[1] T. Kita, T. Misushima and K. Machida , Phys. Rev. A 66 , 061601 (2002) .

1. Vortices lattices in a quenched rotating spinor Bose-Einstein condensate of 23Na —An application of stochastic projected Gross-Pitaevskii equation Shi-Wei Su1, Che-Hsiu. Hsueh2, I-Gung Liu2, Yu-Ching Tsai2, Tzyy-Leng Horng3and Shih-Chuan Gou1 3Department of Physics, National Tsing Hua University, Hsinchu 30047, Taiwan 1Department of Physics, National Changhua University of Education, Changhua 50058, Taiwan2Department of Applied Mathematics, Feng Chia University, Taichung 40724, Taiwan FINESS 2009, Durham, U.K. ABSTRACT NURMERICAL RESULTS By solving the stochastic projected Gross-Pitaevskii (GP) equation, we investigate the dynamics of an F=1 spinor Bose-Einstein condensate (BEC) of 23Na during the rotating evaporative cooling. We find that, during the quench, the condensates described by the order parameter grow upbut in the meantime, quantized vortices nucleate in all three components of the spinor BEC. When the rotating cloud reaches equilibrium at very low temperatures, vortices of each component would closely bind up in pairs and then arrange themselves into some particular spatial structures (see Figure 1) rather than the conventional Abrikosov lattice. Interestingly, if we consider each tightly bound vortex pair as a single entity, then by connecting the center-of-mass locations of these pairs, we find that the vortex pairs in each form a square lattice as a response to the external rotation. (a) INTRODUCTION and MOTIVATIONS • Owing to the multi-component structure of the order parameter and the highly nonlinear nature of the underlying physics, the ground state solutions of a rapid rotating anti-ferromagnetic spinor BEC remains open to question. Kita et. al. [1] concluded that the conventional Abrikosov lattices in such systems are never favored vortex states as a response to the external rotation. • The growth of Bose-Einstein condensates during the evaporative cooling has recently become an important issue. The non-equilibrium dynamics of the cooling process is determined by the thermal fluctuations as well as the binary collisions between the condensate and the thermal cloud, that can be well described by the stochastic projected Gross-Pitaevskii equation (SPGE) [2-4] . • In this work, we investigate the dynamics of vortices nucleation in a rapidly quenched and rotating spinor BEC of 23Na by solving the SPGE. The equilibrium vortex states are of particular interest, which are difficult to be determined by directly solving GP equation at T =0 using the imaginary-time propagation method. (c) Figure 1: The time evolutions of the density profiles of the three components of the order parameter, , are shown in row (a), (b) and (c), respectively. The rotation is kept at a large angular frequency, . During the quench, the rotating condensates grow up and the emergent vortices start to closely bind up in pairs and then assemble into regular structure after a sufficient long period. FORMULATIONS The dynamics of BEC during the rotational evaporative cooling is governed by the coupled stochastic projected Gross-Pitaevskii equations, which can be written in the following dimensionless form in the oscillator units (a) (b) (1) where Figure 2: (a) The map of locations for vortices in all components of when the system reaches equilibrium. The green, red and blue circles represent the vortices in , respectively. (b) The vortex-pairs in each component of the order parameter arrange themselves into a square lattice as a response to the external rotation. (a) (2) (3) T=0 CONCLUDING REMARKS (4) with • The growth of the spinor condensate in rotation can be simulated by applying the stochastic Gross-Piteavskii equation. The nucleation of vortices during the rotational evaporative cooling can be observed in the numerical simulations. • The results of our simulations suggest that the ground state of a rotating spinor BEC of 23Na consists of square lattices of vortex-pairs in all . Note that the vortex states are coreless since the core locations are different among the three components. • Our results are not necessarily comparable with those of Kita et. al. [1], for the latter were obtained based on the Ginzburg-Pitaevskii equations near Tc, instead of the Gross-Pitaevskii equations at T=0. • In our studies, we have assumed that the growth of all are subjected to the same chemical potential. Thus, it is expected that the square lattices in all components are identical, i.e., they have the same lattice constants [see Figure (b)]. It will be interesting to look into the cases when the final chemical potentials are different in all three components, as well as the cases in which an external field is applied. NURMERICAL PROCEDURE The initial state of the i-th component is sampled as [2], where (5) and are the complex Gaussian random variables. The initial temperature and chemical potential of the atomic cloud are specified by and the cut-off energy of the initial state is set by (e) (f) REFERENCES [1] T. Kita, T. Misushima and K. Machida, Phys. Rev. A 66, 061601 (2002). [2] A. S. Bradley, C. W. Gardiner and M. J. Davis, Phys. Rev. A 77, 033616 (2008). [3] C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis and B. P. Anderson, Nature 455, 948-951 [4] C. W. Gardiner, J. R. Anglin and T. I. A. Fudge, J. Phys. B 35, 1555 (2002) Here we choose ; the number of modes of the basis are , and the cut off modes are chosen as . To simulate the situation of rapid quench, we choose the final temperature and the final chemical potentials to be fixed by and .