Ground states of trapped spin-1 condensates in magnetic field

We consider a spin-1 Bose-Einstein condensate trapped in a harmonic potential under the influence of a homogeneous magnetic field. We investigate spatial and spin structure of the mean-field ground states under constraints on the number of atoms and the total magnetization. We show that the trapping potential can make the antiferromagnetic condensate separate into three distinct phases and ferromagnetic condensate into two distinct phases. In the ferromagnetic case, the magnetization is located in the center of the harmonic trap, while in the antiferromagnetic case magnetized phases appear in the outer regions. We describe how the transition from the Thomas-Fermi regime to the single-mode approximation regime with decreasing number of atoms results in the disappearance of the domains. We suggest that the ground states can be created in experiment by adiabatically changing the magnetic-field strength.

Figure 1

Schematic profiles of phase-separated states. (a) The swapping of two domains can change the total energy, even if the number of atoms in each of them is kept constant. (b) The structure of the ground state of an antiferromagnetic condensate composed of three phases. The dashed line shows the local magnetization.

Figure 2

Normalized energy per atom versus magnetization for the possible ground states of the ferromagnetic condensate. The lines correspond to normalized densities (from top to bottom) $\left|c_2\right|n/\delta E=0.1,0.4,0.7,1.0,1.3$. All the paths fulfilling the condition $de/dm=\mathrm{const}$, with the exception of completely magnetized states, arrive at the central point corresponding to the ${\rho }_0$ state in the low-$n$ limit.

Figure 3

Examples of ground states of an antiferromagnetic (a)–(c) and a ferromagnetic (d) condensate in the quasi-1D case. (a) $2\mathrm{C}+{\rho }_{+}$ state for $m=0.8$ and $\delta E/c_2{n}_{\max }=0.05$, (b) ${\rho }_0+2\mathrm{C}+{\rho }_{+}$ state for $m=0.6$ and $\delta E/c_2{n}_{\max }=0.17$, (c) ${\rho }_0+{\rho }_{+}$ state for $m=0.4$ and $\delta E/c_2{n}_{\max }=0.37$, and (d) $\mathrm{PM}+{\rho }_0$ state for $m=0.1$ and $\delta E/c_2{n}_{\max }=-1.13$. The $n_{+}$, $n_0$, and $n_{-}$ components are depicted by dash-dotted, dashed, and dotted lines, respectively, and the solid lines show the total density. Here ${\omega }_{\perp }=2\pi \times {10}^3\hspace{0.5em}$ Hz, other parameters are $N=8.4\times {10}^3$ ${}^{23}\mathrm{Na}$ atoms, ${\omega }_z=2\pi \times 2.5\hspace{0.5em}$ Hz in (a)–(c) and $N=8.4\times {10}^3$ ${}^{87}\mathrm{Rb}$ atoms, ${\omega }_z=2\pi \times 1.7\hspace{0.5em}$ Hz in (d).

Figure 4

Diagrams of phase separation of an antiferromagnetic (a),(b) and a ferromagnetic (c),(d) condensate in the quasi-1D case. The left column corresponds to the untrapped case (spontaneous phase separation), while the right column corresponds to a condensate in a harmonic potential.

Figure 5

The ${\rho }_0+{\rho }_{+}$ state in a quasi-2D harmonic trap. The densities of the (a) $m=+1$, (b) $m=0$, and (c) $m=-1$ components are shown, together with the cross section of the densities along the $x$ axis (d). The parameters are $N=2.5\times {10}^6$ ${}^{23}\mathrm{Na}$ atoms, $m=0.8$, $B=0.15\hspace{0.5em}$G, $\delta E/c_2{n}_{\max }=0.15$, ${\omega }_{\mathit{xy}}=2\pi \times 10$ Hz, ${\omega }_z=2\pi \times {10}^3$ Hz.

Figure 6

Crossover from the TF regime to the single-mode regime and the disappearance of spin domains. The parameters are the same as in Fig. 3b, except that the number of atoms was decreased (a) 4 times and (b) 16 times, while the trap frequency was increased to keep the density constant. Here the spin healing length is ${\xi }_s=17$ $\mu$m and the magnetic healing length is ${\xi }_B=42$ $\mu$m.

Figure 7

Generation of the ground-state profiles by adiabatically changing the magnetic field. The magnetic field is gradually increased from zero to the final value $B=0.15$ G during $t=4$ s for sodium condensate (a), (b) and decreased from $B=0.5$ G to $B=0.35$ G during $t=3$ s (c), (d) for rubidium. Panels (a) and (c) show the time dependence of the atom density in the $m=0$ and $m=+1$ components, respectively, and the right column shows the final domain profiles. The parameters are as in Fig. 3, except $m=0.6$ in (a), (b) and $m=0$ in (c),(d).