Can Waves Be Chaotic?

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 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 ChaoticSystems • The Problem of Non-Universal and System-Specific Effects • ‘De-Coherence’ in Quantum Transport • Conclusions

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

Although this is a deterministic system, Difficulty in making long-term predictions Sensitivity to noise Extreme Sensitivity to Initial Conditions 1-Dimensional Iterated Maps The Logistic Map: Change the initial condition (x0) slightly… x Iteration Number

qi, pi qi+Dqi, pi +Dpi 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

“ray chaos” In the ray-limit it is possible to define chaos 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 Wave Chaos concerns solutions of linear wave equations which, in the semiclassical limit, can be described by chaotic ray trajectories

How Common is Wave Chaos? Consider an infinite square-well potential (i.e. a billiard) that shows chaos in the classical limit: Bunimovich Billiard Hard Walls Sinai billiard 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, 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? YES But how?

Random Matrix Theory (RMT) Wigner; Dyson; Mehta; Bohigas … 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 Some Questions: Is this hypothesis supported by data in other systems? Can losses / decoherence be included? What causes deviations from RMT predictions? 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,…

Microwave Cavity Analog of a 2D Quantum Infinite Square Well ~ 50 cm Table-top experiment! Ez The only propagating mode for f < c/d: d ≈ 8 mm Metal walls Bx By 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)

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

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

TRI Broken (GUE) 18 TRI Preserved (GOE) 15 12 8 4 0 2 A | Y | 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 r = 64.8 cm 10 13.62 GHz 0 b) 0 10 20 30 40 y (cm) Ferrite 20 De-magnetized ferrite 10 13.69 GHz 0 0 10 20 30 40 x (cm) D. H. Wu and S. M. Anlage, Phys. Rev. Lett. 81, 2890 (1998).

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

Chaos and Scattering Compound nuclear reaction Nuclear scattering: Ericson fluctuations Billiard Incoming Channel Outgoing Channel Proton energy 2 mm Transport in 2D quantum dots: Universal Conductance Fluctuations 1 Outgoing Voltage waves B (T) Incoming Voltage waves Resistance (kW) Hypothesis: Random Matrix Theory quantitatively describes the statistical properties of all wave chaotic systems (closed and open) Incoming Channel Outgoing Channel Electromagnetic Cavities: Complicated S11, S22, S21 versus frequency

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

Z-mismatch at interface of port and cavity. Short Orbits “Prompt” Reflection due to Z-Mismatch between antenna and cavity Transmitted wave Universal Fluctuations are Usually Obscured by Non-Universal System-Specific Details • The Most Common Non-Universal Effects: • 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 Port Incoming wave

Z matrix S matrix • Complicated Functions of frequency • Detail Specific (Non-Universal) N-Port Description of an Arbitrary Scattering System V1 , I1 • N Ports • Voltages and Currents, • Incoming and Outgoing Waves N – Port System VN , IN

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: Port Port ZR ZCavity Semiclassical Expansion over Short Orbits Perfectly absorbing boundary Cavity Complex Radiation Impedance (characterizes the non-universal coupling) Action of orbit Index of ‘Short Orbit’ of length l The waves do not return to the port Stability of orbit 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 James Hart, T. Antonsen, E. Ott, Phys. Rev. E 80, 041109 (2009)

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

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) 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 Jen-Hao Yeh, J. Hart, E. Bradshaw, T. Antonsen, E. Ott, S. M. Anlage, Phys. Rev. E 81, 025201(R) (2010)

Graphene Quantum Dot R (kW) DG (e2/h) Washburn+Webb (1986) C. M. Marcus, et al. (1992) T = 4 K Ponomarenko Science (2008) B (T) B (T) Quantum Transport Mesoscopic and Nanoscopic systems show quantum effects in transport: Conductance ~ e2/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

= - G I /( V V ) 2 1 2 Lead 1: Waveguide with N1 modes Lead 2: Waveguide with N2 modes Ballistic Quantum Transport Incoherent Semi-Classical Transport Landauer- Büttiker N1=N2=1 for our experiment Quantum interference  Fluctuations in G ~ e2/h “Universal Conductance Fluctuations” An ensemble of quantum dots has a distribution of conductance values: (N1=N2=1) Quantum vs. Classical Transport in Quantum Dots 2-D Electron Gas Ray-Chaotic 2-Dimensional Quantum Dot C. M. Marcus (1992) electron mean free path >> system size RMT Prediction

De-Phasing in (Chaotic) Quantum Transport Büttiker (1986) De-phasing lead Actually measured We can test these predictions in detail: P(G) Brouwer+Beenakker (1997) g = 0 Pure quantum transport g∞ Incoherent classical limit Conductance measurements through 2-Dimensional quantum dots show behavior that is intermediate between: Ballistic Quantum transport Incoherent Classical transport incoherent Why? “De-Phasing” of the electrons 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.

By comparing the Poynting theorem for a cavity with uniform losses to the continuity equation for probability density, one finds: |S21| |S21| f Many absorbers f No absorbers 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 Quantum De-Phasing Loss Parameter: 3dB bandwidth of resonances g = 0 Pure quantum transport g∞ Classical limit Mean spacing between resonances a is varied by adding microwave absorber to the walls a determined from fits to PDF(Z) g is determined from fits to PDF(eigenvalues of SS+)

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

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

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

Can Waves Be Chaotic?