Is Quantum Mechanics Chaotic?

Steven Anlage

Physics 402

Simple Chaos1-Dimensional Iterated Maps

The Logistic Map: )1(41 nnn xxx −=+ μParameter: μ Initial condition: 0x

10 15 20 25 30

0.2

0.3

0.4

0.5

5.0=μ

100.00 =x

Iteration number

x

10 15 20 25 30

0.2

0.3

0.4

0.5

0.6

0.7

0.8

8.0=μ

Iteration number

x

10 15 20 25 30

0.2

0.4

0.6

0.8

1

0.1=μ

Iteration number

x

Extreme Sensitivity to Initial Conditions1-Dimensional Iterated Maps

The Logistic Map: )1(41 nnn xxx −=+ μ0.1=μ

Change the initial condition (x0) slightly…

10 15 20 25 30

0.2

0.4

0.6

0.8

1

x

Iteration Number

101.00 =x

100.00 =x

Although this is a deterministic system,Difficulty in making long-term predictionsSensitivity to noise

Extreme Sensitivity to Initial ConditionsDouble Pendulum Demo

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."

G1-60: CHAOS - TWO DOUBLE PHYSICAL PENDULA

Start with similar initial conditions The motion of the two pendula diverge

ChaosClassical: Extreme sensitivity to initial conditions

qi, pi qi+Δqi, pi +Δpi

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/2π( )

tiVqAi

m ∂Ψ∂

=Ψ+Ψ−∇− hh2

21

Manifestations of quantum chaos:Breaking of degeneracy, Level repulsion, Strong eigenfunction fluctuations, Scars

ii

ii

qHppHq∂−∂=

∂∂=/

/&

&

Wave Chaos?

Launch 2 waves fromnearby locations

It makes no sense to talk about“diverging trajectories” for waves

However, in the ray-limitit is possible to define chaos

“ray chaos”

2-Dimensional “Sinai billiard”

Wave/Quantum Chaos ???Heisenberg Uncertainty Principle limits knowledge of initial conditions

Δp Δq > ħ( )

tiVqAi

m ∂Ψ∂

=Ψ+Ψ−∇− hh2

21

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

The Difficulty in Making Predictions in Wave Chaotic Systems…

8.7 8.8 8.9 9.0 9.1 9.2 9.3

200

400

600

800

1000

Abs

[Zca

v]

Frequency [GHz]

Perturbation Position 1

8.7 8.8 8.9 9.0 9.1 9.2 9.3

200

400

600

800

1000

Abs

[Zca

v]

Frequency [GHz]

Perturbation Position 2

Antenna Port

(Ω)

05.0~Lλ

Ele

ctro

mag

netic

W

ave

Impe

danc

e

Extreme sensitivityto small perturbations

We must resort to astatistical description

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−−−−−−−−−−−−−−−−

=

M

KH

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 randomlyfrom a Gaussian distribution

Examine the statistical properties of the resulting Hamiltonians

2 Universality classes of interest here:Gaussian Orthogonal Ensemble (GOE): 1 degree of freedom (β=1) [Time-reversal symmetric]Gaussian Unitary Ensemble (GUE): 2 degrees of freedom (β=2) [TRS-Broken]

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?

Hypothesis: Complicated Quantum/Wave systems that have chaotic classical/raycounterparts possess universal statistical properties described byRandom Matrix Theory (RMT)

Cassati, 1980Bohigas, 1984

Ψ=Ψ EH

Integrable

Chaotic TRS

233Th Nucleus

(ds)6 2+ O Ion

Chaotic TRSB

Harm. Osc.Distribution of Eigen Energies

GOE→GUE p(s) crossover experiment: P. So, et al., Phys. Rev. Lett. 74 2662 (1994)

nENormalized Spacing

( ) EEEs nnn Δ−= + /1

Tom

sovi

c, 1

996)

4(

2)( 2sExpssp ππ

−⋅=

)4(32)( 222 sExpssp

ππ−⋅=

)()( sExpsp −=Poisson

TRS (GOE)

TRSB (GUE)

RM

T P

redi

ctio

ns

s

Nearest Neighbor Spacing Distributions

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

integrablechaotic A

s

chaotic BPoisson

TRS

TRSB

p(s)

s

TRSTRSB

Schrödinger – Helmholtz AnalogyOur Experiment: A clean, zero temperature, quantum dot withno Coulomb or correlation effects! Table-top experiment!

Ez

Bx By

( )

boundariesatwith

VEm

n

nnn

0

022

2

=Ψ

=Ψ−+Ψ∇h

boundariesatEwithEkE

nz

nznnz

00

,

,2

,2

=

=+∇

Schrödinger equation

Helmholtz equation

Stöckmann + Stein, 1990Richter, 1992

d ≈ 8 mm

An empty “two-dimensional” electromagnetic resonator

A. Gokirmak, et al. Rev. Sci. Instrum. 69, 3410 (1998).

~ 50 cm

Cryogenic (77 K) Cavity Impedance Statistics Measurement

Examples

Ψα2

2 2 0n n nkψ ψ∇ + =

Circle:Trajectories are not chaotic

Stadium:Trajectories are chaotic

Random Superposition of Plane Waves…

}).([2Re{1

∑=

∞→ +=N

jjjjNn xkjExpa

ANLim θφ

Random Amplitude Random Direction Random Phase

Berry Hypothesis (1977)

http://www.ericjhellergallery.com/

Eric Heller, Harvard

… as Art!

kj is uniformly distributed on a circle |kj|=kn

Random Superposition of Plane Waves as Art

http://www.ericjhellergallery.com/

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

Eric Heller, Harvard

The WaveFunction Imaging Experiment

Quarter bow-tiemicrowave resonator

Measurementsetup

Bow-Tie cavity: All typical ray-trajectory orbits

are chaotic and all periodic orbits are isolated

Perturbationscanning system

( )( )pertz dVEB∫ −+= 2220

2 1ωω

Measure Ez through cavity perturbation(metallic)

Cavity Perturbation Imaging of E2 (|Ψ|2)

24 ′′=r

5.25 ′′=r

0 4 8 12 16

8

4

0

8

4

00 4 8 12 16

x (inches)

y (in

ches

)

181512840

A2||Ψ

Ferrite

a)

b)

13.69 GHz

13.62 GHz

Wave Chaotic Eigenfunctions with and without Time Reversal Symmetry

D. H. Wu and S. M. Anlage, Phys. Rev. Lett. 81, 2890 (1998).

TRS Broken(GUE)

TRS(GOE)

(a)

(b)

(c)

0 10 20 30 40(cm)0

10

20

R=106.7 cm

R=64.8 cm

11.73GHz

10.79GHz

11.05GHz

0.55- 0.50

Ferrite

log10(|Ψ |2ACavity)

GOE

GUE

GOE – GUECrossover

Log10[|Ψ|2] Plots

smallintermediatelarge

0.001

0.01

0.1

1

10

0 1 2 3 4 5 6 7

|Ψ | Α2

P (

)

|Ψ

| Α

2

GUE (TRSB)

GOE (TRS)GUE (TRSB)

GOE (TRS)

0

1

0 1 2

D. H. Wu, et al.Phys. Rev. Lett. 81, 2890 (1998).

Probability Amplitudewith and without Time Reversal Symmetry (TRS)

P(ν) = (2πν)-1/2 e-ν/2 TRS (GOE)e-ν TRSB (GUE)

“Hot Spots”

=νRMT

Prediction:

RMTPredictionc

+c

Chaos and ScatteringHypothesis: Random Matrix Theory quantitatively describes the statistical

properties of all wave chaotic systems

|S|S1111|||S|S2222||

|S|S2121||

Frequency (GHz)

|| xxS

|S|S1111|||S|S2222||

|S|S2121||

Frequency (GHz)

|| xxS

Electromagnetic Cavities: Complicated S11, S22, S21 versus frequency

B (T)

Transport in quantum dots: Universal Conductance Fluctuations

Res

ista

nce

(kΩ

) μm

Nuclear scattering: Ericson fluctuations

ωσ

dd

Proton energy

Compound nuclear reaction

Gaussian Ensembles and Random Matrix Theory

Gaussian Orthogonal Ensemble (GOE): Time Invariant:-

Hamiltonian is real and invariant under orthogonal transformations.

Gaussian Unitary Ensemble (GUE): Not Time Invariant:-

Hamiltonian is Hermitian and invariant under unitary transformations

B=0

)4

(2

)( 2sExpssp ππ−⋅=

)4(32)( 222 sExpssp

ππ−⋅=

Large Contribution from Periodic Ray Paths ?

22 cm

11 cm

• Possible strong reflectionsL = 94.8 cm, Δf =.3GHz

• Single moving perturbation not adequate

47.4 cm

Bow-Tie with diamond scar