Vortices lattices in a quenched rotating spinor Bose-Einstein condensate of Vortices lattices in a quenched rotating spinor Bose-Einstein condensate of 2323Na Na —— AAn application of stochastic projected Gross-Pitaevskii equationn application of stochastic projected Gross-Pitaevskii equation

Shi-Wei Su1, Che-Hsiu. Hsueh2, I-Gung Liu2, Yu-Ching Tsai2, Tzyy-Leng Horng3 and 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

[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)

ABSTRACTABSTRACT

INTRODUCTION and MOTIVATIONSINTRODUCTION and MOTIVATIONS

NURMERICAL PROCEDURENURMERICAL PROCEDURE

REFERENCESREFERENCES

CONCLUDING REMARKSCONCLUDING REMARKS

FORMULATIONSFORMULATIONS

1. 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.

2. 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.

3. 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.

4.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.

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 up but 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.

T=0

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.

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)

(e) (f)

1 0 1, , Ψ

1 0 1

1 1 1

*1 0 1

1 2 2 2 * 21 1 1 0

0 2 2 2 *0 0 0 1

1 2 2 2 21 1 1 0

1 1

2 2

1 12

2 2

1 1

2 2

GP

GP

GP

z n s s

z n s s

z n s s

H x y L g n g n n n g

H x y L g n g n n g

H x y L g n g n n n g

The initial state of the i-th component is sampled as [2], where ,

0x y x y

x y

n n n nn n

i

/1

2

1 , ,

1x y x y x y x y

inin n iniBx yn n n n n n nn k T

N Ne

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

( )

2 2 2 2

( )

2 and

2x y x y

x yn n R n n

x y

k kE k

m L

x ynn

(5)

1T nK 1 0 1 12.5

NURMERICAL RESULTSNURMERICAL RESULTS

25 , 1.5ini iniT nK

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 .

128x yN N 64xc ycN N

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.

(a)

(b)

(c)

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.

,ˆ , , 1,0GP GPB

i iii i i

dW ti P H t H t it k T dt

xx x

(1)

(2)

(3)

(4)

2

0 2

2 0

: complex white noise

: basis functions

22 2

3

2 23

: density pro

ˆ : projection operator restricting the dynamics in the low en

file for t

ergy regim

e

e

h

j B j j

j

i i

n z

s z

dW k Tdt i

a ag

a ag

n

P

x

-th component : total densityii n n

with

FINESS 2009, Durham, U.K.

1m

where

20yxL L

1 0 1, , 0.8

i

(a)

Ψ

0 2, : s-wave scattering lengths in the

channel of 0,2

: aspect ratio of the condensatez

a a

F

i

i