Eniko Madarassy
Reconnections and Turbulence in atomic BEC
with C. F. Barenghi
Durham University,
2006
1
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
2
The Gross-Pitaevskii equation in a rotating system

   2 2
2
(i   )
 
  Vtr  g     Lz  
t
 2m

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:

2
dV  N
D

= wave function

= dissipation [1]

= chemical potential

= rotation frequency of the trap

= reduced Planck constant
Vtr  trapping potential, Vtr (r ) 

1
m 2 1   x x 2  1   Y  y 2
2
m = mass of an atom
g = coupling constant
  LZ = centrifugal term
LZ  angular momentum operator
[1] Tsubota et al, Phys.Rev. A65 023603-1 (2002)
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
Vortices
Vortex: a flow involving rotation about an axis
   e i


= Madelung transformation
= Density = 0, on the axis
= Phase: changes from 0 to 2π
going around the axis

Quantized circulation:
v
S

d l  
4
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)
[4] N.G. Parker, N.P. Proukakis, C.F. Barenghi and C.S. Adams, Phys. Rev. Lett. 92, 160403-1 (2004)
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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:
   ei 
We imprint the phase in two ways:
Case I:
Case II:
in upper two quadrants
in upper left quadrant (x < 0 and y > 0) and
bottom right quadrant (x > 0 and y < 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 profile
The perturbation was created from the phase change
The 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 density
Higher velocity
Sound 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:
Random phase region:
 0
Im   0
Re   0
´   , y  0
i
  e , y  0
´
imprinting
Large fluctuation of the phase:
 Im  

Phase  tan 1 
 Re  
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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:
 2
g
2
2
4
(t )  V (t ) (t )  (t ) dxdy
1. Compute the total energy. ET   
2
 2m

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,   0 and
 0
Phase imprinting applied to vortex lattice in rotating frame
Snapshots of the density and phase profile
at the times:
2
E K  E S  EV
EK 
 dx   
 dx
EV
EK
200
t  200
t  200.4
208
t  201.95
ES
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The sound energy in connection with the total energy
Due to the new level of energy by the discontinuity, the total energy changes.
E S  red
ET  green  18
200
Dimensionless unit:
Time:  ,   2 x 219 Hz
1


208
(The time units is less than 1ms)
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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.
ET  ES  EV
Two contributions to the sound energy. First, from the phase change and
second from the interaction between vortex-antivortex.
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