Is Quantum Mechanics Chaotic? Steven Anlage

Transcription

1 Is Quantum Mechanics Chaotic? Steven Anlage Physics 40

2 0.5 Simple Chaos 1-Dimensional Iterated Maps The Logistic Map: x = 4 x (1 x ) n+ 1 μ n n Parameter: μ Initial condition: 0 = 0.5 μ 0.8 x 0 = = 0.8 μ 1 x =1.0 μ x x x Iteration number Iteration number Iteration number

3 Extreme Sensitivity to Initial Conditions 1-Dimensional Iterated Maps The Logistic Map: x = 4 x (1 x ) μ =1.0 n+ 1 μ n n Change the initial condition (x 0 ) slightly x x = x = Iteration Number Although this is a deterministic system, Difficulty in making long-term predictions Sensitivity to noise

4 Extreme Sensitivity to Initial Conditions Double Pendulum Demo Start with similar initial conditions The motion of the two pendula diverge G1-60: CHAOS - TWO DOUBLE PHYSICAL PENDULA DESCRIPTION: The two pendula are started into apparently identical oscillations, but their motion soon diverges. No matter how closely the motions of the two pendula are started, they eventually must undergo virtually total divergence. This illustrates the modern meaning of "chaos."

5 Chaos Classical: Extreme sensitivity to initial conditions q& i p& i = H / p = H / q i i q i, p i q i +Δq i, p i +Δp i Manifestations of classical chaos: Chaotic oscillations, difficulty in making long-term predictions, sensitivity to noise, etc. Time series, iterated maps, Lyapunov exponents, etc. Wave/Quantum:??? Heisenberg Uncertainty principle limits knowledge of initial conditions Δp Δq > h/π 1 Ψ ( ih qa) Ψ + VΨ = ih m Manifestations of quantum chaos: Breaking of degeneracy, Level repulsion, Strong eigenfunction fluctuations, Scars t

6 Wave Chaos? Launch waves from nearby locations It makes no sense to talk about diverging trajectories for waves Wave/Quantum Chaos??? -Dimensional Sinai billiard Heisenberg Uncertainty Principle limits knowledge of initial conditions Δp Δq > ħ 1 m ( ih qa) Ψ Ψ + VΨ = ih t However, in the ray-limit it is possible to define chaos ray chaos But what is nonlinear here? Maxwell s equations and the Schrödinger equation are linear! One can think of the iterated map of the ray trajectories as providing the diverging orbits

7 (Ω) The Difficulty in Making Predictions in Wave Chaotic Systems 1000 Perturbation Position 1 Antenna Port 800 Abs[Z cav ] Electromagnetic Wave Impedance Frequency [GHz] Perturbation Position λ ~ 0.05 L Extreme sensitivity to small perturbations We must resort to a statistical description

8 Random Matrix Theory (RMT) Wigner; Dyson; Mehta; Bohigas The RMT Approach: Complicated Hamiltonian: e.g. Nucleus: Solve HΨ = EΨ Replace with a Hamiltonian with matrix elements chosen randomly from a Gaussian distribution Examine the statistical properties of the resulting Hamiltonians H = Universality classes of interest here: Gaussian Orthogonal Ensemble (GOE): 1 degree of freedom (β=1) [Time-reversal symmetric] Gaussian Unitary Ensemble (GUE): degrees of freedom (β=) [TRS-Broken] M K Hypothesis: Complicated Quantum/Wave systems that have chaotic classical/ray counterparts possess universal statistical properties described by Random Matrix Theory (RMT) Cassati, 1980 Bohigas, 1984 This hypothesis has been tested in many systems: Nuclei, atoms, molecules, quantum dots, electromagnetic cavities, 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?

9 s n Normalized Spacing ( E E ) ΔE = + / n 1 n Distribution of Eigen Energies E Harm. Osc. n Chaotic TRS Integrable (ds) 6 + O Ion Nearest Neighbor Spacing Distributions Poisson TRS TRSB RMT Predictions p(s) Poisson p( s) = Exp( s) TRS (GOE) TRSB (GUE) 1. 1 integrable chaotic ATRS 0.8 chaotic TRSB B s s π π p( s) = s Exp( s p( s) = s Exp( π π ) s ) 33 Th Nucleus Chaotic TRSB Tomsovic, 1996 GOE GUE p(s) crossover experiment: P. So, et al., Phys. Rev. Lett (1994)

10 Schrödinger Helmholtz Analogy Our Experiment: A clean, zero temperature, quantum dot with no Coulomb or correlation effects! Table-top experiment! E z d 8 mm ~50cm B x B y An empty two-dimensional electromagnetic resonator Ψ with n Ψ + n m ( En V ) Ψn = 0 h = 0 at boundaries Schrödinger equation E with z, n E + z, n k n E z, n = 0 at = 0 boundaries Helmholtz equation A. Gokirmak, et al. Rev. Sci. Instrum. 69, 3410 (1998). Stöckmann + Stein, 1990 Richter, 199

11 Cryogenic (77 K) Cavity Impedance Statistics Measurement

12 Examples ψ + k ψ = n n n 0 Ψ α Circle: Trajectories are not chaotic Stadium: Trajectories are chaotic

13 Random Superposition of Plane Waves Random Amplitude Random Direction Random Phase φ n = Lim N N Re{ a jexp[ j( k j. x + θ j )} AN j= 1 Berry Hypothesis (1977) k j is uniformly distributed on a circle k j =k n as Art! Eric Heller, Harvard

14 Eric Heller, Harvard Random Superposition of Plane Waves as Art Ordered Motion and Crystals Quantum Random Waves Classical Electron Flow Quantum Modes and Classical Analogs Quasi Classical Correspondence, Quantum Scars Quantum Resonances Classical Collisions Quantum Quasi Crystal Maps Caustics Rogue Waves

15 The WaveFunction Imaging Experiment Bow-Tie cavity: All typical ray-trajectory orbits are chaotic and all periodic orbits are isolated Quarter bow-tie microwave resonator Measurement setup

16 Cavity Perturbation Imaging of E ( Ψ ) Measure E z through cavity perturbation (metallic) ω = ω 0 ( ( ) ) 1+ B E z dv pert Perturbation scanning system

17 Wave Chaotic Eigenfunctions with and without Time Reversal Symmetry 8 4 r = 4 r = a) 5.5 TRS Broken (GUE) y (inches) Ferrite x (inches) 13.6 GHz Ψ A b) GHz TRS (GOE) D. H. Wu and S. M. Anlage, Phys. Rev. Lett. 81, 890 (1998).

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19 (a) R=106.7 cm Log 10 [ Ψ ] Plots GOE 0 R=64.8 cm (cm) (b) 11.73GHz GOE GUE Crossover 10.79GHz GUE (c) Ferrite log 10 ( Ψ A Cavity ) large intermediate small 11.05GHz

20 Probability Amplitude with and without Time Reversal Symmetry (TRS) P ( Ψ Α ) GUE (TRSB) GOE (TRS) 1 0 GOE (TRS) GUE (TRSB) 0 1 Hot Spots RMT Prediction: P(ν) = (πν)-1/ e -ν/ e -ν ν = Ψ Α TRS (GOE) TRSB (GUE) D. H. Wu, et al. Phys. Rev. Lett. 81, 890 (1998).

21 +c c RMT Prediction

22 Frequency (GHz) Chaos and Scattering Hypothesis: Random Matrix Theory quantitatively describes the statistical properties of all wave chaotic systems Nuclear scattering: Ericson fluctuations dσ dω Compound nuclear reaction Proton energy Transport in quantum dots: Universal Conductance Fluctuations Resistance (kω) μm B (T) S xx Electromagnetic Cavities: Complicated S 11, S, S 1 versus frequency S 11 S S 1

23 Gaussian Ensembles and Random Matrix Theory Gaussian Orthogonal Ensemble (GOE): Time Invariant:- B=0 Hamiltonian is real and invariant under orthogonal transformations. p( s) = π π s Exp( 4 Gaussian Unitary Ensemble (GUE): Not Time Invariant:- s Hamiltonian is Hermitian and invariant under unitary transformations 3 p( s) = π 4 Exp( π ) s s )

24 Large Contribution from Periodic Ray Paths? cm 47.4 cm 11 cm Possible strong reflections L = 94.8 cm, Δf =.3GHz Single moving perturbation not adequate Bow-Tie with diamond scar