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Quantum Chaos and Sub-Planck Structure
Andrew Jordan
I
Classical Chaos
II
Quantum Chaos
III
Sub-Planck Structure
IV
Decoherence
V
Conclusions
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I Classical Chaos
Jordan (unpublished)
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Lorenz Equations
Lorenz, 63
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Water Wheel Animation
Malkus Howard, 70s
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Lehtihet Miller 86
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Cs atom experiments in laser wedge billiards
Raizen 99
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II Quantum Chaos
Where V is classically chaotic
Detailed analysis usually not analytically
possible
Statistical analysis
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Quantum Chaotic Systems Random Matricies BGS
Conjecture (84)
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SzerediGoodings 93
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Chaotic Wavefunctions
Gaussian Random Variable
Berrys conjecture, 77

• Billiard example

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100,015th state
Li Robnik 96
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Spatial Correlations
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Variance in the GRV Ansatz

• Gaussian Statistics Higher moments are easy.

Define Ccorrelation function, then
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III Sub-Planck Structure

• Smallest scales

L, P are classical scales
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Zurek,01
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IV Decoherence
Quantum Chaotic System Natural Environment Choice
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Zurek claim I
WHY?
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BUT
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Billiard Results
2D Circular Billiard, 1 particle
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Quantum Map Results
N N lattice, N1/
Jordan Srednicki 01
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• 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.

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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.