1 enikö madarassy vortex motion in trapped bose-einstein condensate durham university, march, 2007
TRANSCRIPT
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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:
0x 0y
0y
: 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
' and
d
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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 :Ψ(r,t)
Boundary condition at infinity: Ψ(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)
222
2)( gV
mti tr
D
NdV2
222 112
1)(, yxmrVpotentialtrappingV Yxtrtr
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Vortex-antivortex pair(Without dissipation)
Fig. 1
Fig.1, t = 87.2 Fig.2, t = 93.0 Fig.3, t = 98.8
Fig.4, t = 104.4 Fig.5, t = 110.2 Fig.6, t = 116.0
Period = 28.8 = 0.8
The first vortex has sign +1 and the second sign -1
d
Levels: 0.012…….0.002
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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
[2]M Kobayashi and M. Tsubota, Phys. Rev. Lett. 94, 065302 (2005)
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The sound energy and the vortex energy
vortexkinsound
kin
EEE
xvxm
xdE
,))()((2
22
2
The sound is reabsorbed
Sound energy Vortex energy
Correlation between vortex energy and sound energy
The corelation coefficient:-0.844 which means anticorrelation
Sound Energy
Vor
tex
En
ergy
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The period and frequency of motion for vortex – antivortex pair
2
The period of motion
The frequency of motion
p
p
pp
2
sp 2\
Triangle with Circle with
p\p
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The translation speed for different separation distance
rv pair 2
dxv pair
1
2
1
The translation speed for vortex-antivortex pair:
In our case: t
yv y
'
and
The trajectory for one of thevortices in the pair
In a homogeneous superfluid
Circle: with the formula, Triangle: with numerical calculation
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The trajectory for the one of the vortices in the pair and for one vortex
The trajectory for one of the vortices in the pair (xy)
x - coordinate vs time y - coordinate vs time
The trajectory for one vortex (xy) x - coordinate vs time y - coordinate vs time
are: 0.00 (purple); 0.01 (red); 0.07 (green); 0.10 (blue) (xy)
are: 0.00 (purple); 0.01 (red); 0.07 (green); 0.10 (blue) (xt and yt)
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)
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Two vortices without dissipation andwith dissipation =0.01
Density of the condensate with two vorticesThe initial separation distance d = 1.00
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The trajectory for one vortex set off-centreVarying initial position and dissipation
0x
0x
The trajectory for one vortex (xy) x - coordinate vs time y - coordinate vs time
0x
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
=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) 0x
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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
0x
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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)
0x 0x
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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)
0x
0x
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The frequency of the motion for one vortexas a function of the initial position
p
0x
[3] B.Jackson, J. F. McCann, and C. S. Adams, Phys.Rev. A 61 013604 (1999)
p
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The friction constants for one vortexas a function of the dissipation and initial position
' and
0x
The friction constant
for :0.90 (blu) and 2.00 (red), : 0.001; 0.003; 0.010 and 0.030
0x0x
The friction constant for 0.90 (blu) and 2.00 (red), : 0.001; 0.003; 0.010 and 0.030
sisisi vxzxzvxzvdt
rd \
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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).
',