Enik Madarassy Vortex motion in trapped BoseEinstein condensate

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Enikö Madarassy Vortex motion in trapped
Bose-Einstein condensate

• Durham University,
• March, 2007

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Outline

• Gross - Pitaevskii / Nonlinear Schrödinger
  Equation
• Vortex - Antivortex Pair (Without Dissipation
  and with Dissipation)
• - Sound Energy, Vortex Energy
• - Trajectory
• - Translation Speed
• One vortex (Without Dissipation and With
  Dissipation)
• - Trajectory
• - Frequency of the motion
• - Connection between dissipation and
  friction constants in vortex
• dynamics
• Conclusions

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This work is part of my PhD project with Prof.
Carlo F. Barenghi We are grateful to Brian
Jackson and Andrew Snodin for useful
discussions. Notations
initial position of the vortex from
the centre of the condensate ( 0.0 )
initial separation distance between the
vortex-antivortex pair ( 0.0 )
friction constants model of
dissipation in atomic BEC period
of the vortex motion frequency of the
vortex motion
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The Gross-Pitaevskii equation

• 
  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
• g coupling constant
• 1 Tsubota et al, Phys.Rev. A65 023603-1
  (2002)

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Vortex-antivortex pair(Without dissipation)
Levels 0.012.0.002
Fig.2, t 93.0
Fig.1, t 87.2
Fig.3, t 98.8
Fig. 1
The first vortex has sign 1 and the second sign
-1
Fig.5, t 110.2
Fig.4, t 104.4
Fig.6, t 116.0
Period 28.8
0.8
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Transfer of the energy from the vortices to the
sound field

• Divide the kinetic energy (E) into a component
  due to the sound field Es and a component due to
  the vortices Ev 2
• Procedure to find Ev at a particular time
• 1. Compute the kinetic
  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

2M Kobayashi and M. Tsubota, Phys. Rev. Lett.
94, 065302 (2005)
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The sound energy and the vortex energy
Vortex energy
Sound energy
The sound is reabsorbed
The corelation coefficient -0.844 which means
anticorrelation
Correlation between vortex energy and sound
energy
Vortex Energy
Sound Energy
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The period and frequency of motion for vortex
antivortex pair
The frequency of motion
The period of motion
Triangle with Circle with
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The translation speed for different separation
distance
The trajectory for one of the vortices in the
pair
The translation speed for vortex-antivortex pair
In a homogeneous superfluid
Circle with the formula, Triangle with
numerical calculation
In our case
and
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The trajectory for the one of the vortices in the
pair and for one vortex
are 0.00 (purple) 0.01 (red) 0.07 (green)
0.10 (blue) (xy) are 0.00 (purple) 0.01 (red)
0.07 (green) 0.10 (blue) (xt and yt)

• The trajectory for one of the
• vortices in the pair (xy)
• x - coordinate vs time
• y - coordinate vs time

are 0.00 (purple) 0.01 (red) 0.07 (green)
0.10 (blue) (xy) are 0.01 (purple) 0.07
(green) 0.10 (blue) (xt and yt)

• The trajectory for one vortex (xy)
• x - coordinate vs time
• y - coordinate vs time

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Two vortices without dissipation andwith
dissipation 0.01
Density of the condensate with two vortices The
initial separation distance d 1.00
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The trajectory for one vortex set
off-centre Varying initial position and
dissipation
are 0.90 (y 0.0) and 1.30 (y 0.0)
are 0.00 (purple) 0.01 (red) 0.07 (green)
0.10 (blue) (xy) are 0.01 (red) 0.07
(green) 0.10 (blue) (xt and yt)
1.30

• The trajectory for one vortex (xy)
• x - coordinate vs time
• y - coordinate vs time

0.90
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The x- and y-component of the trajectory for one
vortex (same initial position)
- 2.00
0.030 (purple) 0.010 (blue) 0.003
(aquamarine) and 0.000 (red)
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The x- and y-component of the trajectory for one
vortex (same dissipation)
-0.90 (green) and -
2.00 (red) 0.001
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The trajectory for one vortex (same initial
position)
- 2.00
0.000 (red) and 0.003 (green)
- 2.00
0.030 (red) and 0.010 (green)
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The radius of the trajectory for one vortex (same
initial position)
- 0.90
0.030 (red) 0.010 (purple) 0.003 (blue) and
0.001 (green)
- 2.0
0.030 (green) 0.010 (purple), 0.003
(blue), 0.001(aquamarine) and 0.000 (red)

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The frequency of the motion for one
vortex as a function of the initial position
3 B.Jackson, J. F. McCann, and C. S. Adams,
Phys.Rev. A 61 013604 (1999)
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The friction constants for one
vortex as a function of the dissipation and
initial position
The friction constant for 0.90
(blu) and 2.00 (red), 0.001
0.003 0.010 and 0.030
The friction constant for 0.90 (blu)
and 2.00 (red), 0.001 0.003 0.010
and 0.030
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Conclusions

• Inhomogeneity of the condensate induces vortex
  cyclical motion.
• With dissipation the vortex spirals out to the
  edge of the condensate.
• The cyclical motion of the vortex produces
  acoustic emissions.
• The sound is reabsorbed.
• Relation between (in GP equation) and
  (in vortex dynamics).