Photonic Crystals for Matter Waves: BEC in Optical Lattices

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Photonic Crystals for Matter Waves BEC in
Optical Lattices

• Elena Ostrovskaya and Yuri Kivshar

Nonlinear Physics Group, Australian Research
Council Centre of Excellence for Quantum-Atom
Optics, Australian National University
http//wwwrsphysse.anu.edu.au/nonlinear
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Vortices in BEC
2?m phase change
Vortices exist in a harmonic trap in the presence
of repulsive atom-atom interactions
Density
Vortices described by GP equation
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Two-dimensional optical lattice
What is the effect of periodicity on the
condensate?
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2D band-gap structure
Stationary states
Bloch states
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Gap solitons

• Nonlinearity allows energy transmission in
  the linear bandgap in the form of gap solitons
• Envelope of gap soliton is described in terms
  of Bloch states near the lower band edge
• Critical power for self-focusing of the
  envelope (Townes) soliton in 2D (1960s),
• Strongly localized in the gap

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Imprinting a phase BEC vortex in-band
Bloch vortex at k0
Bloch vortex at kkM
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Discrete Vortices

• A vortex which has discrete steps in its phase
  profile
• Approximated by coupled gap solitons
• Energy flow in the system determined by the phase
  and coupling between lobes

0
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Vortices in the gap
(a)
(b)
(c)
(a) Off-site vortex (b) On-site vortex (c) Wide
off-site vortex
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Evolution of gap vortices robust
Off-site vortex with initial perturbation 5 of
the wavefunction density On-site vortex displays
similar dynamics
t0
t2 ms
t4 ms
t6 ms
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Evolution of gap vortices unstable
Off-site wide vortex with initial perturbation
5 of the wavefunction density
t0
t2 ms
t4 ms
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Evolution of wide gap vortices
On-site wide vortex with initial perturbation
5 of the wavefunction density
t2 ms
t4 ms
t6 ms
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Further directions

• Gap vortices may also exist in robust states with
  
• reduced symmetry, including
• Rhomboidal vortices
• Triangular vortices

Existence depends on coupling between the lobes
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Summary

• In-band vortices exist with Bloch wave tails
• Localised vortices with a discrete phase profile
  may exist in the gap
• Energy flow between the lobes is determined by
  the phase and coupling of the lobes
• Demonstrated the existence of both robust and
  unstable discrete vortex solitons
• Showed the existence of reduced-symmetry vortices.