Bose-Einstein condensation of trapped interacting spin-1 atoms

We investigate Bose-Einstein condensation of trapped spin-1 atoms with ferromagnetic or antiferromagnetic two-body contact interactions. We adopt the mean field theory and develop a Hartree-Fock-Popov type approximation in terms of a semiclassical two-fluid model. For antiferromagnetic interactions, our study reveals double condensations as atoms in the $\mid m_F=0⟩$ state never seem to condense under the constraints of both the conservation of total atom number $N$ and magnetization $M$. For ferromagnetic interactions, however, triple condensations can occur. Our results can be conveniently understood in terms of the interplay of three factors: (anti)ferromagnetic atom-atom interactions, $M$ conservation, and the miscibilities between and among different condensed components.

Figure 1

BEC for a gas of spin-1 atoms with $M>0$ $\left(M<0\right)$. For noninteracting atoms, the $\mid +⟩$ $\left(\mid -⟩\right)$ component condenses first at $T_1$ (dashed line) while the $\mid 0⟩$ and $\mid -⟩$ $\left(\mid +⟩\right)$ components condense simultaneously at $T_2$ (solid line). For ${}^{23}\mathrm{N}\mathrm{a}$ atoms with antiferromagnetic interactions, double condensations persist according to our theoretical study. The $\mid +⟩$ component (denoted by $+$) condenses first, which is then followed by the condensation of the $\mid -⟩$ component (denoted by $\times$). The $\mid 0⟩$ component is unpopulated in the low temperature limit. For ${}^{87}\mathrm{R}\mathrm{b}$ atoms with ferromagnetic interactions, our study reveals the potential for triple condensations. First, the $\mid +⟩$ component condenses (denoted by $◇$), which is then followed by the second condensation for the $\mid -⟩$ component (denoted by $*$), and finally the third condensation for the $\mid 0⟩$ component occurs (denoted by $\circ$).

Figure 2

Double condensations for a spin-1 gas of ${}^{23}\mathrm{N}\mathrm{a}$ atoms. The upper left panel shows the total condensed fraction vs temperature and total magnetization. Similarly, the upper right one shows the fraction of condensed $\mid +⟩$ component, the lower left the condensed $\mid 0⟩$ component, and the lower right the condensed $\mid -⟩$ component.

Figure 3

Typical density distributions for different spin components of a ${}^{23}\mathrm{N}\mathrm{a}$ gas $\left(T∕T_c=0.43,M∕N=0.4\right)$. The upper panel is for the condensate and the lower one for the noncondensed atoms.

Figure 4

The same as in Fig. 2 but for a gas of ${}^{87}\mathrm{R}\mathrm{b}$ atoms.

Figure 5

Double condensations for ${}^{87}\mathrm{R}\mathrm{b}$ atoms when $M=0$ (the upper panel) and triple condensations for $M∕N=0.6$ (the lower panel). The solid line denotes the fractional population of the condensed $\mid +⟩$ component, the dot-dashed line denotes the $\mid 0⟩$ component, and the dashed line denotes the $\mid -⟩$ component.

Figure 6

Typical density distributions for a gas of ${}^{87}\mathrm{R}\mathrm{b}$ atoms at $M∕N=0.6$. The left column is $T∕T_c=0.11$, the middle one is 0.21, and the right one 0.32. The notations are the same as in Fig. 3.

Figure 7

The lowest excitation level for a gas of ${}^{87}\mathrm{R}\mathrm{b}$ atoms at $M∕N=0.6$, $T∕T_c=0.34$ (right before the condensation of the $\mid -⟩$ component). The inset shows the details of a zoomed-in plot near the minimum.