Microwave fidelity studies by varying antenna coupling

Transcription

1 Marburg, May 2010 p. Microwave fidelity studies by varying antenna coupling Hans-Jürgen Stöckmann Fachbereich Physik, Philipps-Universität Marburg, D Marburg, Germany [B. Köber, U. Kuhl, H.-J. St., T. Gorin, T. Seligman, D. Savin, PRE 82, (2010)]

2 Fidelity Marburg, May 2010 p. Introduced by Peres 1985 as a measure of the stability of quantum motion: Calculate the propagation of an initial pulse ψ 0 under the influence of two slightly different Hamiltonians H = H 0 and H λ = H 0 + λv Fidelity amplitude defined as overlap integral f λ (t) = ψ λ (t) ψ(t) = ψ 0 e ıh λt e ıh 0t ψ 0 Fidelity, as originally introduced by Peres: F λ (t) = f λ (t) 2

3 Experimental realisations Marburg, May 2010 p. Spin-Echo experiments in NMR (Levstein, Usaj, Pastawski 1998) Measures the nuclear magnetisation averaged over the whole sample forward and backward in time is measured! But wave functions are not accessible! Microwave billiards (Marburg group ) Allows in principle to measure wave functions by scanning with a probe antenna through the system Drawback: the probe antenna introduces a perturbation comparable in size with the effect to be studied!

4 Scattering matrix Marburg, May 2010 p. a i bi b n = S nm a m m Scattering matrix S = (S nm ): S nn : reflection amplitude at antenna n S nm, n m: transmission amplitude between antennas n and m Unique property of microwave experiments: All components of S accessible! Standard scattering experiments usually yield cross-sections only!

5 The scattering fidelity Marburg, May 2010 p. Introduced by us as a substitute for the ordinary fidelity: S ab (ω), S (λ) ab (ω): scattering matrix elements for the unperturbed and the perturbed system, resp. Scattering fidelity defined as f (λ) ab (t) = Ŝ(λ) ab (t)ŝ ab (t) : ensemble average Ŝ ab (t) = S ab (ω)e ıωt dω / Ŝ (λ) ab (t)ŝ (λ) ab (t) Ŝ ab (t)ŝ ab (t) For weak antenna coupling and chaotic systems the scattering fidelity reduces to the ordinary fidelity: f (λ) ab (t) f λ(t)

6 Possible parameter variations Marburg, May 2010 p. Shift of a wall (with T. Seligman, T. Gorin) Global variation, modeled by H = H 0 + λv, with H 0, V from GOE Gaussian or exponential decay Shift of an impurity Local variation, modeled by H = H 0 + λv V, with H 0, V from GOE Algebraic decay Wall deformation (with K. Richter, A. Goussev) Local variation, semi-classical description Algebraic decay

7 Variation of a channel coupling Marburg, May 2010 p. bouncing balls suppressed by insets ensemble average by rotatable ellipse

8 Variation of a channel coupling Marburg, May 2010 p. Antenna with different terminators: hard wall reflection open end reflection 50 Ω load bouncing balls suppressed by insets ensemble average by rotatable ellipse terminators from calibration kit

9 Marburg, May 2010 p. The effective Hamiltonian a i bi b n = S nm a m m Scattering theory yields S = 1 ıw GW 1 + ıw GW, G = 1 E H W = (W nk ): Matrix carrying the information on the coupling of the kth antenna to the nth eigenfunction Point-like coupling: W nk ψ n (r k )

10 The effective Hamiltonian (cont.) Marburg, May 2010 p. b A b C = S AA S CA A: measuring antenna C: variable antenna S AC S CC a A a C Terminator relates a C to b C : a C = e (α+ıϕ) b C = S matrix reduces to a 1D matrix for the measuring antenna: S AA = 1 ıw A G effw A 1 + ıw A G effw A, G eff = 1 E H eff H eff = H ıλ T W C W C, λ T = tanh α + ıϕ 2

11 Marburg, May 2010 p. 1 The effective Hamiltonian (cont.) H eff = H ıλ T W C W C, λ T = tanh α + 2 ıϕ Normalized coupling matrix: V = W C / W C W C With λ C = W C W C it follows H eff = H ıλv V, λ = λ T λ C λ C can be determined experimentally via transmission coefficient T C = 1 S CC 2 = 4λ C (1 + λ C ) 2

12 Marburg, May 2010 p. 1 Special cases of H eff H eff = H ıλv V, λ = λ T λ C λ T = tanh α + 2 ıϕ, λ C = W C W C 50Ω load nothing comes back, α : λ T = tanh α + 2 ıϕ H eff = H ıλ C V V 1 open end or hard wall reflection everything comes back, α = 0: λ T = tanh ıϕ 2 = ı tan ϕ 2 H eff = H + tan ϕ 2 λ CV V

13 Marburg, May 2010 p. 1 Special cases of H eff H eff = H ıλv V, λ = λ T λ C λ T = tanh α + 2 ıϕ, λ C = W C W C 50Ω load nothing comes back, α : λ T = tanh α + 2 ıϕ H eff = H ıλ C V V 1 open end or hard wall reflection everything comes back, α = 0: λ T = tanh ıϕ 2 = ı tan ϕ 2 H eff = H + tan ϕ 2 λ CV V ϕ unknown, but ϕ oe = ϕ hw + π! Hence (λ T ) hw = 1/ (λ T ) oe Yields relation between the three coupling constants: λ hw λ oe = λ 2 50Ω = λ2 C

14 Marburg, May 2010 p. 1 Theoretical description Fidelity f (λ ab (t) Ŝ(λ) ab (t)ŝ ab (t) Parametric cross-correlation function! Main result (D. Savin): Parametric cross-correlation function can be expressed in terms of an autocorrelation function with an effective parameter: Ŝ (λ 1) ab (t)ŝ (λ 2) ab (t) = Ŝ (λ eff) ab λ eff related to λ 1, λ 2 via (t)ŝ (λ eff) ab (t) 4λ eff (1 + λ eff ) 2 = 2(λ 1 + λ 2 ) (1 + λ 1)(1 + λ 2 ) Results from VWZ paper (Verbarschoot et al. 1985) applicable!

15 Results for fidelity amplitude Marburg, May 2010 p. 1 unperturbed system: no antenna perturbed system: antenna with terminator From fidelity decay: λ oe = 0.19ı λ hw = 0.23ı λ 50Ω = 0.20 λoe λ hw = 0.21 From reflection: λ C = 0.20

16 Fidelity for reflecting terminator Marburg, May 2010 p. 1 hard wall reflection: λ T = ı tan ϕ 2 open end reflection: λ T = ı cot ϕ 2 ϕ = 2πl/λ = 2πνl/c, l: effective length : hard wall : open end solid: experiment dashed: VWZ model

17 Fidelity for reflecting terminator Marburg, May 2010 p. 1 hard wall reflection: λ T = ı tan ϕ 2 open end reflection: λ T = ı cot ϕ 2 ϕ = 2πl/λ = 2πνl/c, l: effective length : hard wall : open end solid: experiment dashed: VWZ model

18 Fidelity for reflecting terminator Marburg, May 2010 p. 1 hard wall reflection: λ T = ı tan ϕ 2 open end reflection: λ T = ı cot ϕ 2 ϕ = 2πl/λ = 2πνl/c, l: effective length : hard wall : open end solid: experiment dashed: VWZ model

19 Marburg, May 2010 p. 1 Collected results λ oe = 0.65ı λ hw = 0.04ı λ 50Ω = 0.37 λ oe = 0.19ı λ hw = 0.23ı λ 50Ω = 0.20 λ oe = 0.05ı λ hw = 0.83ı λ 50Ω = 0.21 λoe λ hw = 0.16 λ C = 0.19 λoe λ hw = 0.21 λ C = 0.21 λoe λ hw = 0.20 λ C = 0.24

20 Marburg, May 2010 p. 1 Conclusions Description of the billiard with variable antenna in terms of an effective Hamiltonian Explicit expressions of the coupling parameters in terms of terminator properties Description of the scattering fidelity, a parametric cross-correlation function, in terms of an autocorrelation function with an effective parameter, thus reduction to the VWZ problem Quantitative agreement between experiment and theory

21 Marburg, May 2010 p. 1 Conclusions Description of the billiard with variable antenna in terms of an effective Hamiltonian Explicit expressions of the coupling parameters in terms of terminator properties Description of the scattering fidelity, a parametric cross-correlation function, in terms of an autocorrelation function with an effective parameter, thus reduction to the VWZ problem Quantitative agreement between experiment and theory General problem: Fidelity decays for closed and open systems hardly discernible Reliable results only for a perfectly controllable situation Usually an open channel simultaneously acts as a scatterer

22 Marburg, May 2010 p. 1 Thanks! Coworkers: B. Köber U. Kuhl Cooperations: T. Gorin, Guadalajara, Mexico T. Seligman, Cuernavaca, Mexico D. Savin, Brunel, UK The experiments have been supported by the DFG via the FG 760 Scattering Systems with Complex Dynamics.