Classical and Quantum Chaos in the SinaiAndreev billiard

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Classical and Quantum Chaos in the Sinai-Andreev
billiard
Andor Kormányos
Collaborators Zoltán Kaufmann (Eötvös
University) József Cserti (Eötvös
University) Colin Lambert (Lancaster University)
cond-mat/0506671
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Outline of the talk

• Motivation less detailed knowledge of ballistic
  N-S systems from quantum-chaos point of view
• Classical dynamics in N-S hybrid systems and in
  the Sinai-Andreev billiard
• Quantum description and the quantal-classical
  correspondence, Wigner transform
• Wavefunctions

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Classical dynamics (1)
Andreev billiard (AB) normal dot
superconductor Andreev reflection conversion of
an electron of pepF v(1e/E F) to a hole of
phpF v(1 - e/E F) if elt? ( ? pair
potential) retracing approximation ? all AB
integrable
Silvestrov et al. Phys. Rev Lett. 90
116801 Adiabatically integrable
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Classical dynamics (2)
In a more general case non-exact retracing
diffraction at the critical points destroys
the integrability for e?0 Poincaré section of
surface at the N-S interface

• chaotic sea
• intermittent-like motion
• integrable islands

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Quantum calculations and the Wigner function
QM description Bogoliubov-de Gennes equations
single-particle Hamiltonian
pair potential
Quantal-classical correspondence projection of
the Wigner function onto the PS
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Picture gallery chaotic states
scarred states
ergodic states
Z scaled probability density
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Picture gallery regular states
intermittent states
tori states
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Summary
M. C. Escher Order and Chaos