EPs in Microwave Billiards: Eigenvectors and the Full Hamiltonian for T-invariant and T-noninvariant Systems Dresden 2011

Transcription

1 EPs in Microwave Billiards: Eigenvectors and the Full Hamiltonian for T-invariant and T-noninvariant Systems Dresden 2011 Precision experiment with microwave billiard extraction of full EP Hamiltonian from scattering matrix Properties of eigenvalues and eigenvectors at and close to an EP EPs in systems with violated T invariance Encircling the EP: geometric phases and amplitudes PT symmetry of the EP Hamiltonian Supported by DFG within SFB 634 S. Bittner, B. D., M. Miski-Oglu, A. Richter, F. Schäfer H. L. Harney, O. N. Kirillov, U. Günther Dietz

2 Microwave Resonator for the Observation of Exceptional Points Divide a circular microwave billiard into two approximately equal parts The coalescence at the EP is accomplished by the variation of two parameters The opening s controls the coupling of the eigenmodes of the two billiard parts F The position δ of the Teflon disk mainly effects the resonance frequencies of the left part Insert a ferrite F and magnetize it with an exterior magnetic field B to induce T violation

3 Experimental setup (B. Dietz et al., Phys. Rev. Lett. 106, (2011)) variation of δ variation of B variation of s Parameter plane (s,δ) is scanned on a very fine grid

4 Resonance Spectra Close to an EP (B=0) Frequency (GHz) Frequency (GHz) ˆ T (E Hˆ ) 1 W ˆ Scattering matrix: Sˆ = ˆI 2 π i W eff Ĥeff : two-state Hamiltonian including dissipation and coupling to the exterior Ŵ : coupling of the resonator modes to the antenna states

5 Two-State Matrix Model for the T-invariant Case Determine the non-hermitian complex symmetric 2 2 matrix Ĥeff and its eigenvalues and eigenvectors for each (s,δ) from the measured Ŝ matrix E1 ˆ H eff (s,δ) = S H 12 S H12 E 2 ries are functions of δ and s Eigenvalues: E + E2 e1, 2 = 1 ±ℜ 2 S ℜ = H12 Z2 + 1 ; Z = EPs: E1 E 2 S 2H12 ℜ = 0 : Z = ± i δ = δ EP, s = s EP

6 Resonance Shape at the EP At the EP the Ĥeff is given in terms of a Jordan normal form ˆ (s,δ ) = λ0 H eff EP EP 0 1 λ0 with E1 + E 2 λ0 = 2 Sab has two poles of 1st order and one pole of 2nd order Sab = δ ab V1a 1 1 V1b V2a V2 b ( f λ0 ) ( f λ0 ) H12S { i ( V1a V1b V2a V2b ) + V1a V2b + V2a V1b } 2 ( f λ0 ) at the EP the resonance shape is not described by a Breit-Wigner form Note: this lineshape leads to t2-behavior ( first talk)

7 Localization of an EP (B=0) Change of the real and the imaginary part of the eigenvalues e1,2=f1,2+i Γ1,2. They cross at s=sep=1.68 mm and δ= δep=41.19 mm. δ (mm) Change of modulus and phase of the ratio of the components rj,1, rj,2 of the eigenvector rj νj = At (sep,δep) rj,1 rj,2 =νje iφ j rj,1 i rj = r 1 j, 2 s=1.68 mm

8 Eigenvalues and Ratios of Eigenvector Components in the Parameter Plane (B=0) t=0,t1,t2 S matrix is measured for each point of a grid with s = δ = 0.01 mm Note the dark line, where e1-e2 is either real or purely imaginary. There, Ĥeff PT-symmetric Ĥ Parameterize the contour by the variable t with t=0 at start point, t=t1 after one loop, t=t2 after second one

9 Encircling the EP in the Parameter Plane (T-invariant Case) The biorthonormalized eigenvectors lj(t) and rj(t) with lj(t) rj(t) =1 are defined up to a geometric factor e± iγ j(t) The geometric phases γ j(t) are fixed by the condition of parallel transport l j ( t )e iγ j ( t ) d iγ ( t ) rj ( t ) e j = 0 dt γ1 (t) = γ 2 (t) 0 With tan θ ( t ) = 1 + Z 2 Z the eigenvectors are cos θ ( t ) sin θ ( t ), r2 ( t ) = r1 ( t ) = sin θ ( t ) cos θ ( t ) π Encircling the EP once: θ θ + e1 e 2, 2 r1 r2 r r 2 1

10 Change of the Eigenvalues along the Contour (B=0) t1 t2 t1 The real, respectively, the imaginary parts of the eigenvalues cross once during each encircling at different t e1 e 2 The eigenvalues are interchanged e 2 e1 Note: Im(e1) + Im(e2) const. dissipation depends weakly on (s,δ) t2

11 Evolution of the Eigenvector Components along the Contour (B=0) r2,1 r2,1 r1,1 t1 r1,1 t2 t1 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 π r1 r 2 r2 r 1 r1 r 2 Phase does not depend on choice of circuit topological phase t2

12 Summary for T-invariant case Full EP Hamiltonian extracted from measured scattering matrix direct determination of its eigenvalues and eigenvectors possible Behavior of eigenvalues and eigenvectors for T-invariant case as expected confirms validity of the procedure used for data analysis Next step: Investigation of the T-noninvariant case following the same procedure

13 Microwave Billiard for the Study of Induced T Violation N S 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

14 Induced Violation of T Invariance with a Ferrite Spins of magnetized ferrite precess collectively with their Larmor frequency about the external magnetic field ( first talk) Coupling of rf magnetic field to the ferromagnetic resonance depends on the direction a b Sab F a b Sba T-invariant system principle of reciprocity Sab = Sba detailed balance Sab 2 = Sba 2

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

16 Two-State Matrix Model for Broken T invariance Ĥeff : non-hermitian and non-symmetric complex 2 2 matrix ( first talk) ˆ (s,δ) = E1 H eff HS 12 S 0 H12A H12 +i A E 2 H12 0 A H12 : T-breaking matrix element Eigenvalues: E + E2 e1, 2 = 1 ±ℜ 2 S 2 12 ℜ= H EPs: +H A2 12 Z +1; Z = E1 E 2 2 S H +H A2 12 ℜ = 0 : Z = ± i δ = δ EP, s = s EP

17 T-Violation Parameter τ For each set of parameters (s,δ) Ĥeff is obtained from the measured Ŝ matrix ˆ T (E H ˆ ) 1 W ˆ Sˆ = ˆI 2 π i W eff Ĥeff and Ŵ are determined up to common real orthogonal transformations Choose real orthogonal transformation such that (H12S + i H12A ) 2i τ =e with τ [0,π[ real S A (H12 i H12 ) T violation is expressed by a real phase usual practice in nuclear physics

18 Localization of an EP (B=53mT) Change of the real and the imaginary part of the eigenvalues e1,2=f1,2+i Γ1,2. They cross at s=sep=1.66 mm and δ= δep=41.25 mm. δ (mm) Change of modulus and phase of the ratio of the components rj,1, rj,2 of the eigenvector rj νj = τ At (sep,δep) rj,1 r j, 2 = νj e iφ j rj,1 ie iτ rj = r j, 2 1 s=1.66 mm

19 T-Violation Parameter τ at the EP ΦEP=π/2+τ Φ and also the T-violating matrix element shows resonance like structure π ωm2 A r H12 (B) = λ B T ( first talk) r 2 ω0 (B) ω i / T coupling strength spin relaxation time magnetic susceptibility

20 Eigenvalues and Ratios of Eigenvector Components in Parameter Plane (B=53mT) t=0,t1,t2 S matrix is measured for each point of a grid with s = δ = 0.01 mm Note the dark line, where e1-e2 is either real or purely imaginary. There, Ĥeff PT-symmetric Ĥ Parameterize the contour by the variable t with t=0 at starting point, t=t1 after one loop, t=t2 after second one

21 Encircling the EP in the Parameter Space (with T violation) L j (t ) = l j (t ) e Eigenvectors along contour: - iγ j ( t ), R j ( t ) = rj ( t ) e Biorthonormality is defined up to a geometric factor Condition of parallel transport L j (t) e iγ j (t ) ± iγ j (t) d R j ( t ) = 0 yields dt dγ1 dγ dτ = 2 and γ1,2 (t ) 0 for 0 dt dt dt r1 r2 as in T-invariant case Encircling EP once: e1 e 2, r r 2 1 Additional geometric factor: e iγ j (t) =e i Re( γ j (t)) e Im(γ j (t))

22 T-violation Parameter τ along Contour (B=53mT) t1 t2 T-violation parameter τ varies along the contour even though B is fixed because the electromagnetic field at the ferrite changes τ increases (decreases) with increasing (decreasing) parameters s and δ τ returns after each loop around the EP to its initial value

23 Change of the Eigenvalues along the Contour (B=53mT) t1 t2 t1 The real and the imaginary parts of the eigenvalues cross once during each encircling at different values of t the eigenvalues are interchanged e1 e 2 e 2 e1 t2

24 Evolution of the Eigenvector Components along the Contour (B=53mT) r2,1 r1,1 r2,1 r1,1 tt1 1 Measured transformation scheme: t2 tt11 r1 r2 e iγ 1 (t1 ) r1 e iγ 1 ( t2 ) i γ ( t ) i γ ( t ) r r e 1 1 r e No general rule exists for the transformation scheme of the γj t2

25 Geometric Phase Re (γ j(t)) Gathered along Two Loops Two different loops t1 t Twice the same loop t2 t t1 t2 Clearly visible that Re(γ 2(t)) = Re(γ 1(t)) t = 0: Re(γ 1(0)) = Re(γ 2(0)) = 0 t = 0: Re(γ 1(0)) = Re(γ 2(0)) = 0 t = t1: Re(γ 1(t1)) = Re(γ 2(t1)) = t = t1: Re(γ 1(t1)) = Re(γ 2(t1)) = t = t2: Re(γ 1(t2)) = Re(γ 2(t2)) = t = t2: Re(γ 1(t2)) = Re(γ 2(t2)) = Same behavior observed for the imaginary part of γj(t)

26 Complex Phase γ 1(t) Gathered along Two Loops Two different loops Twice the same loop Δ : start point : γ 1(t1) γ 1(0) : γ 1(t2) γ 1(0) if EP is encircled along different loops e iγ 1(t) 1 possible γ 1(t)2011 depends on choice of path geometrical phase Dresden Institut für Kernphysik, Darmstadt SFB 634 Barbara

27 Complex Phases γ j(t) Gathered when Encircling EP Twice along Double Loop Δ : start point : γ 1(nt2) γ 1(0), n=1,2, Encircle the EP 4 times along the contour with two different loops In the complex plane γ 1,2(t) drift away from the origin

28 Difference of Eigenvalues in the Parameter Plane 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(γ1-γ2) is purely imaginary for s<sep / purely real for s>sep ( ) 1 ± (e1-e2) are the eigenvalues of Hˆ DL = Hˆ eff Tr Hˆ eff Ιˆ 2

29 Eigenvalues of ĤDL along Dark Line Eigenvalues of ĤDL: ε ± = ± Vr Vi + 2i Vri, 2 2 Vr, Vi, Vri real Dark Line: Vri=0 radicand is real Vr2 and Vi2 cross at the EP

30 PT symmetry of ĤDL along Dark Line General form of Ĥ, which fulfills [Ĥ, PT]=0: B ia ˆ, A, B real H = B ia P : parity operator P = σˆ x, T : time-reversal operator T=K exact PT symmetry: the eigenvalues of Ĥ are real For Vri=0 ĤDL can be brought to the form of Ĥ with the unitary transformation i φ σˆ Uˆ = e y ei τ/2 σˆ z At the EP its eigenvalues change from purely real to purely imaginary exact PT symmetry is spontaneously broken Talk by Uwe Günther

31 Summary High precision experiments were performed in microwave billiards with and without T violation at and in the vicinity of an EP The behavior of the complex eigenvalues and ratios of the eigenvector components of the associated two-state Hamiltonian were investigated Encircling an EP: Eigenvalues: e1 e 2 e 2 e1 Eigenvectors: r1 r2 e iγ1 (t1 ) r1 e iγ1 (t 2 ), γ (t) = γ (t) 2 r r e iγ1 (t1 ) r e iγ1 (t 2 ) γ 1(t) 0 T-invariant case: Violated T invariance: γ 1,2(t1) γ1,2(0), different loops: γ 1,2(t2) γ1,2(0) The size of T violation at the EP is determined from the phase of the ratio of the eigenvector components Exact PT symmetry is observed along a line in the parameter plane Exact PT symmetry is spontaneously broken at the EP

32 Localization of EP for B=0 and B=53mT B = 0: sep=1.68 mm δep=41.19 mm B = 53 mt: sep=1.66 mm δep=41.25 mm

33 Geometric Amplitude e Im(γ 1(t)) Gathered Along Two Different/Equal Loops t1 t1 t = 0: Im(γ 1(0)) = Im(γ 2(0)) = 0 t = 0: Im(γ 1(0)) = Im(γ 2(0)) = 0 t = t1: Im(γ 1(t1)) = Im(γ 2(t1)) = t = t1: Im(γ 1(t1)) = Im(γ 2(t1)) = t = t2: Im(γ 1(t2)) = Im(γ 2(t2)) = t = t2: Im(γ 1(t2)) = Im(γ 2(t2)) =

34 Difference of Eigenvalues in the parameter plane 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(γ1-γ2) is real for s>sep and purely imaginary for s<sep ± (e1-e2) are the eigenvalues of Hˆ PT = Hˆ eff E1 E 2 S H12 i H12A E + E2 ˆ 2 1 Ι= S E1 E 2 2 A H + i H

35 Eigenvalues of HPT along Dark Line S ε ± = ± H12 Vr2 Vi2 + 2i Vri, Eigenvalues of ĤPT: Dark Line: ( S S A A 0 = Vri Re H12 Im H12 + Re H12 Im H12 + Re( E1 E 2 ) Im( E1 E 2 ) / 4 2 (( = ( ( Im H S S H12 Vr2 = Re H12 S 2 12 H Vi2 ) + ( Re H ) + ( Re [ E E ] ) ) + ( Im H ) + ( Im [ E E ] ) 2 S 2 12 A 2 12 A ) / 4) /4 Vr2 and Vi2 cross at the EP )

36 PT symmetry of HPT along Dark Line For Vri=0 ĤPT can be transformed into a PT-symmetric Hamiltonian S Im H12 i ˆH ˆ U ˆ 1 = 1 sin 2φ U PT S cosτ Re H12 cos 2φ Unitary transformation: Uˆ = e i φ σˆ y i τ/2 σˆ z e S Re H12 cos 2φ S Im H12 i sin 2φ S 2 ImH12 with tan 2φ = cos τ Im( E1 E 2 ) Talk by Uwe Günther