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------------- ADDITIONAL FIGURES ----------------
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Included here are additional hi-resolution versions of the figures in the paper, and movies detailing the
evolution of our simulation.

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---------------- GENERAL REMARKS ----------------
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1. Presentation of results.

   Background of movies presents magnitude of the order parameter expressed in Fermi energy units eF
   in the center of the initial axially symmetric solution, which for all movies is eF=0.5.
   By arrows we plot velocity field, where magnitude of the velocity expressed in units of Fermi velocity vF
   (computed as vF=pF/m=1) is encoded in color of arrows. Lengths of arrows are scaled separately for each frame
   by constant factor to avoid extensive overlap of vectors and they do not display actual value 
   of the magnitude of the velocity.

2. Trap parameters.

   All simulations we initialize with self-consistent solution in an axially symmetric trap,
   where the radial frequency of the trap omega_r is chosen in a such way to get value of the trapping potential
   equal 2.5 of chemical potential at the radial edge of the box. In time interval t*eF in [0-100] we smoothly
   turn on trapping potential in axial direction where axial frequency omega_z is chosen using the same prescription
   as for radial frequency, i.e. value of the trapping potential is equal 2.5 of chemical potential for the axial edge
   of the box.

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------------- SLDA SIMULATION MOVIES -------------
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1. nx24ny24nz96_movie1.mpeg
   
   Lattice: 24 x 24 x 96 with lattice spacing equal 1.0.
   
   Stages:
   a) t*eF in [0-100]: Turning on axial confining, cooling algorithm is turn on.
   b) t*eF in [100-140]: Cooling of the cloud using quantum-friction algorithm.
   c) t*eF in [140-190]: Turning on knife-edge potential (Gaussian potential of height 2.5*chemical potential and sigma=2.2),
                         shifted by 5 lattice units with respect to the center, cooling algorithm is turn on.
   d) t*eF in [190-220]: Cooling of the cloud using quantum-friction algorithm. 
                         For t*eF=220 phase difference between two clouds approaches pi and we turn off then cooling algorithm.
   e) t*eF in [220-800]: We quickly remove knife-edge potential, and observe formation and dynamic of the vortex ring.
   
   
2. nx24ny24nz96_movie2.mpeg
   
   Lattice: 24 x 24 x 96 with lattice spacing equal 1.0. 
   
   Stages:
   a) t*eF in [0-100]: Turning on axial confining together with knife-edge potential (Gaussian potential of height 2.5*chemical potential and sigma=2.2), 
                       located in the center of the trap, cooling algorithm is turn on.
   b) t*eF in [100-200]: Cooling of the cloud using quantum-friction algorithm. At the end of this stage we turn off the cooling algorithm.
   c) t*eF in [200-250]: For left half of the cloud we apply constant potential to obtain phase difference between two clouds equal pi.
   d) t*eF in [250-750]: We quickly remove knife-edge potential, and observe formation and dynamic of the vortex ring.
   
   
3. nx32ny32nz128_movie1.mpeg   
   
   Lattice: 32 x 32 x 128 with lattice spacing equal 1.0.
   
   Stages:
   a) t*eF in [0-100]: Turning on axial confining, cooling algorithm is turn on.
   b) t*eF in [100-175]: Cooling of the cloud using quantum-friction algorithm.
   c) t*eF in [175-215]: Turning on knife-edge potential (Gaussian potential of height 2.5*chemical potential and sigma=2.5),
                         shifted by 7 lattice units with respect to the center, cooling algorithm is turn on.
   d) t*eF in [215-310]: Cooling of the cloud using quantum-friction algorithm. We turn on cooling algorithm for t*eF=300.
                         For t*eF=310 phase difference between two clouds approaches pi.
   e) t*eF in [310-1100]: We quickly remove knife-edge potential, and observe formation and dynamic of the vortex ring.
   

4. nx32ny32nz128_movie2.mpeg   
   
   Lattice: 32 x 32 x 128 with lattice spacing equal 1.0.
   
   Stages:
   a) t*eF in [0-100]: Turning on axial confining, cooling algorithm is turn on.
   b) t*eF in [100-150]: Cooling of the cloud using quantum-friction algorithm.
   c) t*eF in [150-190]: Turning on knife-edge potential (Gaussian potential of height 2.5*chemical potential and sigma=2.5),
                         located in the center of the trap, cooling algorithm is turn on.
   d) t*eF in [190-235]: Cooling of the cloud using quantum-friction algorithm. At the end of this stage we turn off the cooling algorithm.
   e) t*eF in [235-275]: For left half of the cloud we apply constant potential to obtain phase difference between two clouds equal pi.
   f) t*eF in [275-675]: We quickly remove knife-edge potential, and observe formation and dynamic of the vortex ring.


5. nx48ny48nz128_movie2.mpeg   
   
   Lattice: 48 x 48 x 128 with lattice spacing equal 1.0.
   
   Stages:
   a) t*eF in [0-100]: Turning on axial confining, cooling algorithm is turn on.
   b) t*eF in [100-150]: Cooling of the cloud using quantum-friction algorithm.
   c) t*eF in [150-190]: Turning on knife-edge potential (Gaussian potential of height 2.5*chemical potential and sigma=2.5),
                         located in the center of the trap, cooling algorithm is turn on.
   d) t*eF in [190-235]: Cooling of the cloud using quantum-friction algorithm. At the end of this stage we turn off the cooling algorithm.
   e) t*eF in [235-275]: For left half of the cloud we apply constant potential to obtain phase difference between two clouds equal pi.
   f) t*eF in [275-675]: We quickly remove knife-edge potential, and observe formation and dynamic of the vortex ring.

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------------- EXPANSION MOVIES (ETF) -------------
--------------------------------------------------

These movies demonstrate the expansion procedure used to image the objects.
These simulations were performed with the ETF in cylindrical coordinates on a
2048 x 156 grid.  The procedure is as follows:

  A) Imprint and cool (with imaginary time evolution) a vortex ring with radius
     r_0 as a fraction of R_perp -- the Thomas Fermi radius of the trap waist
  B) 0  - 10ms: Linearly decrease the magnetic field from resonance to 750G
  C)      10ms: Remove the external trapping potential
  D) 10 - 11ms: Expansion at 750G.
  E) 11 - 12ms: Expansion while linearly ramping magnetic field to B_min
  F) 12 - 16ms: Expansion at B_min
  G) 16 - 17ms: Expansion while linearly ramping magnetic field to 750G
  H) 17 - 19ms: Expansion at 750G
  I)      19ms: Record image.

6. expansion_movie_small_ring.mp4

   Expansion of a small vortex ring r_0/R_perp = 0.2 at B_min = 580G

7. expansion_movie_large_ring.mp4

   Expansion of a small vortex ring r_0/R_perp = 0.48 at B_min = 580G

8. expansion_movie_small_ring_Bmin_702.mp4

   Expansion of a small vortex ring r_0/R_perp = 0.2 at B_min = 702G
