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50 UEC Int’l Mini-Conference No.53

The University of Electro-Communications SAITO LABORATORY
Ground State of Spin-1 and Spin-2 Bose-Einstein Condensates

Uyen Ha Phuong MAI and Hiroki SAITO
Department of Engineering Science, The University of Electro-Communications
Tokyo 182-8585, Japan

Abstract Ground State
Spinor Bose-Einstein condensate (BEC) was first realized
Table: Scattering lengths coefficient of Na, Rb
23
87
in 1988 and has inspired many new researches on BEC
spin-1 and spin-2
atomic gases, especially the ground-state properties of
23 Na c = 1.87a [1]
spinor BEC systems. Monte Carlo is an efficient method 1 B
Spin-1
for finding the minimal value in many physics problems. 87 Rb c = 1 − 0.47a B [1]
In this work, Monte Carlo is used to find the energy of 23 Na c = 2.67a B , c = − 2.89a B [1]
2
1
spin-1 and spin-2 BEC systems' ground states, Spin-2
a
87 Rb c = 0.99 , c = − 0.53a [2,3]
respectively. Their spin states are demonstrated by 1 B 2 B
spherical-harmonic representation. (a) (b)
Spin-1 and Spin-2 BEC
Spinor order parameters of spin-1 and spin-2 BEC:
 ( ) 1  (  1 ( ) 1 , 0 ( ) 1 , − ( ) 1 1 ) ;

 ( ) 2  (  2 ( ) 2 , 1 ( ) 2 , 0 ( ) 2 , − ( ) 2 1 , − ( ) 2 2 )
(c)
The spherical-harmonic representation: visualizing the
phase
symmetry of spin states. 2
) 
S ( ,   = f  m ( ) f Y f m ( ,   )
m= − f
The energy of uniform spin-1 and spin-2 BEC:
0
c 2 c 2 c 2
E ( ) 1 = 1 F ; E ( ) 2 = 1 F + 2 A
2 2 2 00
Figure: Spherical harmonic representations of the
23
Monte Carlo method spin-1 and spin-2 states. (a) Polar state of spin-1 Na
87
and (b) ferromagnetic state of spin-1 Rb of spin-1;
87
23
Create random complex wave functions (c) biaxial nematic state of spin-2 Na and Rb.
23
For spin-1, the ground state is polar when c > 0 ( Na)
1
87
or ferromagnetic when c < 0 ( Rb). For spin-2, when
1
Compute the energy of the system
c > 0 and c < 0, the ground state is biaxial nematic for
1 2
23
87
both Na and Rb.
Larger than Smaller than References
the initial energy the initial energy
[1] Y. Kawaguchi and M. Ueda. “Spinor Bose-Einstein
Condensates.” Physics Reports, 2012.
[2] A. Widera et. al., Precision measurement of spin-dependent
Minimal energy interaction strengths for spin-1 and spin-2 Rb atoms, New
87
(new initial energy) J. Phys. 8, 152 (2006).
[3] M. Egorov, et. al., Measurement of s-wave scattering
lengths in a two-component Bose-Einstein condensate,
Phys. Rev. A 87, 053614 (2013).