Andrew Jordan

Quantum Chaos and Sub-Planck Structure

I

II

III

IV

Classical Chaos

Quantum Chaos

Decoherence

Sub-Planck Structure

V Conclusions

I Classical Chaos

01

10J

H

J

p

q

Jordan (unpublished)

Lorenz Equations

bzxyz

xzyrxy

xyx

)(

Lorenz, ‘63

Water Wheel Animation

Malkus & Howard, ’70s

Lehtihet

&

Miller

‘86

Raizen

’99

Cs atom experiments

in laser wedge

billiards

II Quantum Chaos

ErV

m)(

2

22

Where V is classically chaotic

Detailed analysis usually not

analytically possible

Statistical analysis

Quantum Chaotic Systems ~ Random Matricies

BGS Conjecture (’84):

Chaotic

Integrable2s

s

es

e

)(sP {

2, No Time Reversal invarience1, Time Reversal invarient{

Integrable45

Chaotic49

Szeredi&Goodings ‘93

Chaotic Wavefunctions

Billiard example:

Gaussian Random VariableBerry’s conjecture, ‘77

]exp[)( rkiAr jj

j

mEk j 222

)exp()(2

j

jAP then

100,015th state

Li &

Robnik

‘96

)(P

Spatial Correlations

XAPdAdAX jji

i )(,

)(exp)()(,

ykxkiAAyx jijji

i

ji ,~

)(J1

0 yxkA

Variance in the GRV Ansatz

Gaussian Statistics Higher moments are easy.

)(J)( 0 kssC

Define C=correlation function,

then

)(J)(J#

)()()()( 2102102121 ssksskkR

sCsCsCsC

Srednicki and Stiernelof, ‘96

III Sub-Planck Structure

2/2//exp, sxsxspih

sdpxW

d

d

Smallest scales:

Px /~min,

relation Heisenburg the,/~ 2

min LPV

Lp /~min, L, P are classical scales

Zurek,’01

IV Decoherence

nigH zIˆ

envenvloffdiagona pxxpipxC |)ˆˆ(exp|,

Quantum Chaotic System ~Natural

Environment Choice

LpPxpxC /and/or /for 0),( Zurek claim I:

Zurek claim II:

This is because of Sub-Planck structure in W(x,p).

WHY?

),(),(|),(| 2 ppxxWpxWdpxdhpxC ddd

BUT

),()(exp),( pxWxppxipdxdpxC dd

Large scale structure of W important

)(),( HEpxW

Berry-Voros ‘76

Billiard Results

2/

)/(J)/(J 1

0 Lp

LpPx

),( pxC

2D Circular Billiard, 1 particle:

222222 24/exp6/exp LpPx ),( pxC 3D Box, Many-Body limit:

Jordan & Srednicki ‘01

Quantum Map Results:

)or ( pxC

Nx

NxxC

|sin|

|)(|

N N lattice, N~1/

px or

Jordan & Srednicki ‘01

This tells us:Zurek Claim I:

-Yes, if Many-Body environment.Zurek Claim II:

This is because of Sub-Planck structure in W(x,p).

-Only in the Wigner Representation, otherwise a “Classical” effect.

LpPxpxC /and/or /for 0),(

V Conclusions

1. Quantum Chaotic Systems ~ Random Matricies

2. Chaotic Wavefunctions ~ Gaussian Random Variables

3. Sub-Planck scales have a physical interpretation in the context of decoherence.

4. Many-Body environments are efficient at decoherence.