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. Schäfer H. L. Harney, O. N. Kirillov, U. Günther

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 2 Microwave and Quantum Billiards microwave billiardquantum billiard normal conducting resonators  ~700 eigenfunctions superconducting resonators  ~1000 eigenvalues resonance frequencies  eigenvalues electric field strengths  eigenfunctions

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 3 Microwave Resonator as a 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 inrf power out  

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 4 Transmission Spectrum Relative power transmitted from a to b Scattering matrix Ĥ : resonator Hamiltonian Ŵ : coupling of resonator states to antenna states and to the walls

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 5 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  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

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 6 Experimental setup variation of  variation of s variation of B (B. Dietz et al., Phys. Rev. Lett. 106, (2011)) Parameter plane ( s,  ) is scanned on a very fine grid

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 7 Experimental Setup to Induce T Violation NSNS NSNS 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

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 8 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 S ab = S ba → detailed balance |S ab | 2 = |S ba | 2 S ab S ba a b F

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 9 Test of Reciprocity Clear violation of the principle of reciprocity for nonzero magnetic field B=0B=53mT

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 10 Resonance Spectra Close to an EP for B=0 Scattering matrix : Ŵ : coupling of the resonator modes to the antenna states Ĥ eff : two-state Hamiltonian including dissipation and coupling to the exterior Frequency (GHz)

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 11 Two-State Effective Hamiltonian Eigenvalues : EPs : T- breaking matrix element. It vanishes for B=0. Ĥ eff : non-Hermitian and complex non-symmetric 2  2 matrix B=0 B0B0

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 12 Resonance Shape at the EP At the EP the Ĥ eff is given in terms of a Jordan normal form with → at the EP the resonance shape is not described by a Breit-Wigner form Note: this lineshape leads to a temporal t 2 -decay behavior at the EP S ab has two poles of 1st order and one pole of 2nd order

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 13 Time Decay of the Resonances near and at the EP Time spectrum = |FT{S ba (f)}| 2 Near the EP the time spectrum exhibits besides the decay of the resonances Rabi oscillations with frequency Ω   = ( e 1 -e 2 )/2 At the EP  and  vanish  no oscillations 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, (2007))

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 14 Time Decay of the Resonances near and at the EP

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 15 T -Violation Parameter  For each set of parameters ( s,  ) Ĥ eff is obtained from the measured Ŝ matrix Ĥ eff 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 with  real

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 16 Localization of an EP (B=0) Change of the real and the imaginary part of the eigenvalues e 1,2 =f 1,2 +i  1,2 /2. They cross at s=s EP =1.68 mm and  EP =41.19 mm. δ (mm) At (s EP,  EP ) Change of modulus and phase of the ratio of the components r j,1, r j,2 of the eigenvector |r j  s=1.68 mm

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 17 Localization of an EP (B=53mT) Change of the real and the imaginary part of the eigenvalues e 1,2 =f 1,2 +i  1,2 /2. They cross at s=s EP =1.66 mm and  EP =41.25 mm. Change of modulus and phase of the ratio of the components r j,1, r j,2 of the eigenvector |r j  At (s EP,  EP ) τ δ (mm) s=1.66 mm

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 18 T -Violation Parameter  at the EP (B.Dietz et al. PRL 98, (2007))  and also the T -violating matrix element shows resonance like structure coupling strength spin relaxation time magnetic susceptibility  EP =  /2+ 

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 19 Eigenvalue Differences ( e 1 -e 2 ) 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 e 1 -e 2 is either real or purely imaginary Ĥ eff  PT -symmetric Ĥ  (mm) s (mm)  (mm) s (mm) B=0 B=53mT

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 20 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=t 1 after one loop, t=t 2 after second one B=53mT t=0,t 1,t 2  (mm) s (mm) t=0,t 1,t 2  mm) s (mm) B=0 t=0, t 1, t 2

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 21 Encircling the EP in the Parameter Space Encircling EP once: Biorthonormality defines eigenvectors  l j (t)| r j (t)  =1 along contour up to a geometric factor : For B  0 additional geometric factor: and for Condition of parallel transport yields for all  for B=0

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 22 Change of the Eigenvalues along the Contour (B=0) Note: Im(e 1 ) + Im(e 2 ) ≈ 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 t1t1 t1t1 t2t2 t2t2

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 23 Evolution of the Eigenvector Components along the Contour (B=0) Evolution of the first component r j,1 of the eigenvector |r j  as function of t After each loop the eigenvectors are interchanged and the first one picks up a geometric phase  t1t1 t2t2 r 1,1 Phase does not depend on choice of circuit  topological phase r 2,1 t1t1 t2t2 r 1,1 r 2,1

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 24 T -violation Parameter  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  τ returns after each loop around the EP to its initial value t1t1 t2t2

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 25 Evolution of the Eigenvector Components along the Contour (B=53mT) Measured transformation scheme : No general rule exists for the transformation scheme of the γ j How does the geometric phase γ j evolve along two different and equal loops?

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 26 Geometric Phase Re ( j (t)) Gathered along Two Loops t  0: Re(  1 (0))  Re(  2 (0))  0 t  t 1 : Re(  1 (t 1 ))   Re(  2 (t 1 ))  t  t 2 : Re(  1 (t 2 ))   Re(  2 (t 2 ))  t  0: Re(  1 (0))  Re(  2 (0))  0 t  t 1 : Re(  1 (t 1 ))   Re(  2 (t 1 ))  t  t 2 : Re(  1 (t 2 ))   Re(  2 (t 2 ))  3.3  Clearly visible that Re(  2 (t))  Re(  1 (t)) t2t2 Twice the same loop t1t1 t t1t1 t2t2 Two different loops t Same behavior observed for the imaginary part of γ j (t)

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 27 Complex Phase  1 (t) Gathered along Two Loops Δ : start point  :  1 (t 1 )  1 (0)  :  1 (t 2 )  1 (0) if EP is encircled along different loops → | e i  1 (t) |  1 →  1 (t) depends on choice of path  geometrical phase Two different loops Twice the same loop

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 28 Complex Phases  j (t) Gathered when Encircling EP Twice along Double Loop Δ : start point  :  1 (nt 2 )  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

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 29 Difference of Eigenvalues in the Parameter Plane (S.Bittner et al., Phys.Rev.Lett. 108,024101(2012)) Dark line: |f 1 -f 2 |=0 for s s EP → (e 1 -e 2 ) = (f 1 -f 2 )+i (  1 -  2 )/2 is purely imaginary for s s EP  (e 1 -e 2 ) are the eigenvalues of B=0 B=38mT B=61mT

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 30 Eigenvalues of Ĥ DL along Dark Line Eigenvalues of Ĥ DL : real Dark Line: V ri =0 → radicand is real V r 2 and V i 2 cross at the EP Behavior reminds on that of the eigenvalues of a PT symmetric Hamiltonian

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 31 PT symmetry of Ĥ DL along Dark Line For V ri =0 Ĥ DL can be brought to the form of Ĥ with the unitary transformation → Ĥ DL 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: real P : parity operator, T : time-reversal operator T=K PT symmetry: the eigenvectors of Ĥ commute with PT

2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 32 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 (t 1 )  γ 1,2 (0), different loops :  1,2 (t 2 )  γ 1,2 (0) PT symmetry is observed along a line in the parameter plane. It is spontaneously broken at the EP Summary