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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:

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 1121)(, 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

Ene

rgy

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The period and frequency of motion for vortex – antivortex pair

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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

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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).

',