Vortex Generation and Dynamics in a Unitary Fermi Gas
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Quantized vortices are the hallmark of superfluidity in both Bose and Fermi systems. The theoretical description of the generation and of the dynamics of vortices in cold atomic Fermi gases is essentially non-existing, apart from an early study, based on an incomplete theory, performed by Giulia Tonini, Felix Werner, and Yvan Castin in 2005 [1]. In 2003, Aurel Bulgac and Yongle Yu [2] have predicted that one can observe a vortex by observing the density depletion in the vortex core caused by the very fast vortical motion. Two years later Zwierlein, Abo-Shaeer, Schirotzek, Schunck, and Ketterle [3] observed the images of the trapped Li6 atoms  and confirmed the large density depletion of the vortex cores, thus providing the definitive evidence of the superfluidity. However, the vortex generation and dynamics has not yet been observed experimentally, and thus their formation mechanism is still unknown. We extended the density functional description to superfluid and time-dependent phenomena and apply it here to study these aspects. We perform simulations under conditions close to those in Ref. [3], namely for an axially symmetric system, when vortex axes are parallel to the trap symmetry axis and under which conditions the emerging vortex lattice appears stable. We use either one or two stirrers  whose intensity are turned on and subsequently off adiabatically. The stirring potential models the laser beams used in Ref. [3], it has a Gaussian shape, with its center rotating on a circular orbit with constant angular velocity. The stirring velocity is chosen to match the velocity profile of the vortex calculated in [2].
 
The time-independent version of the density functional theory applied to a unitary Fermi gas was used for the first time by Bulgac and Yu [2], and was subsequently refined [4], and then applied to the time-dependent case by Bulgac and Yoon [5]. The parallel numerical implementation developed  by Bulgac, Roche, and Yu into parallel codes for 3+1 dimensional systems was briefly described in Ref. [6]. In the present  simulations we use a somewhat simplified version of that code, in which matter distribution is homogeneous in the z-direction and this version was implemented numerically by Alan (Yuan Lung) Luo and Aurel Bulgac. The initial conditions were generated by determining the ground state properties of the trapped system with a code developed by Piotr Magierski and parallelized by Alan Luo , with some input from Aurel Bulgac. 

The physical system we simulate here is placed on X3 spatial grid, where X be 32 or 64. The particles are confined by an axially symmetric potential with a nearly flat bottom.   The number of quasi-particle wave functions is of order of 323 and we follow the time evolution of the system for a few million time steps in each case.  This corresponds to a total time of the order of 1000-2000 characteristic periods of the system, which is defined as the inverse of the system Fermi energy. We record the mean field potential, the particle density, the pairing field, and the current density. Usually we take 1500 samples during each simulation and are used to generate a movie of the entire time evolution. We show also the plots of the total internal energy of the system and of the strength of the stirring potential as a function of time. For each simulation we show six different movies, from upper left to the lower right are the total potential, the particle density, the pairing field, the phase of the pairing field, the magnitude of the current density in x-y plane, and the phase of the current density.

The 32-cube systems are simulated on University of Washington's ATHENA supercomputer. The 48-cube systems are simulated on NERSC's Franklin supercomputer.

      
References

            [1]  G.  Tonini, F. Werner, and Y. Castin, arXiv:cond-mat/0504612, Eur. J. Phys. D 39, 283 (2006)
          [2] A. Bulgac and Y. Yu, Phys. Rev. Lett.  91, 190404 (2003),  cond-mat/0303235
          [3] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck and W. Ketterle, Nature 435, 1047 (2005), arXiv:cond-mat/0505635
          [4] A. Bulgac,  Phys. Rev. A 76 040502(R) (2007),  cond-mat/0703526
          [5] A. Bulgac and S. Yoon,  Phys. Rev. Lett.  102, 085302 (2009), arXiv:0812:3643
          [6] A. Bulgac and K. Roche, J. Phys.:  Conf. Ser. 125, 012064 (2008)         

Movies:

Number of vortices

View on YouTube(TM)

Higher resolution

(.mov format)

Internal Energy and Stirring Potential strength
 vs time plot
The following simulations are done on a 32-cube system

2

link

here

here

4(small stirring radius)

link

here

here

4(large stirring radius)

link

here

here

5

link

here

here

7(small stirring radius)

link

here

here

7(large stirring radius)

link

here

here

9

link

here

here
The following simulations are done on a 48-cube system

13
 link
 here
 here
No vortices
(v_rot too high)
 		link
 here
 here
5
 link
 here
 here
No vortices
(v_rot too slow)
 link here here


NB. The movies were created as Matlab(TM)  .avi file and subsequently converted to .mp4 file using ffmpeg. If anyone knows of a stable Matlab 3rd-party script (besides mpgwrite) that are able to produce .mpeg (or others) file please let me know: yuanl@u.washington.edu