Random Wave Model in theory and experiment
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1 Random Wave Model in theory and experiment Ulrich Kuhl Fachbereich Physik, Philipps-Universität Marburg, D Marburg, Deutschland Maribor, February 2009
2 Literature on Quantum Chaos H.-J. Stöckmann, Quantum Chaos - An Introduction F. Haake, Quantum Signatures of Chaos M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics M. L. Mehta, Random Matrices B. Eckhardt, Quantum mechanics of classically non-integrable systems, Phys. Rep. 163, 205 (1988) O. Bohigas in: Proc. Les Houches Summer School on Chaos and Quantum Physics T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Random matrix theories in quantum physics: common concepts, Phys. Rep. 299, 189 (1998) M.V. Berry, in: M.J. Gianonni, A. Voros, J. Zinn-Justin (Eds.), Chaos and Quantum Physics FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 1
3 Literature on Classical Chaos A. J. Lichtenberg and M. A. Liebermann. Regular and Stochastic Motion E. Ott, Chaos in Dynamical Systems H.G. Schuster, Deterministic chaos: An introduction P. Cvitanovic et al, Classical and Quantum Chaos, ( FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 2
4 Papers M. S. Longuet-Higgins, Phil. Trans. R. Soc. Lond. A 249, 321 (1957) M. S. Longuet-Higgins, Proc. R. Soc. Lond. A 246, 99 (1958) M. V. Berry, J. Phys. A 10 (1977) 2083) K. J. Ebeling, Optica Acta 26, 1345 (1979) M. V. Berry and M. R. Dennis, Proc. R. Soc. Lond. A 456, 2059 (2000). Y.-H. Kim, M. Barth, U. Kuhl, and H.-J. Stöckmann, Prog. Theor. Phys. Suppl. 150, 105 (2003) J. D. Urbina and K. Richter, J. Phys. A 36, L495 (2003) U. Kuhl, Eur. Phys. J. Special Topics 145, 103 (2007) M. R. Dennis, Eur. Phys. J. Special Topics 145, 191 (2007) S. Tomsovic, D. Ullmo, and A. Bäcker. Phys. Rev. Lett. 100, (2008) R. Höhmann, U. Kuhl, H.-J. Stöckmann, J. D. Urbina, and M. Dennis, Phys. Rev. E 79, (2009) FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 3
5 Classical billiards I There exist only a few billiards with integrable dynamics: There are some with fully chaotic dynamics: FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 4
6 Classical billiards II Mostly dynamics have a mixed phase space Limaçon billiard (ρ = z + λz 2,z C, z = 1) and Poincaré map λ = 0.00 λ = 0.05 λ = 0.20 λ = 0.50 FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 5
7 Hierarchy of classical chaos Name Definition Example Recurrent Trajectory reoccurs infinitely often to its neighborhood Ergodic Mixing K-system C-System Bernoullisystem time average phase space average Correlation function declines for infinitely long times Nearly all trajectories are exponentially separated The system is hyperbolic in all phase space points System with full symbolic dynamics and finite number of symbols and a full shift All Hamiltonian systems, with a finite phase space (not chaotic). x n+1 = (x n + b) mod 1 b : irrational (not necessarily chaotic) always chaotic, cat map (x n+1 = (x n + y n ) mod 1, y n+1 = (x n + 2y n ) mod 1) Stadium billiard (not C!), cat map Billiard with constant negativ curvature Bernoulli-shift-map FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 6
8 Classical waves Experiments can be performed with different classical waves microwaves acoustics in solids light water surface waves ultrasound in water FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 7
9 Microwave resonators cylindrical symmetry in z-direction z-component can be separated transverse magnetic (TM) modes E = (0,0,E z (x,y,z)) metallic top and bottom plates cut-off frequency ν c = c/2h E z (x,y) two-dimensional Helmholtz equation. FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 8
10 Microwave resonators cylindrical symmetry in z-direction z-component can be separated transverse magnetic (TM) modes E = (0,0,E z (x,y,z)) metallic top and bottom plates cut-off frequency ν c = c/2h E z (x,y) two-dimensional Helmholtz equation. Typical set-up Frequency: 1 20 GHz Wave length: cm FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 8
11 Microwaves versus quantum mechanics There is a one-to-one correspondence between the stationary Schrödinger equation ( 2 2 ) 2m x y 2 ψ n = E n ψ n with the boundary condition ψ n S = 0 (billiard), and the two-dimensional Helmholtz equation ( ) 2 x y 2 E n = kn 2 E n. (E n : electric field strength of z-component) In quasi-two-dimensional microwave billiards even the electromagnetic and the quantum mechanical boundary conditions are equivalent. FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 9
12 Advantages of microwave experiment Real quantum systems: antidot structures (Weiss et al.1991) mesoscopic billiards (Marcus et al.1992) quantum corrals (Crommie et al.1993) tunnelling barriers (Fromhold et al.1994) Aspects of microwave billiards: corresponds to mesoscopic physics daily-life sizes, parameters are easy to control no Coulomb interaction test bed for scattering theory (nuclear physics) commercial measurement equipment (VNA) Magnetoresistance of quantum dots FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 10
13 Husimi representation I Example of a pulse as a projection on a Poincaré-Husimi function with minimal uncertainty : [R. Schäfer et al. NJP 8, 46 (2006)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 11
14 Husimi representation II Classical Poincaré map vs. Poincaré-Husimi distribution of an open quadrupolar billiard (time averaged): [R. Schäfer et al. NJP 8, 46 (2006)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 12
15 Random plane wave model Which is an eigenfunction of a billiard and which is a superposition of random plane waves? FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 13
16 Random plane wave model Which is an eigenfunction of a billiard and which is a superposition of random plane waves? Random superposition of plane waves by courtesy of Arnd Bäcker 6000 eigenfunction of a Limaçon billard FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 13
17 RWM - Distribution of ψ (closed system) where A : Area of the Billiard ) A P(ψ) = ( 2π exp Aψ2 2 FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 14
18 RWM - Distribution of ψ (closed system) where A : Area of the Billiard ) A P(ψ) = ( 2π exp Aψ2 2 by courtesy of Arnd Bäcker FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 14
19 Light fiber (k direction) Near field (left) and far field pattern of a D-shaped fiber. The far field corresponds to a Fourier-transform of the nearfield and thus shows the distribution of k vectors. [Doya etal. Phys. Rev. E 65, (2002)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 15
20 Intensity distribution (acoustic billiards) Vibration amplitude pattern for two eigenfrequencies of a plate of a quarter Sinai-stadium billiard (left column) and corresponding distribution function for the squared amplitudes. The solid line is a Porter-Thomas function. [K. Schaadt, PhD-thesis, NBI, Copenhagen, 1997, H.-J. Stöckmann, Quantum Chaos - An Introduction (University Press, Cambridge, 1999).] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 16
21 Partial width (microwaves) The partial width distribution of a two dimensional microwave cavity. The solid line corresponds to a Porter-Thomas distribution. [Alt et al. Phys. Rev. Lett. 74, 62 (1995)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 17
22 3-dimensional microwave billiard metallic sphere is moved inside the cavity Measured spectra as a function of frequency and sphere position. [Eckhardt et al. Europhys. Lett. 46, 134 (1999)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 18
23 Frequency shift distribution (3D billiard) Shift-eigenmode (δν 2E 2 + B 2 ) Distribution function of frequency shift ν Assuming 6 independent Gaussian ( modes: 2α 2 P( ν) = 3π ν exp α ν ( ) 34 )K 1 4 α ν [Dörr et al. Phys. Rev. Lett. 80, 1030 (1998)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 19
24 Correlations function Correlation function C Ψ and C Ψ 2 for the stadium billiard (average over the 30 lowest eigenstates) C Ψ = 1 A J 0( k r ) C Ψ 2 = 2 3 (J 0( k r )) [Eckhardt et al. Europhys. Lett. 46, 134 (1999)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 20
25 Parametric dependence (global perturbation) Sketch of the Sinai billiards Corresponding level dynamics Global velocity distribution (Theory from RMT: Gaussian) [Barth etal. Phys. Rev. Lett. 82, 2026 (1999), Bart etal. Ann. Phys. (Leipzig) 8, 733 (1999)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 21
26 Parametric dependencies (local perturbation) Rectangular billiard with scatterers, one is moved Dynamics of the normalized eigenvalues [Barth etal. Phys. Rev. Lett. 82, 2026 (1999), Bart etal. Ann. Phys. (Leipzig) 8, 733 (1999)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 22
27 Local velocity distribution Local velocity distribution for different δ ranges: 0.35 < δ < 0.65 (a), 1.4 < δ < 2.6 (b) bzw. 5.1 < δ < 5.9 (c) [Barth etal. Phys. Rev. Lett. 82, 2026 (1999), Bart etal. Ann. Phys. (Leipzig) 8, 733 (1999)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 23
28 Velocity autocorrelation function (global) Global perturbation Dotted line corresponds to theory by Simons und Altshuler (RMT) [Barth etal. Phys. Rev. Lett. 82, 2026 (1999), Bart etal. Ann. Phys. (Leipzig) 8, 733 (1999)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 24
29 Velocity autocorrelation (local) rescaled parameter kr for different δ ranges [Barth etal. Phys. Rev. Lett. 82, 2026 (1999), Bart etal. Ann. Phys. (Leipzig) 8, 733 (1999)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 25
30 Wavefunctions in open systems Real ψ R (a), imaginary part ψ I (b), modulus ψ 2 = ψr 2 + ψ2 I (c), and phase φ (d) of a wave function ψ at a frequency ν = 5.64 GHz. Nodallines (for ψ R und ψ I ) and nodal points (for ψ 2 ) are marked. White corresponds to the phase φ = 0 in d). [U. Kuhl, Eur. Phys. J. Special Topics 145, 103 (2007)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 26
31 Global phase rotation Imaginary- vs. real part of the wavefunction at a frequeny ν = GHz. a) Directly measured and b) after a decorrelation via a global phase rotation, ψ R + iψ I = e iϕ g,0 (ψ R + iψ I ). ϕ g,0 is a globale phase that comes in the experiment from the antenna and the channel. [U. Kuhl, Eur. Phys. J. Special Topics 145, 103 (2007)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 27
32 Phase rigidity Phase rigidity ρ 2 = <ψ2 R <ψ2 I 2 <ψ as a function of frequency R 2 + <ψ2 I N corresponds to the number of open channels [U. Kuhl, Eur. Phys. J. Special Topics 145, 103 (2007)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 28
33 Flow Probability current density j. The color scale corresponds to the modulus j and the arrows give the modulus and direction of j. In the zoom the vortices (dots) and saddles (crosses) are clearly seen with their sense of rotation. [U. Kuhl, Eur. Phys. J. Special Topics 145, 103 (2007)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 29
34 Intensity distribution (open systems) Intensity distribution for four different wave functions at frequencies ν = 8.0 (a), 16.9 (b), 15.6 (c), and 15.4 GHz (d) with different phase rigidities ρ 2. The solid lines corresponds to the theoretical prediction from the RWM. The modulus of the wave function is shown in the inset. [U. Kuhl, Eur. Phys. J. Special Topics 145, 103 (2007)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 30
35 Current distribution P( j ) and as an inset the x component of j (P(j x )) is plotted. The distortion for j x comes from the main transport direction. [U. Kuhl, Eur. Phys. J. Special Topics 145, 103 (2007)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 31
36 Vortex dynamics Vortices as function of frequency: Top: Transmission S 12 into the billiard Dots are vortices (clockwise: red) Crosses are saddles (yellow) Below: Zoom into regions of creation and annihilation [U. Kuhl, Eur. Phys. J. Special Topics 145, 103 (2007)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 32
37 Vortex pair correlation function (theory) Defining: C(R) = J 0 (R) with R = k r und J 0 the Bessel function. And with the following abbreviations: C = C(R), E = C (R), H = C (R)/R, F = C (R), F 0 = C (0) D 1 = [E 2 (1 + C)(F 0 F)][E 2 (1 C)(F 0 + F)], D 2 = F 2 0 H 2 Y = H2 (CE 2 F(1 C 2 )) 2 F0 2(E2 F 0 (1 C 2 )) 2, Z = D 1 D 2 (1 C 2 ) F0 2(E2 F 0 (1 C 2 )) 2 where denotes the derivative. Finally we can write the result as the following integral which can be evaluated numerically: g vv (R) = 2(E2 F 0 (1 C 2 )) πf 0 (1 C 2 dt 3 Z + 2Y + (3 + Z 2Y )t2 + 2Zt 4 ) 0 (1 + t 2 ) (1 + Z Y )t 2 + Zt 4 The charge correlation function g Q (R) which accounts also for the chirality of the vortex points, can be expressed in a much nicer way: g Q (R) = 4 [ ] 2 d d arcsin(j0 (R)) R dr dr [R. Höhmann etal. Phys. Rev. E 79, (2009)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 33
38 Vortex pair correlation function Vortex pair correlation function g and charged correlation function g Q for low (5-9 GHz, a) and high ( GHz b) frequencies [R. Höhmann et al. Phys. Rev. E 79, (2009)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 34
39 Vortex nearest neighbor spacing Nearest neighbor spacing for vortices without (a) and with consideration (b) of the different sense of rotation. Solid lines correspond to prediction using the Poisson approximation. Dashed lines are numerical calculations using the RWM. [R. Höhmann et al. Phys. Rev. E 79, (2009)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 35
40 Vortex-saddle pair correlation function (a) Saddle-saddle correlation function g ss (R) (b) Vortex-saddle correlations function g vs (R) Experimental data (histogram) and asymptotic approximation [R. Höhmann et al. Phys. Rev. E 79, (2009)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 36
41 Vortex nearest neighbor spacing Histogramms display the different nearest neighbor spacings of critical points. Solid lines correspond to the predictions using the Poisson approximation. Dashed lines correspond to numerical calculations using the RWM. Distribution of (a) Vortices with the same, (b) different, (c) and without consideration of the sense of rotation. (d) saddle-points and (e) between vortices and saddles. [R. Höhmann et al. Phys. Rev. E 79, (2009)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 37
42 Correlations function including size effects Spatial correlation function of the real part of Ψ. Left at 5.4 GHz and right at GHz. Red line corresponds to the RWM prediction. Blue to the RWM prediction including the corrections. FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 38
43 Scaling factor Scaling factor s for the experimental pair correlation function to reproduce the RWM prediction as a function of frequency. FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 39
44 Vortex density (close to a straight wall) Density fluctuations of critical points as a function of the scaled distance Y from the straight wall: (left) vortex density; (right) saddle density. [R. Höhmann et al. Phys. Rev. E 79, (2009)] FG 760: Scattering systems with complex dynamics IV. SOCRATES Workshop, Maribor, February 2009, p. 40
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