Exceptional Points in Microwave Billiards: Eigenvalues and Eigenfunctions

Transcription

1 Exceptional Points in Microwave Billiards: Eigenvalues and Eigenfunctions Dresden 011 Microwave billiards and quantum billiards Microwave billiards as a scattering system Eigenvalues and eigenfunctions of a dissipative system near an exceptional point Properties of exceptional points in the time domain Induced violation of time-reversal invariance determination of Ĥ eff at the EP Supported by DFG within SFB 634 S. Bittner, C. Dembowski, B. Dietz, J. Metz, M. Miski-Oglu, A. R., F. Schäfer H. L. Harney, H. A. Weidenmüller, D. Heiss, C. A. Stafford 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 1

2 Cylindrical Microwave Billiards y z x vectorial Helmholtz equation cylindrical resonators scalar Helmholtz equation ( ) Δ + k f μ r E f max = c d r E μ r (r) = E ( ) Δ + k E (x, y) = 0, E (x, y) = 0 μ μ r (r) μ = 0, r r n E μ r (r) μ G = 0, G μ k μ r (x, y) e = z πf c μ 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter

3 Microwave and Quantum Billiards microwave billiard quantum billiard ( Δ + k )Ez (x, y) = 0, Ez (x, y) G = 0 ( Δ + k ) Ψ(x, y) = 0, Ψ(x, y) G = 0 resonance frequencies eigenvalues electric field strengths eigenfunctions normal conducting resonators ~700 eigenfunctions superconducting resonators ~1000 eigenvalues 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 3

4 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 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 4

5 Transmission Spectrum Transmission measurements: relative power transmitted from a to b P out, b / P in,a S ba Scattering matrix Ŝ = Î-πiŴ T T (E-Ĥ+iπŴŴ ) -1 Ŵ Ĥ : resonator Hamiltonian Ŵ : coupling of resonator states to antenna states and to the walls 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 5

6 Resonance Parameters Use eigenrepresentation of S ba = δ Ĥ ba eff -i = Ĥ -iπŵŵ and obtain for a scattering system with isolated resonances a resonator b μ f -f μ Γ μa Γ T μb + (i/)γ μ Here: f µ = real part Γ µ = imaginary part of eigenvalues e µ of Ĥ eff Partial widths Γ µ,a, Γ µ,a and total width Γ µ 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 6

7 Typical Transmission Spectrum a) b) c) Transmission measurements: relative power from antenna a b S ba = P out,b / P in,a 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 7

8 Eigenvalues and Eigenfunctions of a Dissipative System near an EP At an exceptional point (EP) two (or more) complex eigenvalues and the corresponding eigenfunctions of a dissipative system coalesce The crossing of two eigenvalues is accomplished by the variation of two parameters Sketch of the experimental setup: Divide a circular microwave billiard into two approximately equal parts The opening s controls the coupling of the eigenmodes of the two billiard parts The position δ of the Teflon disc determines the resonance frequencies of the left part 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 8

9 Two-state Matrix Model (C. Dembowski et al., Phys. Rev. E 69, (004)) IsolatedEP in its vicinity the dynamics is determined by the two eigenstates model system with a d non-hermitian symmetric matrix All entries are functions of δ and s Eigenvalues: Ĥ e eff 1, R = E1 (s,δ) = H E1 + E = H S 1 Z S 1 + 1; H E ± R S 1 Z = E1 E H S 1 EPs: R = 0 : Z = ± i δ = δ, s = EP s EP 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 9

10 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 10 Encircling the EP in the Parameter Space Encircling the EP located at the parameter values s EP and δ EP Theeigenvalues are interchanged and one of the eigenfunctions in addition picks up a topological phase π ψ ψ ψ ψ, E E E E Ψ 1 Ψ Ψ Ψ e e e e

11 Experimental Setup (C. Dembowski et al., Phys. Rev. Lett. 86, 787 (001)) 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 11

12 Frequencies and Widths s < s EP s > s EP Change of the real (resonance frequency) and the imaginary (resonance width) part of the eigenvalue e 1, =f 1, +i. Γ 1, for varying δ and fixed s 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 1

13 Measuring Field Distributions perturbation body guiding magnet Use dielectric perturbation body, e.g. magnetic rubber Maier-Slater theorem: δ f(x,y) = c1 E (x,y) Reconstruct the eigenfunctions from the pattern of nodal lines, where E(x,y)=0 extraction of Ψ(x, y) 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 13

14 Evolution of Wave Functions 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 14

15 Chirality of Eigenfunctions Eigenfunctions are interchanged and one eigenfunction picks up a topological phase of π Surrounding the EP four times restores the original situation 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 15

16 What happens at the EP? (C. Dembowski et al., Phys. Rev. Lett. 90, (003)) At the exceptional point the two eigenmodes coalesce with a phase difference π/ Ψ (s = 0) s = 0 Ψ EP = Ψ 1 (s=0) + i Ψ (s=0) Ψ 1 (s = 0) at s = s EP and δ = δ EP Can this phase difference be measured? Choose two eigenmodes which are each localized in one of the semicircular parts of the cavity for s < s EP and δ = δ EP (i.e. f 1 =f ) these can be excited separately 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 16

17 Measured Phase Difference Modes Ψ 1 and Ψ are excited separately by antennas 1 and Measurement of the phase difference φ 0 between the oscillating fields Ψ 1 and Ψ at the positions of the antennas 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 17

18 Time Decay of the Resonances near and at the EP (Dietz et al., Phys. Rev. E 75, 0701 (007)) In the vicinity of an EP the two eigenmodes can be described as a pair of coupled damped oscillators Near the EP the time spectrum exhibits besides the decay of the resonances oscillations Rabi oscillations with frequency Ω = πf sin ( Ωt) (t) exp( Γt) ; Ω P Ω R AttheEP R and Ω vanish no oscillations P(t) t exp( Γt) In distinction, an isolated resonance decays simply exponentially line-shape at EP is not of a Breit-Wigner form 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 18

19 Time Decay of the Resonances "far" from the EP P(t) = FT{S 1 } (db) S 1 (db) f (GHz) Г 1 = ± MHz Г = 1.0 ± 0.00 MHz Region of interaction at about µs Rabi oscillations 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 19

20 Time Decay of the Resonances near and at the EP 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 0

21 Detailed Balance in Nuclear Reactions Search for Time-reversal (T-) Invariance Violation (TIV) in nuclear reactions 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 1

22 Detailed Balance in Nuclear Reactions Search for TIV in nuclear reactions upper limits 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter

23 Induced T-Invariance Violation in Microwave Billiards (B. Dietz et al., Phys. Rev. Lett. 98, (007)) T-invariance violation caused by a magnetized ferrite Ferrite features Ferromagnetic Resonance (FMR) Coupling of microwaves to the ferrite depends on the direction a b a b S ab b a S ba Principle of detailed balance: S ab = S ba Principle of reciprocity: S ab = S ba 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 3

24 Scattering Matrix Description Remember: Scattering matrix formalism Ŝ(f) = I - πiŵ T ( T f I - Ĥ + iπ ŴŴ )-1 Ŵ Investigation of T-invariant T-noninvariant systems replace Ĥ by a Note: Ĥ eff = Ĥ iπŵŵ T is non-hermitian in both cases Isolated resonances: singlets and doublets Ĥ is 1D or D GOE Overlapping resonances (Γ > D) Ĥ = in RMT GUE (B. Dietz et al., Phys. Rev. Lett. 103, (009) real symmetric Hermitian matrix 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 4

25 Isolated Resonances - Setup 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 5

26 Singlet with T-Violation S S ab S ba Arg(S) 0 -π Frequency (GHz) Reciprocity holds T-violation cannot be detected this way 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 6

27 Doublet with T-Violation Violation of reciprocity due to interference of two resonances 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 7

28 Scattering Matrix and T-violation Scattering matrix element (ω = πf) S ab (ω) = δ ab - πi a Ŵ (ω - Ĥ Decomposition of effective Hamiltonian Hˆ eff = ihˆ a + Hˆ s + eff ) -1 Ŵ b 0 ih a -ih H 1 0 H a 1 Ansatz for T-violation incorporating the ferromagnetic resonance and its selective coupling to the microwaves Note: Ĥ s conserves T-invariance and Ĥ a violates it 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 8

29 T-Violating Matrix Element H a 1 (B) = π λ B T r ω 0 ω M (B) - ω - i/t r coupling strength external field spin relaxation time magnetic susceptibility Fit parameters: λ and ω 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 9

30 Measured T-Violating Matrix Element π/ 0 -π/ T-violating matrix element shows resonance like structure Successful description of dependence on magnetic field Ĥ eff was determined and will be used to describe TIV at the EP 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 30

31 Relative Strength of T-Violation Compare: TIV matrix a element H 1 to the energy difference of two eigenvalues of the T-invariant system ξ = E H s 1 a 1 s - E 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 31

32 Summary and Outlook Microwave billiards are precisely controlled, parameter-dependent dissipative systems Complex eigenvalues and eigenfunctions can be determined from the billiard scattering matrix Observation of EPs in microwave billiards Behavior of eigenvalues close to and at an EP investigated experimentally Also time-behavior close to and at an EP was studied disappearance of echoes at the EP t -behavior T-invariance violation induced in microwave billiards and observed for doublets of resonances determination of complete Ĥ eff possible at an exceptional point Second talk: Measurement and interpretation of T-violation 011 Institute of Nuclear Physics Darmstadt/SFB 634 & ECT*, Trento Achim Richter 3