Eniko%20Madarassy%20%20Reconnections%20and%20Turbulence%20in%20atomic%20BEC%20with%20C.%20F.%20Barenghi

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Eniko Madarassy Reconnections and Turbulence
in atomic BECwith C. F. Barenghi

• Durham University,
• 2006

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Outline

• Gross - Pitaevskii / Nonlinear Schrödinger
  Equation
• Vortices (phase, density, quantized circulation)
• Phase imprinting produces a soliton-like
  disturbance which
• decays into vortices
• Sound energy and Kinetic energy
• Conclusions

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The Gross-Pitaevskii equation in a rotating system

• 
  also
  called Nonlinear Schrödinger Equation
• The GPE governs the time evolution of the
  (macroscopic) complex wave function
• ?(r,t)
• Boundary condition at infinity ?(x,y) 0
• The wave function is normalized
• wave function
• 
  
  reduced Planck constant
• dissipation 1
• 
  
• chemical potential
• 
  m
  mass of an atom
• rotation frequency of the trap
• 
  
  centrifugal term

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Vortices

• Vortex a flow involving rotation about an axis
  
• Madelung transformation
• Density 0, on the axis
• Phase changes from 0 to 2p
• going around the axis
• Quantized circulation

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Aim / motivations

• Creation of mini-turbulent vortex system
• Large scale turbulence of quantized vortices
  is studied in superfluid 3He-B
• and 4He.
• Disadvantage of turbulence in BEC
• small system and few vortices
• Advantage relatively good visualization of
  individual vortices,
• more detail
• Particularly can study detail of
  transformation of kinetic energy
• into acustic energy 2 , (which occurs in
  liquid helium too).
• Because of 1) vortex reconnection 3
• 2) vortex
  acceleration 4
• 2 C. Nore, M. Abid, and M.E. Brachet.,
  Phys. Rev. Lett. 78, 3896 (1997 )
• 3 M.Leadbeater, T. Winiecki, D.S.
  Samuels, C.F. Barenghi, C.S. Adam, Phys. Rev.
  Lett. 86, 1410
• (2001)

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Decay of soliton-like perturbation into vortices

• Dark solitons are observed in BECs 5,6 ,
  they are produced with the Phase Imprinting
  method 7.
• For example
• We imprint the phase in two ways
• Case I
  Case II
• in upper two quadrants
  in upper left quadrant (x lt 0 and y gt 0) and
• 
  bottom right quadrant (x gt 0 and y lt
  0)
• In both cases soliton-like perturbations are
  produced.
• Solitary waves in matter waves are characterized
  by a particular local density minimum and a sharp
  phase gradient of the wave function at the
  position of the minimum.
• 5 S. Burger et al., Phys.Rev. Lett. 83,5198
  (1999) J. Denschlag et al., Science 287, 97,
  (2000)
• 6 N.P. Proukakis, N.G. Parker, C.F. Barenghi,
  C.S. Adams, Phys. Rev. Lett. 93, 130408-1, (2004)
• 7 L. Dobrek et al., Phys. Rev. A 60, R3381
  (1999)

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Case I. Snapshots of the density profileThe
perturbation was created from the phase
changeThe original sound wave
The perturbation bends and decays into the vortex
pair Sound waves due to the decay of the
perturbation
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Case I. (continued) The perturbation starts to
move and bends because of the difference in the
densityHigher velocitySound waves due to the
vortex pair production
Five pairs of vortices
Three pairs go into boundary. Two pairs survive
.
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(Case I. Continued) Another view Sound waves
due to the decay of the perturbation.
The
perturbation bends and starts to move.
The perturbation
decays into the vortex pair. The soliton like
perturbation.
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(Case I continued)
Phase
imprinting
Random phase region ? 0
Large fluctuation of the phase
Im ? 0
Re ? 0
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Transfer of the energy from the vortices to the
sound field

• Divide the total energy into a component due to
  the sound field Es and a component due to the
  vortices Ev 8
• Procedure to find Ev at a particular time
• 1. Compute the total energy.
• 2. Take the real-time
  vortex distribution and impose this on a separate
  state
• with the same a)
  potential and
• 
  b) number of particles
• 3. By propagating the GPE in
  imaginary time, the lowest energy state is
• obtained with this vortex
  distribution but without sound.
• 4. The energy of this state
  is Ev.
• Finally, the the sound energy is Es E Ev

8 N.G. Parker and C.S. Adams, Phys. Rev. Lett.
95, 145301 (2005)
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Case II,Phase imprinting
applied to vortex lattice in rotating frame
Snapshots of the density and phase profile at the
times

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The sound energy in connection with the total
energyDue to the new level of energy by the
discontinuity, the total energy changes.
Dimensionless unit (The
time units is less than 1ms) Time
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Conclusions

• By generating a discontinuity in the phase, the
  system tries to smooth out this change and
  generate a soliton-like perturbation, which
  decays into vortices.
• We observe transformation of kinetic energy into
  sound energy.
• The sound energy is the biggest contribution to
  the change of the total energy.
• Two contributions to the sound energy. First,
  from the phase change and second from the
  interaction between vortex-antivortex.