Can Waves Be Chaotic JenHao Yeh Biniyam Taddese

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

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 2

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

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

Classical Chaos in Billiards Best characterized as “extreme sensitivity to initial conditions” Chaotic system Regular system Newtonian particle trajectories qi, pi qi+Dqi, pi +Dpi 2 -Dimensional “billiard” tables Hamiltonian 5

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 6

Wave Chaos? 1) Waves do not have trajectories It makes no sense to talk about “diverging trajectories” for waves 2) Linear wave systems can’t be chaotic Maxwell’s equations, Schrödinger’s equation are linear 3) However in the semiclassical limit, you can think about rays In the ray-limit it is possible to define chaos “ray chaos” Wave Chaos concerns solutions of linear wave equations which, in the semiclassical limit, can be described by chaotic ray trajectories 7

How Common is Wave Chaos? Consider an infinite square-well potential (i. e. a billiard) that shows chaos in the classical limit: Bunimovich Billiard Sinai billiard Hard Walls Bow-tie Bunimovich stadium L Solve the wave equation in the same potential well Examine the solutions in the semiclassical regime: 0 < l << L Some example physical systems: Nuclei, 2 D 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? YES 8 But how?

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 9

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 Hypothesis: Complicated Quantum/Wave systems that have chaotic classical/ray Cassati, 1980 counterparts possess universal statistical properties described by Bohigas, 1984 Random Matrix Theory (RMT) “BGS Conjecture” 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? 10

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 11

Microwave Cavity Analog of a 2 D Quantum Infinite Square Well Table-top experiment! The only propagating mode for f < c/d: Bx Ez ~ 50 cm d ≈ 8 mm By Metal walls An empty “two-dimensional” electromagnetic resonator Schrödinger equation Helmholtz equation Stöckmann + Stein, 1990 Doron+Smilansky+Frenkel, 1990 Sridhar, 1991 Richter, 1992 A. Gokirmak, et al. Rev. Sci. Instrum. 69, 3410 (1998) 12

The Experiment: A simplified model of wave-chaotic scattering systems ports 21. 6 cm λ 43. 2 cm A thin metal box Side view 0. 8 cm 13

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

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 15

Wave Chaotic Eigenfunctions (~ closed system) with and without Time Reversal Invariance (TRI) r = 106. 7 cm a) Bext 20 A magnetized ferrite in the cavity breaks TRI 10 y (cm) De-magnetized ferrite r = 64. 8 cm 0 13. 62 GHz 0 10 20 30 b) 40 Ferrite 20 10 0 2 |Y| A 18 15 12 8 4 0 TRI Preserved (GOE) 13. 69 GHz 0 10 20 30 x (cm) 16 TRI Broken (GUE) 40 D. H. Wu and S. M. Anlage, Phys. Rev. Lett. 81, 2890 (1998).

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

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 18

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

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

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 21

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

N-Port Description of an Arbitrary Scattering System N Ports V 1 , I 1 § Voltages and Currents, N – Port § Incoming and Outgoing Waves System V N , IN S matrix Z matrix § Complicated Functions of frequency § Detail Specific (Non-Universal) 23

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

Testing Insensitivity to System Details Coaxial Cable CAVITY LID Cross Section View § Freq. Range : 9 to 9. 75 GHz § Cavity Height : h= 7. 87 mm § Statistics drawn from 100, 125 pts. Radius (a) CAVITY BASE Probability Density RAW Impedance PDF 25 2 a=0. 635 mm 2 a=1. 27 mm NORMALIZED Impedance PDF 2 a=1. 27 mm 2 a=0. 635 mm

A universal property: uniform phase of the scattering parameter RMT prediction: The distribution of the phase of S should be uniform from 0 ~ 2π Independent of Loss! From 100 realizations 10. 0 ~ 10. 5 GHz (about 14 modes) 26 Guhr, Müller-Groeling, Weidenmüller, Physics Reports 299, 189 (1998)

A universal property: uniform phase of the scattering parameter RMT prediction: The distribution of the phase of S should be uniform from 0 ~ 2π Independent of Loss! From 100 realizations 10. 0 ~ 10. 5 GHz (about 14 modes) 27 Guhr, Müller-Groeling, Weidenmüller, Physics Reports 299, 189 (1998)

A universal property: uniform phase of the scattering parameter RMT prediction: The distribution of the phase of S should be uniform from 0 ~ 2π Independent of Loss! From 100 realizations 10. 0 ~ 10. 5 GHz (about 14 modes) Short-orbit correction up to 200 cm 28 Jen-Hao Yeh, J. Hart, E. Bradshaw, T. Antonsen, E. Ott, S. M. Anlage, Phys. Rev. E 81, 025201(R) (2010)

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 29

B (T) T = 4 K C. M. Marcus, et al. (1992) R (k. W) DG (e 2/h) Washburn+Webb (1986) Mesoscopic and Nanoscopic systems show quantum effects in transport: Conductance ~ e 2/h per channel Graphene Quantum Dot Wave interference effects “Universal” statistical properties Ponomarenko Science (2008) Quantum Transport B (T) 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 30

Quantum vs. Classical Transport in Quantum Dots G = I 2 /(V 1 - V 2 ) 2 -D Electron Gas Lead 1: Waveguide with N 1 modes C. M. Marcus (1992) electron mean free path >> system size Ballistic Quantum Transport Landauer. Büttiker Quantum interference Fluctuations in G ~ e 2/h “Universal Conductance Fluctuations” An ensemble of quantum dots has a distribution of conductance values: RMT Prediction 31 (N 1=N 2=1) 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

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 incoherent Büttiker (1986) De-phasing lead Actually measured We can test these predictions in detail: One class of models: Add a “de-phasing lead” with Nf modes with transparency Gf. . Electrons that visit the lead are re-injected with random phase. g = 0 Pure quantum transport g ∞ Incoherent classical limit 32 P(G) Brouwer+Beenakker (1997) Why? “De-Phasing” of the electrons

The Microwave Cavity Mimics the Scattering Properties of a 2 -Dimensional Quantum Dot Uniformly-distributed microwave losses are equivalent to quantum “de-phasing” Brouwer+Beenakker (1997) Microwave Losses Loss Parameter: Quantum De-Phasing 3 d. B bandwidth of resonances Mean spacing between resonances g = 0 Pure quantum transport g ∞ Classical limit By comparing the Poynting theorem for a cavity with uniform losses |S 21| to the continuity equation for probability density, one finds: f |S 21| Many absorbers f 33 No absorbers

Conductance Fluctuations of the Surrogate Quantum Dot Ensemble Measurements of the Microwave Cavity High Loss / Dephasing Surrogate Conductance Low Loss / Dephasing Ordinary Transmission Data (symbols) RMT predictions (solid lines) (valid only for g >> 1) RM Monte Carlo computation RMT prediction (valid only for g >> 1) Data (symbols) Correction for waves that visit the “parasitic channels” Beenakker RMP (97) 34

Conclusions Chaos does play a role in Electromagnetism and Quantum Mechanics in the semi-classical limit When the wave properties of ray-chaotic systems are studied, one finds certain universal properties: Eigenvalue repulsion, statistics Strong Eigenfunction fluctuations Scattering fluctuations 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 35

The Maryland Wave Chaos Group Elliott Bradshaw Ed Ott 36 Jen-Hao Yeh Tom Antonsen James Hart Biniyam Taddese Steve Anlage