mT • Clear violation of the principle of reciprocity for nonzero magnetic field

10. Frequency (GHz) Frequency (GHz) • Scattering matrix: ResonanceSpectra Close to an EP for B=0 • Ŵ :coupling of the resonator modes to the antenna states • Ĥeff : two-state Hamiltonian including dissipation and coupling to the exterior

11. T-breaking matrix element. It vanishes for B=0. Two-State Effective Hamiltonian • Ĥeff : non-Hermitian and complex non-symmetric 22 matrix B0 B=0 • Eigenvalues: • EPs:

12. Resonance Shape at the EP • At the EP the Ĥeff is given in terms of a Jordan normal form with • Sab has two poles of 1st order and one pole of 2nd order • → 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

13. Time Decay of the Resonancesnear and at the EP (Dietz et al., Phys. Rev. E 75, 027201 (2007)) • Time spectrum = |FT{Sba(f)}|2 • Near the EP the time spectrum exhibits besides the decay of the resonances Rabi oscillationswith frequency Ω   = (e1-e2)/2 • Atthe EP andvanish  nooscillations • In distinction, an isolated resonance decays simply exponentially→ line-shape at EP is not of a Breit-Wigner form

14. Time Decay of the Resonancesnear and at the EP

15. witht[- p/2,p/2[real T-Violation Parameter t • For each set of parameters (s,d) Ĥeffis obtained from the measuredŜmatrix • ĤeffandŴare determined up to common real orthogonal transformations • Choose real orthogonal transformation such that • Tviolation is expressed by a real phase → usual practice in nuclear physics

16. At (sEP,dEP) Localization of an EP (B=0) • Change of the real and the imaginary part of • the eigenvaluese1,2=f1,2+iG1,2/2. They cross at • s=sEP=1.68 mmandd=dEP=41.19 mm. δ (mm) • Change of modulus and phase of the ratio • of the components rj,1, rj,2of the eigenvector |rj s=1.68 mm

17. Localization of an EP (B=53mT) • Change of the real and the imaginary part of the eigenvaluese1,2=f1,2+iG1,2/2. They cross ats=sEP=1.66 mmandd=dEP=41.25 mm. δ (mm) • Change of modulus and phase of the ratio of the components rj,1, rj,2 of theeigenvector|rj τ • At (sEP,dEP) s=1.66 mm

18. spinrelaxationtime couplingstrength magneticsusceptibility T-Violation Parameter t attheEP(B.Dietz et al. PRL 98, 074103 (2007)) • FEP=p/2+t • Fand also theT-violating matrix element shows resonance like structure

19. d (mm) d (mm) s (mm) s (mm) Eigenvalue Differences (e1-e2) in the Parameter Plane B=53mT B=0 • S matrix is measured for each point of a grid withDs= Dd = 0.01 mm • The darker the color, the smaller is the respective difference • Along the darkest line e1-e2iseither real or purely imaginary • Ĥeff PT-symmetricĤ

20. d (mm) d mm) t=0, t1, t2 t=0,t1,t2 t=0, t1, t2 t=0,t1,t2 s (mm) s (mm) Differences (|n1|-|n2|) and (F1-F2) of the Eigenvector Ratios in the Parameter Plane B=53mT B=0 • The darker the color, the smaller the respective difference • Encircle the EP twice along different loops • Parameterize the contour by the variabletwitht=0at start point, t=t1after oneloop,t=t2after second one

21. Condition of parallel transport yields and for Encircling the EP in the Parameter Space • Biorthonormality defines eigenvectors lj(t)| rj(t)=1along contour up to a geometric factor : for B=0 for all t • Encircling EP once: • For B0 additionalgeometric factor:

22. Change of the Eigenvalues along the Contour (B=0) t1 t2 t2 t1 • 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

23. Evolution of the Eigenvector Components along the Contour (B=0) r2,1 r2,1 r1,1 r1,1 t1 t2 t1 t2 • 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  • Phase does not depend on choice of circuit  topological phase

24. T-violation Parameter t along Contour (B=53mT) t1 t2 • 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 sand d • τ returns after each loop around the EP to its initial value

25. Evolution of the Eigenvector Components along the Contour (B=53mT) • Measured transformation scheme: • No general rule exists for the transformation scheme of the γj • How does the geometric phase γj evolve along two different and equal loops?

26. Two different loops Twice the same loop t1 t t2 t1 t t2 Geometric PhaseRe (gj(t))Gathered along Two Loops • Clearly visible that Re(g2(t)) = - Re(g1(t)) t = 0: Re(g1(0)) = Re(g2(0)) = 0 t = 0: Re(g1(0)) = Re(g2(0)) = 0 t = t1: Re(g1(t1))  Re(g2(t1)) = 0.31778 t = t1: Re(g1(t1))  Re(g2(t1)) = 0.22468 t = t2: Re(g1(t2))  Re(g2(t2)) = 0.09311 t = t2: Re(g1(t2))  Re(g2(t2)) = -3.310-7 • Same behavior observed for the imaginary part of γj(t)

27. Complex Phase g1(t) Gathered along Two Loops Twice the same loop Two different loops • Δ: startpoint: g1(t1)  g1(0)  : g1(t2)  g1(0) if EP isencircledalong different loops → | e ig1(t) | 1 • → g1(t) depends on choiceofpathgeometricalphase

28. Complex Phases gj(t) Gathered when Encircling EP Twice along Double Loop Δ: start point : g1(nt2)  g1(0), n=1,2,… • Encircle the EP 4 times along the contour with two different loops • In the complex plane g1(t) drifts away from g2(t) and the origin →‘geometric instability’ • The geometric instability is physically reversible

29. Difference of Eigenvalues in the Parameter Plane (S.Bittner et al., Phys.Rev.Lett. 108,024101(2012)) B=0 B=61mT B=38mT • Dark line: |f1-f2|=0 for s<sEP, |Γ1-Γ2|=0 for s>sEP • → (e1-e2) = (f1-f2)+i(G1-G2)/2is purely imaginary for s<sEP / purelyreal for s>sEP •  (e1-e2)are the eigenvalues of

30. Eigenvalues of ĤDL: real Eigenvalues of ĤDL along Dark Line • Dark Line: Vri=0 → radicand is real • Vr2 and Vi2 cross at the EP • Behavior reminds on that of the eigenvalues of a PT symmetric Hamiltonian

31. real ,T: time-reversal operatorT=K • P: parity operator PT symmetry of ĤDL along Dark Line • General form of Ĥ, which fulfills [Ĥ, PT]=0: • PT symmetry: the eigenvectors of Ĥcommute with PT • For Vri=0ĤDL can be brought to the form of Ĥwiththe unitary transformation • → ĤDLis invariant under the antilinear operator • At the EP the eigenvalues change from purely real to purely imaginary • →PTsymmetry is spontaneously broken at the EP

32. Summary • 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:g1(t)  0 • Violated T invariance: g1,2(t1)  γ1,2(0), different loops: g1,2(t2)  γ1,2(0) • PT symmetry is observed along a line in the parameter plane. It is spontaneously broken at the EP