Quantum Chaos and SubPlanck Structure Andrew Jordan I

Quantum Chaos and Sub-Planck Structure Andrew Jordan I Classical Chaos II Quantum Chaos III Sub-Planck Structure IV Decoherence V Conclusions

I Classical Chaos Jordan (unpublished)

Lorenz Equations Lorenz, ‘ 63

Water Wheel Animation Malkus & Howard, ’ 70 s

Lehtihet & Miller ‘ 86

Cs atom experiments in laser wedge billiards Raizen ’ 99

II Quantum Chaos Where V is classically chaotic Detailed analysis usually not analytically possible Statistical analysis

Quantum Chaotic Systems ~ Random Matricies BGS Conjecture (’ 84): 1, Time Reversal invarient 2, No Time Reversal invarience

Szeredi&Goodings ‘ 93

Chaotic Wavefunctions Gaussian Random Variable Berry’s conjecture, ‘ 77 Billiard example: then

100, 015 th state Li & Robnik ‘ 96

Spatial Correlations

Variance in the GRV Ansatz Gaussian Statistics Higher moments are easy. Define C=correlation function, then Srednicki and Stiernelof, ‘ 96

III Sub-Planck Structure Smallest scales: lx , min ~ h/ P L, P are classical scales

Zurek, ’ 01

IV Decoherence Quantum Chaotic System ~Natural Environment Choice

Zurek claim I: Zurek claim II: This is because of Sub-Planck structure in W(x, p). WHY?

BUT Large scale structure of W important Berry-Voros ‘ 76

Billiard Results 2 D Circular Billiard, 1 particle: 3 D Box, Many-Body limit: Jordan & Srednicki ‘ 01

Quantum Map Results: N N lattice, N~1/ 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.

V Conclusions 1. 2. 3. 4. Quantum Chaotic Systems ~ Random Matricies Chaotic Wavefunctions ~ Gaussian Random Variables Sub-Planck scales have a physical interpretation in the context of decoherence. Many-Body environments are efficient at decoherence.