1 Universal Statistical Electromagnetic Properties of Ray-Chaotic Enclosures Steven M. Anlage, Sameer Hemmady, Xing Zheng, James Hart, Elliott Bradshaw, Edward Ott, Thomas Antonsen Physics 606, 14 April, 2008

2 OUTLINE Wave Chaos – What is it? Universality in Wave Chaotic Systems Experiments do not show universality… Must address two issues: Coupling Short Orbits Theory Experimental Results Discussion Conclusions

3 It makes no sense to talk about “diverging trajectories” for waves 1) Classical chaotic systems have diverging trajectories q i, p i q i +  q i, p i +  p i Regular system 2-Dimensional “billiard” tables with hard wall boundaries Newtonian particle trajectories q i, p i q i +  q i, p i +  p i Chaotic system Wave Chaos? 2) Linear wave systems can’t be chaotic 3) However in the semiclassical limit, you can think about rays Wave Chaos concerns solutions of wave equations which, in the semiclassical limit, can be described by ray trajectories In the ray-limit it is possible to define chaos “ray chaos”

4 The Difficulty in Making Predictions in Wave Chaotic Systems… Perturbation Position 1 Perturbation Position 2 Antenna Port  Electromagnetic Wave Impedance Extreme sensitivity to small perturbations We must resort to a statistical description In our experiments we systematically move the perturber to generate many “realizations” of the system

5 Wave Chaotic Systems are expected to show universal statistical properties, as predicted by Random Matrix Theory (RMT) Motivation: - UNIVERSALITY RMT predicts universal statistical properties: Closed Systems Open Systems Eigenvalue nearest neighbor spacing Eigenvalue long-range correlations Eigenfunction 1-pt, 2-pt correlations etc. Scattering matrix statistics: |S|,  S Impedance matrix (Z) statistics (R, K matrix) Transmission matrix (T = SS † ), conductance statistics etc. Microwave Cavity Standing Wave Pattern Map of

6 What Universality? The universal scattering fluctuations predicted by RMT for wave-chaotic systems are hard to measure in real life. 2a=0.635mm 2a=1.27mm Manifestation of Non-ideal coupling in measured scattering fluctuations. Port Chaotic Cavity Incoming wave “Prompt” Reflection due to Potential Mismatch between port and cavity Tunnel barrier at interface of port and cavity. Transmitted wave Cavity Base Cavity Lid Inner Diameter (2a) Hemmady et.al. PRE 71, (2005). Perturbers Port Short Orbits We must remove “Non-Ideal Coupling” and “Short Orbits” from the data Numerical simulation of ¼-bow-tie Poisson Kernel:

7 Remove the Non-Universal Coupling Form the Normalized Impedance (z) Port Z Cavity Port Z Rad The waves do not return to the port Radiation Losses Reactive Impedance of Antenna Perfectly absorbing boundary Cavity Combine

8 Uncovering Universal Statistical Properties Removing the Non-Universal Effects of the Coupling (Previous Work) z is the impedance in the ‘perfectly coupled’ limit and shows universal fluctuations S. Hemmady, et al., Phys. Rev. Lett. 94, (2005). Before Normalization PDF(Im[Z Cav ]) After Normalization PDF(Im[z])

9 Uncovering Universal Statistical Properties Removing the Non-Universal Effects of Short Orbits (New Work) Re-Define the radiation impedance to include short orbits Theory by James Hart Length of the m th orbit Atten. of the m th orbit wavenumberNumber of bounces of the m th orbit Geometrical Factor: ►Segment length ►Angle of incidence ►Radius of curvature of wall Plot corrections of the form:

10 Our Experimental Setup : 21.6cm Circular Arc R=106.9cm Circular Arc R=64.7cm 43.2cm Radiation Case Diameter (2a) Coaxial Cable Height (h=7.8mm) Antenna Profile Schematic One port S, Z measurements. Frequency Range 3-18 GHz (~800 modes) Ensemble Average: Move and Rotate perturbations to 100 locations within cavity. Metallic Perturbations Port 1 Microwave Absorber The bow-tie billiard Microwave Cavity Billiard

11 Microwave Cavity Analog of a 2DEG Our Experiment: A clean, zero temperature, quantum dot with no Coulomb or correlation effects! Table-top experiment! EzEz BxBx ByBy Schrödinger equation Helmholtz equation Stöckmann + Stein, 1990 Doron+Smilansky+Frenkel, 1990 Sridhar, 1991 Richter, 1992 d ≈ 8 mm An empty “two-dimensional” electromagnetic resonator A. Gokirmak, et al. Rev. Sci. Instrum. 69, 3410 (1998). ~ 50 cm

12 Short Orbits Involving 1, 2, 3 Walls Data (Red): Theory (Blue): or with  m = 0

13 Why are there differences between experiment and theory? 1) Finite width of the wall The calculation is done for a wall of great extent (length >> ) 2) Imperfections of the microwave absorber The absorbers reflect some of the incident signal 3) Presence of the absorber in the near-field of the antenna The short-orbit calculation assumes far-field wave pattern 4) Difficulty in reproducing the antenna geometry with each measurement 5) Attenuation of long-path orbits due to the finite conductivity of the parallel plates 6) Inclusion of only a finite number of orbits in the sums:

14 Improvement in Statistical Properties 6 – 7.1 GHz data The Phase of the perfectly-coupled Scattering parameter is predicted to be uniformly distributed: Data averaged over a very broad frequency range (> 1 GHz) and 100 realizations

15 We have identified two systematic errors present in experiments: Non-ideal coupling (characterized by the radiation impedance Z Rad ) Short-Orbits between the port and the fixed walls ( ) A corrected radiation impedance theory, including short orbits, was proposed and tested Clear improvement in statistical property measurement was observed after short orbits were removed Conclusions Acknowledgements: Sameer Hemmady– early experiments Florian Schaefer – early experiments and analysis Richard Prange – theory NSF - Funding

16 S. Hemmady, et al., Phys. Rev. Lett. 94, (2005) S. Hemmady, et al., Phys. Rev. E 71, (2005) X. Zheng, T. M. Antonsen Jr., E. Ott, Electromagnetics 26, 3 (2006) X. Zheng, T. M. Antonsen Jr., E. Ott, Electromagnetics 26, 37 (2006) X. Zheng, et al., Phys. Rev. E 73, (2006) S. Hemmady, et al., Phys. Rev. E 74, (2006) S. Hemmady, et al., Phys. Rev. B 74, (2006) S. M. Anlage, et al., Acta Physica Polonica A 112, 569 (2007) Publication List: THANK YOU!

17 New Research Areas Use of a superconducting 2D microwave cavity to explore predictions in the low-loss limit (  << 1) Z, S and conductance G statistics Test P(Re[z]) and P(Im[z]) statistics vs.  Test dependence of the variance of the Re[z] and Im[z] distributions on  Test the dependence of P(|s|) and P(  s ) on  Test the dependence of P(G) and and  G on  Hauser-Feshbach relations for S and Z

18 New Research Areas Time-Reversed Electromagnetics Time Reversal Invariance Fidelity Decay of Quantum Systems Search for deviations from power-law decay at long times in superconducting cavity  decay >> t Heisenberg

19 Improvement in Statistical Properties I Poisson Kernel Ensemble created by moving 2 perturbers to 100 different locations

20 Cavity Loss Parameter Port 1 Cavity Port 2 1) Perfectly absorbing boundary Port 2 Port 1 2) Random Coupling Model. Think of Impedances !! Normalized Scattering Matrix Statistics Determined by Zheng, Antonsen, Ott. Electromag. 26, 3 (2006).

21 Potential with uniform imaginary part Efetov (1995) McCann+Lerner (1996) Why does the Microwave Cavity ↔ Quantum Dot Analogy Work? It is known that for electromagnetic systems: Uniformly distributed losses are equivalent to a large number of “ports”, each with small transmission Physical Ports Uniformly Lossy Cavity Lewenkopf, Müller, Doron (1992) Schanze, et al. (2005) Potential with zero imaginary part Brouwer+Beenakker (1997) channels Physical Ports Lossless cavity Parasitic Ports (Equivalent Loss Channels) “Locally weak absorbing limit” Zirnbauer (93)

22 Conductance Fluctuations of the Surrogate Quantum Dot RMT predictions (solid lines) (valid only for  >> 1) Data (symbols) High Loss / Dephasing Low Loss / Dephasing RMT prediction (valid only for  >> 1) Data (symbols) RM Monte Carlo computation Scaling Prediction for P(G) Brouwer+Beenakker (1997) Ordinary Transmission Correction for waves that visit the “parasitic channels” Surrogate Conductance Beenakker RMP (97)

23 Microwave Scattering Experiment Port 1 Port 2 The statistics of z are “universal” and are described by Random Matrix Theory (RMT) S. Hemmady, et al. PRL 94, (2005). Cavity Base Cavity Lid Diameter (2a) Coaxial Cable (Z 0 ) Height (h=7.8mm) Antenna Cross-Section Measured with outgoing boundary condition Wigner+Eisenbud (1947) Reaction matrix Works for any number of ports/channels

24 Determining Loss Parameter  From The Data Stars, +: Re[z] Circles, x: Im[z] PDF Fit Parameter Random matrix Monte Carlo computation  = 7.6  = 4.2  = 0.8 Single parameter (  ) fits to P(Re[z]) and P(Im[z]) Correct for  > 5 Correct for all  Statistics compiled from 100 realizations and averaging over 1 GHz-wide frequency windows Typically 10 5 data points Independent determination of 

25 Data Theory Loss Case 0 Determining  From The Data 1)Maximum likelihood estimate fit to P  (T 1, T 2 ) 1-T 1 and 1-T 2 are the eigenvalues of ss † 2) Value of Both methods based on: Brouwer+Beenakker (1997) Hemmady (2006) Statistics compiled from 100 realizations and averaging over 1 GHz-wide frequency windows to remove effects of short periodic orbits Typically 10 5 data points

26 Variation of Loss Parameter  with Frequency Loss Case 0 (No absorbers) Loss Case 1 (16 2”-absorbers) Loss Case 2 (32 2”-absorbers) Also “Dry Ice” case Variation of  Contribution to  from coupling through the physical ports:  coup ~ 0.03 – 0.12 (7 – 9 GHz)

27 sub-unitary From overall current conservation: Conductance in the Presence of De-phasing From Landauer / Büttiker: k =1,2,  with Baranger+Mello (95) Beenakker RMP (97) Brouwer+Beenakker (97)

28 Antenna Entry Points  wave Absorber x 16 : Loss Case 2 x 32 : Loss Case 4 Loss Case 0 Loss Case 2 Loss Case 4 Marginal PDFs of Real and Imaginary parts of Eigenvalues of Normalized 2x2 Impedance and Admittance Frequency Range: GHz Equivalence of y and z statistics: iK → 1/iK symmetry Fyodorov, Savin, Sommers: cond-mat/ PDF of Im[z] related to LDOS Mirlin + Fyodorov, Europhys Lett. (1994) Mirlin, Phys. Rep. (2000)  are the eigenvalues of the 2x2 z and y matrices

29 Random Matrix Monte Carlo Computation N = # channels  = loss parameter W ij = coupling between channel-i and mode-j (Gaussian random variable) = diagonal matrix of random matrix eigenvalues

30 Chaos and Scattering Hypothesis: Random Matrix Theory quantitatively describes the statistical properties of all wave chaotic systems Electromagnetic Cavities: Complicated S 11, S 22, S 21 versus frequency B (T) Transport in quantum dots: Universal Conductance Fluctuations Resistance (k  ) mm C. M. Marcus, et al. (1992) O. Häusser, Nucl. Phys. A (1968) Nuclear scattering: Ericson fluctuations Proton energy Compound nuclear reaction

31 Integrable Chaotic TRS 233 Th Nucleus (ds) O Ion Chaotic TRSB Harm. Osc. Distribution of Eigen Energies GOE→GUE p(s) crossover experiment: P. So, et al., Phys. Rev. Lett (1994) Normalized Spacing Tomsovic, 1996 Poisson TRS (GOE) TRSB (GUE) RMT Predictions s Nearest Neighbor Spacing Distributions Poisson TRS TRSB p(s) s TRS TRSB

32 D. H. Wu, et al. Phys. Rev. Lett. 81, 2890 (1998). Probability Amplitude with and without Time Reversal Symmetry (TRS) P( ) = (2  ) -1/2 e - /2 TRS (GOE) e -  TRSB (GUE) “Hot Spots” RMT Prediction:

33 Wave Chaotic Eigenfunctions with and without Time Reversal Symmetry (TRS) D. H. Wu and S. M. Anlage, Phys. Rev. Lett. 81, 2890 (1998). TRS Broken (GUE) TRS (GOE) r = cm r = 64.8 cm x (cm) y (cm) A 2 ||  Ferrite a) b) GHz GHz

34 RMT Prediction c +c+c

35 Random Superposition of Plane Waves… Random Amplitude Random Direction Random Phase Berry Hypothesis (1977) Eric Heller, Harvard … as Art! k j is uniformly distributed on a circle |k j |=k n

36 Nuclear Scattering and Microwave Scattering potential Scattering states discrete interior states  in,n  out E. P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947) Nuclear Physics: R (K) matrix Another Nuclear Microwave connection (for the experts!) An “improved” Hauser-Feshbach relation X. Zheng, et al.

37 Comparison of Conductance Distributions to the Predictions of Brouwer+Beenakker (1997)

38 Buttiker PRB ’86 Marcus PRL ’92 Baranger + Mello, PRB ‘95 Locally weak absorption limit Zirnbauer Nuc. Phys. A ‘93 Joint distribution for T 1 and T 2 for  = 1 One mode in each lead sub-unitary s T 1 and T 2 give the “absorption strength” of the fictitious lead ( ) Polar Decomposition of s : 1-T 1 and 1-T 2 are eigenvalues of ss † s

39 For high energies (semiclassical (ray) limit) Understanding the origin of “Chaos” in Quantum Chaotic Systems : Electron in Infinite Square Well-potential d For low energies Linear Maps for “Integrable” systems !! Non-Linear Maps for “Chaotic” systems !! So, in Quantum Chaos, we are interested in studying quantum systems in the semi-classical limit which are known to be classically chaotic. The “Chaos” arises due to the shape of the boundaries enclosing the system. 0

40 Testing Predictions on Variance of z PDFs  Frequency Range for loss cases: 7.2 to 8.4GHz  Antenna Diameter (2a) = 1.27mm 0  1  2  3  4 Open: Re[z] Closed: Im[z] Prediction (for Q >> 1): 0.33  0.02 Varied k, loss and h in the experiment PDF Fit Parameter

41 Experimental Tests of the Random Coupling Model Single-parameter fits to PDF of Re[z], Im[z] Equivalence of variances of PDFs and single fitting parameter Insensitivity of Re[z] and Im[z] to irrelevant details Frequency, volume, loss dependence of Re[z] and Im[z] PDFs Single-parameter fits to PDF of |s|, uniform distribution of Arg[s] Independence of |s| and Arg[s] z = normalized impedance s = normalized scattering matrix Good Agreement! These results are general, and their applicability is not limited in any way by the nature of our “bow-tie” cavity (dimensionality, shape, etc.), or the exact coupling structure

42 N-Port Description of an Arbitrary Enclosure N – Port System N Ports  Voltages and Currents,  Incoming and Outgoing Waves Z matrixS matrix V 1, I 1 V N, I N  Complicated Functions of frequency  Detail Specific (Non-Universal)

43 Perturbation scanning system Measure E z through cavity perturbation (metallic) Cavity Perturbation Imaging of E 2 (|  | 2 )

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45 Schrödinger equation for a charged particle in a magnetic field Helmholtz equation for a microwave cavity including a magnetized ferrite A Magnetized Ferrite in the Cavity Produces Time-Reversal Symmetry-Breaking (TRSB) Analogous to a QM particle in a magnetic field è The magnetized ferrite problem and the magnetized Schrödinger problem are in the same TRSB universality class (GUE) B B

46 B ferrite B x y Incident plane wave Reflection coefficient Note that  changes upon reversal of k y, giving rise to a non-reciprocal phase shift xy Forward: k = k x x + k y y xy Backward: k = k x x - k y y d conducting wall How do we Break Time-Reversal Symmetry? A microwave ferrite in the cavity magnetized ferrite permeability tensor