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UEC Int’l Mini-Conference No.54 23

Ground State of Spin-1 Bose–Einstein Condensates: A Many-Body Approach

Beyond the Single-Mode Approximation

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

Keywords: Bose-Einstein Condensate (BEC), many-body theory, phase separation, spinor

1. Introduction
Bose–Einstein condensates (BECs) occur when bosonic
particles are cooled to near absolute zero temperatures,
resulting in macroscopic occupation of the lowest quantum-
mechanical ground state. A BEC with a spin internal degree of
freedom, which is called a spinor BEC, was first observed in a
23
gas of spin-1 Na atoms confined in an optical dipole trap in
1998 [1], opening up a new research arena of ultracold atomic
systems. The mean-field theory has been the principal Fig. 1. Schematic illustration of a quasi-one-dimensional torus [4].
framework used to investigate spinor BECs due to its relative annihilation operator of a boson with angular momentum l. We
simplicity and capability of capturing essential behaviors [2]. employ a numerical diagonalization method to investigate
However, it neglects critical quantum phenomena, including many-body eigenstates of the Hamiltonian H by restricting
ˆ
many-body quantum fluctuations, correlations, and the Hilbert space to that spanned by the angular momentum
fragmentation effects, which can be particularly pronounced in l = 0, 1 states and obtain the ground state of the system.

multi-component and spinor systems and can be investigated 3. Expected results
by many-body theory [3]. Most of the previous studies on
mixtures of spinor BECs have been restricted to the single- The confinement and spatial degrees of freedom are
mode approximation (SMA), where the spatial degrees of expected to have profound effects on the ground state of the
freedom are frozen [2, 3]. However, there is a possibility that spin-1 system. Specifically, we anticipate observing
phase separation occurs in a system much larger than the spin fragmentation phenomena that differ substantially from mean-
healing length, which cannot be captured by the SMA. The field and SMA predictions due to these spatial effects. The
presence of orbital angular momentum modes might induce
purpose of the present research is to explore the possibility of
phase separation in the spin-1 BEC using many-body theory and stabilize fragmented states. Our detailed numerical
beyond SMA. simulations will provide precise angular momentum
2. Model distributions, spatial density profiles, spin correlation functions
enabling a comprehensive understanding of the quantum nature
We consider a system of bosons on a quasi-torus with of phase separation, fragmentation, and their dependence on
radius as schematically illustrated in Fig. 1. The spatial effects.
ˆ
ˆ
ˆ
ˆ
Hamiltonian of the system is given by H = H + H + H , Through our many-body numerical approach, we aim to
s
0
a
where capture both similarities and potentially significant deviations
h
H = ˆ 0  0 2 d    2M 2  2 + V  −    † ˆ ˆ (1) from mean-field theory predictions regarding the nature and

extent of phase separation. Detailed spatial structures such as
spin domains, immiscibility and dynamics will be carefully


2
ˆ
†2 ˆ
ˆ
ˆ
H = 0  2 d  † ˆ    2  +  2 g   2   , (2) examined to reveal the precise role of quantum many-body

a

correlations.
ˆ
ˆ
† 0 ˆ
ˆ
H = 2 d   g   † ˆ    References
s 0   m 1 m 2 m 2 m 1 
m 1 ,m 1 ,m 2 ,m 2  2   (3) [1]. D.M. Stamper-Kurn et. al., Optical confinement of a Bose–
 g   Einstein condensate, Phys. Rev. Lett. 80 (1998).
ˆ
ˆ
† 1 ˆ
+    † ˆ m f  f   m    . [2]. Y. Eto et. al., Observation of dipole-induced spin texture in
m
 2 1 2 m m  m 2 m  2 m  2 1  
1 1
an Rb Bose-Einstein condensate, Phys. Rev. Lett. 112 (2014).
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with  is the azimuthal angle. The field operator is [3]. Y. Shi, Ground states of a mixture of two species of spinor Bose
1
ˆ
 ( )  = 1  c ˆ e il , (4) gases with interspecies spin exchange, Phys. Rev. A 82 (2010).
2 l=− 1 l [4]. R. Kanamoto et. al., Critical fluctuations in a soliton formation of
attractive Bose-Einstein condensates, Phys. Rev. A 73 (2006).
where magnetic quantum number m = − 1,0,1 and ˆ c is the
l

*The author is supported by (SESS) MEXT Scholarship