1. 1 Can Waves Be Chaotic? Jen-Hao Yeh, Biniyam Taddese, James Hart, Elliott Bradshaw Edward Ott, Thomas Antonsen, Steven M. Anlage Research funded by AFOSR and the ONR-MURI and DURIP programs Karlsruhe Institute of Technology 16 April, 2010

2. 2 Outline Classical Chaos What is Wave Chaos? Universal Statistical Properties of Wave Chaotic Systems Our Microwave Analog Experiment Statistical Properties of Closed Wave Chaotic Systems Properties of Open / Scattering Wave Chaotic Systems The Problem of Non-Universal and System-Specific Effects ‘De-Coherence’ in Quantum Transport Conclusions

3. 3 Simple Chaos 1-Dimensional Iterated Maps The Logistic Map: Parameter:Initial condition: Iteration number x x x

4. 4 Extreme Sensitivity to Initial Conditions 1-Dimensional Iterated Maps The Logistic Map: Change the initial condition (x 0 ) slightly… x Iteration Number Although this is a deterministic system, Difficulty in making long-term predictions Sensitivity to noise

5. 5 Classical Chaos in Billiards Best characterized as “extreme sensitivity to initial conditions” q i, p i q i +  q i, p i +  p i Regular system 2-Dimensional “billiard” tables Newtonian particle trajectories Hamiltonian q i +  q i, p i +  p i q i, p i Chaotic system

6. 6 Outline Classical Chaos What is Wave Chaos? Universal Statistical Properties of Wave Chaotic Systems Our Microwave Analog Experiment Statistical Properties of Closed Wave Chaotic Systems Properties of Open / Scattering Wave Chaotic Systems The Problem of Non-Universal and System-Specific Effects ‘De-Coherence’ in Quantum Transport Conclusions

7. 7 It makes no sense to talk about “diverging trajectories” for waves 1) Waves do not have trajectories 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 linear wave equations which, in the semiclassical limit, can be described by chaotic ray trajectories In the ray-limit it is possible to define chaos “ray chaos” Maxwell’s equations, Schrödinger’s equation are linear

8. 8 How Common is Wave Chaos? Consider an infinite square-well potential (i.e. a billiard) that shows chaos in the classical limit: Solve the wave equation in the same potential well Examine the solutions in the semiclassical regime: 0 < << L Hard Walls Bow-tie L Sinai billiard Bunimovich stadium YES But how? Bunimovich Billiard Some example physical systems: Nuclei, 2D electron gas billiards, acoustic waves in irregular blocks or rooms, electromagnetic waves in enclosures Will the chaos present in the classical limit have an affect on the wave properties?

9. 9 Outline Classical Chaos What is Wave Chaos? Universal Statistical Properties of Wave Chaotic Systems Our Microwave Analog Experiment Statistical Properties of Closed Wave Chaotic Systems Properties of Open / Scattering Wave Chaotic Systems The Problem of Non-Universal and System-Specific Effects ‘De-Coherence’ in Quantum Transport Conclusions

10. 10 Random Matrix Theory (RMT) Wigner; Dyson; Mehta; Bohigas … The RMT Approach: Complicated Hamiltonian: e.g. Nucleus: Solve Replace with a Hamiltonian with matrix elements chosen randomly from a Gaussian distribution Examine the statistical properties of the resulting Hamiltonians This hypothesis has been tested in many systems: Nuclei, atoms, molecules, quantum dots, acoustics (room, solid body, seismic), optical resonators, random lasers,… Some Questions: Is this hypothesis supported by data in other systems? Can losses / decoherence be included? What causes deviations from RMT predictions? Hypothesis: Complicated Quantum/Wave systems that have chaotic classical/ray counterparts possess universal statistical properties described by Random Matrix Theory (RMT)“BGS Conjecture” Cassati, 1980 Bohigas, 1984

11. 11 Outline Classical Chaos What is Wave Chaos? Universal Statistical Properties of Wave Chaotic Systems Our Microwave Analog Experiment Statistical Properties of Closed Wave Chaotic Systems Properties of Open / Scattering Wave Chaotic Systems The Problem of Non-Universal and System-Specific Effects ‘De-Coherence’ in Quantum Transport Conclusions

12. 12 Microwave Cavity Analog of a 2D Quantum Infinite Square Well 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 The only propagating mode for f < c/d: Metal walls

13. 13 The Experiment: A simplified model of wave-chaotic scattering systems A thin metal box ports Side view 21.6 cm 43.2 cm 0.8 cm λ

14. 14 Microwave-Cavity Analog of a 2D Infinite Square Well with Coupling to Scattering States Network Analyzer (measures S-matrix vs. frequency) Thin Microwave CavityPorts Electromagnet We measure from 500 MHz – 19 GHz, covering about 750 modes in the semi-classical limit

15. 15 Outline Classical Chaos What is Wave Chaos? Universal Statistical Properties of Wave Chaotic Systems Our Microwave Analog Experiment Statistical Properties of Closed Wave Chaotic Systems Properties of Open / Scattering Wave Chaotic Systems The Problem of Non-Universal and System-Specific Effects ‘De-Coherence’ in Quantum Transport Conclusions

16. 16 Wave Chaotic Eigenfunctions (~ closed system) with and without Time Reversal Invariance (TRI) D. H. Wu and S. M. Anlage, Phys. Rev. Lett. 81, 2890 (1998). TRI Broken (GUE) TRI Preserved (GOE) r = 106.7 cm r = 64.8 cm 0 10 20 30 40 20 10 0 20 10 0 0 10 20 30 40 x (cm) y (cm) 18 15 12 8 4 0 A 2 ||  Ferrite a) b) 13.69 GHz 13.62 GHz A magnetized ferrite in the cavity breaks TRI De-magnetized ferrite B ext

17. 17 D. H. Wu, et al. Phys. Rev. Lett. 81, 2890 (1998). Probability Amplitude Fluctuations with and without Time Reversal Invariance (TRI) P( ) = (2  ) -1/2 e - /2 TRI (GOE) e -  TRI Broken (GUE) “Hot Spots” RMT Prediction: Broken TRI (TRI) (Broken TRI)

18. 18 Outline Classical Chaos What is Wave Chaos? Universal Statistical Properties of Wave Chaotic Systems Our Microwave Analog Experiment Statistical Properties of Closed Wave Chaotic Systems Properties of Open / Scattering Wave Chaotic Systems The Problem of Non-Universal and System-Specific Effects ‘De-Coherence’ in Quantum Transport Conclusions

19. 19 Billiard Incoming Channel Outgoing Channel Chaos and Scattering Hypothesis: Random Matrix Theory quantitatively describes the statistical properties of all wave chaotic systems (closed and open) Electromagnetic Cavities: Complicated S 11, S 22, S 21 versus frequency B (T) Transport in 2D quantum dots: Universal Conductance Fluctuations Resistance (k  ) mm Incoming Voltage waves Outgoing Voltage waves Nuclear scattering: Ericson fluctuations Proton energy Compound nuclear reaction 1 2 Incoming Channel Outgoing Channel

20. 20 Universal Scattering Statistics Despite the very different physical circumstances, these measured scattering fluctuations have a common underlying origin! Universal Properties of the Scattering Matrix: Re[S] Im[S] RMT prediction: Eigenphases of S uniformly distributed on the unit circle Eigenphase repulsion Nuclear Scattering Cross Section 2D Electron Gas Quantum Dot Resistance Microwave Cavity Scattering Matrix, Impedance, Admittance, etc. Unitary Case

21. 21 Outline Classical Chaos What is Wave Chaos? Universal Statistical Properties of Wave Chaotic Systems Our Microwave Analog Experiment Statistical Properties of Closed Wave Chaotic Systems Properties of Open / Scattering Wave Chaotic Systems The Problem of Non-Universal and System-Specific Effects ‘De-Coherence’ in Quantum Transport Conclusions

22. 22 The Most Common Non-Universal Effects: 1)Non-Ideal Coupling between external scattering states and internal modes (i.e. Port properties) Universal Fluctuations are Usually Obscured by Non-Universal System-Specific Details Port Ray-Chaotic Cavity Incoming wave “Prompt” Reflection due to Z-Mismatch between antenna and cavity Z-mismatch at interface of port and cavity. Transmitted wave Short Orbits 2) Short-Orbits between the port and fixed walls of the billiard

23. 23 N-Port Description of an Arbitrary Scattering System 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)

24. 24 Universally Fluctuating Complex Quantity with Mean 1 (0) for the Real (Imaginary) Part. Predicted by RMT Semiclassical Expansion over Short Orbits Complex Radiation Impedance (characterizes the non-universal coupling) Index of ‘Short Orbit’ of length l Stability of orbit Action of orbit Theory of Non-Universal Wave Scattering Properties James Hart, T. Antonsen, E. Ott, Phys. Rev. E 80, 041109 (2009) Port Z Cavity Port ZRZR The waves do not return to the port Perfectly absorbing boundary Cavity Orbit Stability Factor: ►Segment length ►Angle of incidence ►Radius of curvature of wall Assumes foci and caustics are absent! Orbit Action: ►Segment length ►Wavenumber ►Number of Wall Bounces 1-Port, Loss-less case:

25. 25 Probability Density 2a=0.635mm 2a=1.27mm 2a=0.635mm Testing Insensitivity to System Details CAVITY BASE Cross Section View CAVITY LID Radius (a) Coaxial Cable  Freq. Range : 9 to 9.75 GHz  Cavity Height : h= 7.87mm  Statistics drawn from 100,125 pts. RAW Impedance PDF NORMALIZED Impedance PDF

26. 26 RMT prediction: The distribution of the phase of S should be uniform from 0 ~ 2π Independent of Loss! Guhr, Müller-Groeling, Weidenmüller, Physics Reports 299, 189 (1998) A universal property: uniform phase of the scattering parameter From 100 realizations 10.0 ~ 10.5 GHz (about 14 modes)

27. 27 From 100 realizations 10.0 ~ 10.5 GHz (about 14 modes) RMT prediction: The distribution of the phase of S should be uniform from 0 ~ 2π Independent of Loss! Guhr, Müller-Groeling, Weidenmüller, Physics Reports 299, 189 (1998) A universal property: uniform phase of the scattering parameter

28. 28 Jen-Hao Yeh, J. Hart, E. Bradshaw, T. Antonsen, E. Ott, S. M. Anlage, Phys. Rev. E 81, 025201(R) (2010) From 100 realizations 10.0 ~ 10.5 GHz (about 14 modes) Short-orbit correction up to 200 cm RMT prediction: The distribution of the phase of S should be uniform from 0 ~ 2π Independent of Loss! A universal property: uniform phase of the scattering parameter

29. 29 Outline Classical Chaos What is Wave Chaos? Universal Statistical Properties of Wave Chaotic Systems Our Microwave Analog Experiment Statistical Properties of Closed Wave Chaotic Systems Properties of Open / Scattering Wave Chaotic Systems The Problem of Non-Universal and System-Specific Effects ‘De-Coherence’ in Quantum Transport Conclusions

30. 30 Quantum Transport Mesoscopic and Nanoscopic systems show quantum effects in transport : Conductance ~ e 2 /h per channel Wave interference effects “Universal” statistical properties However these effects are partially hidden by finite-temperatures, electron de-phasing, and electron-electron interactions Also theory calculates many quantities that are difficult to observe experimentally, e.g. scattering matrix elements complex wavefunctions correlation functions Develop a simpler experiment that demonstrates the wave properties without all of the complications ► Electromagnetic resonator analog of quantum transport Washburn+Webb (1986) B (T)  G (e 2 /h) C. M. Marcus, et al. (1992) B (T) R (k  ) Ponomarenko Science (2008) Graphene Quantum Dot T = 4 K

31. 31 Quantum vs. Classical Transport in Quantum Dots )/( 212 VVIG  Lead 1: Waveguide with N 1 modes Lead 2: Waveguide with N 2 modes Ray-Chaotic 2-Dimensional Quantum Dot Incoherent Semi-Classical Transport N 1 =N 2 =1 for our experiment C. M. Marcus (1992) 2-D Electron Gas electron mean free path >> system size Ballistic Quantum Transport Quantum interference  Fluctuations in G ~ e 2 /h “Universal Conductance Fluctuations” An ensemble of quantum dots has a distribution of conductance values: (N 1 =N 2 =1) Landauer- Büttiker RMT Prediction

32. 32 De-Phasing in (Chaotic) Quantum Transport Conductance measurements through 2-Dimensional quantum dots show behavior that is intermediate between: Ballistic Quantum transport Incoherent Classical transport Why? “De-Phasing” of the electrons One class of models: Add a “de-phasing lead” with N  modes with transparency    Electrons that visit the lead are re-injected with random phase. Actually measured incoherent P(G) Brouwer+Beenakker (1997) We can test these predictions in detail: De-phasing lead Büttiker (1986)  = 0 Pure quantum transport   ∞ Incoherent classical limit

33. 33 The Microwave Cavity Mimics the Scattering Properties of a 2-Dimensional Quantum Dot Uniformly-distributed microwave losses are equivalent to quantum “de-phasing” Microwave LossesQuantum De-Phasing 3dB bandwidth of resonances Loss Parameter: Mean spacing between resonances  = 0 Pure quantum transport   ∞ Classical limit  is varied by adding microwave absorber to the walls  determined from fits to PDF(Z)  is determined from fits to PDF(eigenvalues of SS + ) By comparing the Poynting theorem for a cavity with uniform losses to the continuity equation for probability density, one finds: Brouwer+Beenakker (1997) No absorbers Many absorbers f |S 21 | f

34. 34 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 Ordinary Transmission Correction for waves that visit the “parasitic channels” Surrogate Conductance Beenakker RMP (97) Ensemble Measurements of the Microwave Cavity

35. 35 Conclusions When the wave properties of ray-chaotic systems are studied, one finds certain universal properties: Eigenvalue repulsion, statistics Strong Eigenfunction fluctuations Scattering fluctuations Chaos does play a role in Electromagnetism and Quantum Mechanics in the semi-classical limit Our microwave analog experiment directly simulates quantum mechanical systems with “de-phasing” Ongoing experiments: Tests of RMT in the loss-less limit: Superconducting cavity Pulse-propagation and tests of RMT in the time-domain Classical analogs of quantum fidelity and the Loschmidt echo Time-reversed electromagnetics and quantum mechanics Many thanks to: P. Brouwer, M. Fink, S. Fishman, Y. Fyodorov, T. Guhr, U. Kuhl, P. Mello, R. Prange, A. Richter, D. Savin, F. Schafer, L. Sirko, H.-J. Stöckmann, J.-P. Parmantier

36. 36 The Maryland Wave Chaos Group Tom AntonsenSteve Anlage Ed Ott Elliott BradshawJen-Hao Yeh James HartBiniyam Taddese