Exceptional Points in Microwave Billiards with and without
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 | 1
Microwave and Quantum Billiards microwave billiard resonance frequencies electric field strengths normal conducting resonators superconducting resonators 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 2 quantum billiard eigenvalues eigenfunctions ~700 eigenfunctions ~1000 eigenvalues
Microwave Resonator as a Scattering System rf power in rf power out • 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 3
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 | 4
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 F • 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 5
Experimental setup (B. Dietz et al. , Phys. Rev. Lett. 106, 150403 (2011)) variation of d variation of B variation of s • Parameter plane (s, d) is scanned on a very fine grid 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 6
Experimental Setup to Induce T Violation N S • 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 | 7
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 Sab b F Sba • T-invariant system → principle of reciprocity Sab = Sba → detailed balance |Sab|2 = |Sba|2 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 8 a
Test of Reciprocity B=0 B=53 m. T • Clear violation of the principle of reciprocity for nonzero magnetic field 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 9
Resonance Spectra Close to an EP for B=0 Frequency (GHz) • Scattering matrix: • Ŵ : coupling of the resonator modes to the antenna states • Ĥeff : two-state Hamiltonian including dissipation and coupling to the exterior 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 10
Two-State Effective Hamiltonian • Ĥeff : non-Hermitian and complex non-symmetric 2 2 matrix B 0 B=0 • T-breaking matrix element. It vanishes for B=0. • Eigenvalues: • EPs: 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 11
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 1 st order and one pole of 2 nd order → 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 12
Time Decay of the Resonances near 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 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 13
Time Decay of the Resonances near and at the EP 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 14
T-Violation Parameter t • 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 with t [ /2, /2[ real • T violation is expressed by a real phase → usual practice in nuclear physics 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 15
Localization of an EP (B=0) • Change of the real and the imaginary part of the eigenvalues e 1, 2=f 1, 2+i G 1, 2/2. They cross at s=s. EP=1. 68 mm and d d. EP=41. 19 mm. δ (mm) • Change of modulus and phase of the ratio of the components rj, 1, rj, 2 of the eigenvector |rj • At (s. EP, d. EP) s=1. 68 mm 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 16
Localization of an EP (B=53 m. T) • Change of the real and the imaginary part of the eigenvalues e 1, 2=f 1, 2+i G 1, 2/2. They cross at s=s. EP=1. 66 mm and d d. EP=41. 25 mm. δ (mm) • Change of modulus and phase of the ratio of the components rj, 1, rj, 2 of the eigenvector |rj τ • At (s. EP, d. EP) s=1. 66 mm 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 17
T-Violation Parameter t at the EP (B. Dietz et al. PRL 98, 074103 (2007)) • FEP= /2+t • F and also the T-violating matrix element shows resonance like structure coupling strength spin relaxation time 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 18 magnetic susceptibility
Eigenvalue Differences (e 1 -e 2) in the Parameter Plane B=53 m. T d (mm) B=0 s (mm) • S matrix is measured for each point of a grid with Ds = Dd = 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 • Ĥeff PT-symmetric Ĥ 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 19
Differences (|n 1|-|n 2|) and (F 1 -F 2) of the Eigenvector Ratios in the Parameter Plane B=53 m. T d mm) d (mm) B=0 t=0, t 1, t 2 t=0, t 1 t=0, t 2 1, t 2 s (mm) • 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 20
Encircling the EP in the Parameter Space • Biorthonormality defines eigenvectors lj(t)| rj(t) =1 along contour up to a geometric factor : • Condition of parallel transport and yields for • Encircling EP once: • For B 0 additional geometric factor: 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 21 for B=0 for all t
Change of the Eigenvalues along the Contour (B=0) t 1 t 2 • 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=53 m. T 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 22
Evolution of the Eigenvector Components along the Contour (B=0) r 2, 1 r 1, 1 t 2 t 1 • 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 23 t 2
T-violation Parameter t along Contour (B=53 m. T) t 1 t 2 • 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 24
Evolution of the Eigenvector Components along the Contour (B=53 m. T) • 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? 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 25
Geometric Phase Re (gj(t)) Gathered along Two Loops Two different loops t 1 t Twice the same loop t 2 t t 1 t 2 • Clearly visible that Re(g 2(t)) Re(g 1(t)) t 0: Re(g 1(0)) Re(g 2(0)) 0 t t 1: Re(g 1(t 1)) Re(g 2(t 1)) 0. 31778 t t 1: Re(g 1(t 1)) Re(g 2(t 1)) 0. 22468 t t 2: Re(g 1(t 2)) Re(g 2(t 2)) 0. 09311 t t 2: Re(g 1(t 2)) Re(g 2(t 2)) 3. 3 10 -7 • Same behavior observed for the imaginary part of γj(t) 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 26
Complex Phase g 1(t) Gathered along Two Loops Two different loops Twice the same loop Δ : start point : g 1(t 1) g 1(0) : g 1(t 2) g 1(0) if EP is encircled along different loops → | e ig 1(t) | 1 → g 1(t) depends on choice of path geometrical phase 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 27
Complex Phases gj(t) Gathered when Encircling EP Twice along Double Loop Δ : start point : g 1(nt 2) g 1(0), n=1, 2, … • Encircle the EP 4 times along the contour with two different loops • In the complex plane g 1(t) drifts away from g 2(t) and the origin → ‘geometric instability’ • The geometric instability is physically reversible 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 28
Difference of Eigenvalues in the Parameter Plane (S. Bittner et al. , Phys. Rev. Lett. 108, 024101(2012)) B=0 B=38 m. T B=61 m. T • Dark line: |f 1 -f 2|=0 for s<s. EP, |Γ 1 -Γ 2|=0 for s>s. EP → (e 1 -e 2) = (f 1 -f 2)+i(G 1 -G 2)/2 is purely imaginary for s<s. EP / purely real for s>s. EP • (e 1 -e 2) are the eigenvalues of 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 29
Eigenvalues of ĤDL along Dark Line • Eigenvalues of ĤDL: real • Dark Line: Vri=0 → radicand is real • Vr 2 and Vi 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 | 30
PT symmetry of ĤDL along Dark Line • P : parity operator , T : time-reversal operator T=K • General form of Ĥ, which fulfills [Ĥ, PT]=0: real • PT symmetry: the eigenvectors of Ĥ commute with PT • For Vri=0 ĤDL can be brought to the form of Ĥ with the unitary transformation → 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 31
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: g 1(t) 0 • Violated T invariance: g 1, 2(t 1) γ 1, 2(0), different loops: g 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 2013 | Institute of Nuclear Physics Darmstadt/SFB 634 | Achim Richter | 32
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