Spinor condensate of ${}^{87}\mathrm{Rb}$ as a dipolar gas

We consider a spinor condensate of ${}^{87}\mathrm{Rb}$ atoms in the $F=1$ hyperfine state confined in an optical dipole trap. Putting initially all atoms in the $m_F=0$ component, we find that the system evolves toward a state of thermal equilibrium with kinetic energy equally distributed among all magnetic components. We show that this process is dominated by the dipolar interaction of magnetic spins rather than spin-mixing contact potential. Our results show that because of a dynamical separation of magnetic components, the spin-mixing dynamics in the ${}^{87}\mathrm{Rb}$ condensate is governed by the dipolar interaction which plays no role in a single-component rubidium system in a magnetic trap.

Figure 1

Populations of thermal clouds for $m_F=+1$ (red/gray), $m_F=0$ (black), and $m_F=-1$ (green/light gray) states as a function of time with dipole-dipole interactions (main panel) and without them (inset). The parameters are $N=3\times {10}^5$, $\beta =1$. Total populations of $m_F=\pm 1$ components are identical when the dipole-dipole interaction is turned off. The external magnetic field is equal to zero.

Figure 2

Kinetic energy of $m_F=+1$ (red/gray), $m_F=0$ (black), $m_F=-1$ (green/light gray) states as a function of time with dipole interactions (left panel) and without them (right panel) for $N=3\times {10}^5$ and $\beta =1$. The thermalization time is shorter if dipolar interactions are included. The external magnetic field is equal to zero.

Figure 3

The atomic density on the $z=0$ plane of $m_F=+1$ (left column), $m_F=0$ (middle column), and $m_F=-1$ (right column) components for the spherically symmetric trap and $N=3\times {10}^5$ atoms at time $t=0.25$ s (top row) and $t=0.95$ s (bottom row). The external magnetic field is equal to zero.

Figure 4

Ratio of averaged dipole-dipole energy to the mean value of the contact spin-mixing term ${\varepsilon }_d/{\varepsilon }_c$ as a function of time. The external magnetic field is equal to zero.

Figure 5

Kinetic energy of $m_F=+1$, 0, $-1$ components for $N={10}^5$ atoms as a function of time. (Top) The oblate geometry, $\beta =2$, (bottom) the prolate shape, $\beta =0.5$. Results were obtained from dynamics generated by the full Hamiltonian (main panels) and by the Hamiltonian without the dipole-dipole term (insets). The external magnetic field is equal to zero.

Figure 6

Densities of $m_F=+1$ (left column), $m_F=0$ (middle column), and $m_F=-1$ (right column) components for the case of $N={10}^5$ atoms. The top row is for $\beta =2$ at $t=0.7$ s, and the bottom row is for $\beta =0.5$ at $t=0.35$ s. The densities of $m_F=+1$ and $m_F=-1$ components for $\beta =2$ are spatially separated, while for $\beta =0.5$ they overlap.

Figure 7

Populations of thermal clouds for $m_F=+1$ (red/gray), $m_F=0$ (black), and $m_F=-1$ (green/light gray) states as a function of time with dipole-dipole interactions (main panel) and without them (inset). The parameters are $N=3\times {10}^5$, $\beta =1$, and $B_z=1$ mG. The thermalization time is $t\approx 0.4$ s.

Figure 8

Populations of thermal clouds for the $m_F=+1$ (red/gray), $m_F=0$ (black), and $m_F=-1$ (green/light gray) states as a function of time with dipole-dipole interactions (main panel) and without them (inset) for spherical symmetry and for $B_z=100$ mG. The time of thermalization at $B_z=100$ mG is longer then for $B_z=1$ mG; however, a role of dipolar forces is still important (compare inset). The system thermalizes after $t\approx 0.7$ s.

Figure 9

The atomic density on the $z=0$ plane of the $m_F=+1$ (left column), $m_F=0$ (middle column), and $m_F=-1$ (right column) components for the spherically symmetric trap and $N=3\times {10}^5$ atoms at $B_z=1$ mG at time $t=0.45$ s with dipole-dipole interactions (top row) and without them (bottom row).

Figure 10

A sequence of frames showing the evolution of spin textures under dipole-dipole interactions. The value of the external magnetic field is $B_z=1$ mG and the Larmor precession frequency is ${\omega }_L^{1\hspace{0.5em}\mathrm{mG}}=6.9$ and $t_L=1.45\times {10}^{-3}$ s. The sequence of frames starts at $t=1.2$ s and ends after a single cycle.

Figure 11

A sequence of frames showing the evolution of spin textures under dipole-dipole interactions. Here, the value of external magnetic field is $B_z=100$ mG and Larmor precession frequency is ${\omega }_L^{100\mathrm{\hspace{0.5em}}\mathrm{mG}}=690$ and $t_L=1.45\times {10}^{-5}$ s. The sequence of frames starts from 0.75 s and ends after a single cycle.

Figure 12

Kinetic energy of $m_F=+1$ (red/gray), $m_F=0$ (black), $m_F=-1$ (green/light gray) states as a function of time with dipole interactions (main panel) and without them (inset) for spherical symmetry. The thermalization time at $B_z=1$ mG remains shorter if dipolar interactions are included.

Figure 13

Kinetic energy of the $m_F=+1$ (red/gray), $m_F=0$ (black), $m_F=-1$ (green/light gray) states as a function of time with dipole interactions (main panel) and without them (inset) for spherical symmetry and magnetic field $B_z=100$ mG. The system thermalizes after $t\approx 0.7$ s.