Tunable dipolar resonances and Einstein-de Haas effect in a ${}^{87}\mathrm{Rb}$-atom condensate

We theoretically study a spinor condensate of ${}^{87}\mathrm{Rb}$ atoms in a $F=1$ hyperfine state confined in an optical dipole trap. Putting initially all atoms in an $m_F=1$, component we observe a significant transfer of atoms to other, initially empty Zeeman states exclusively due to dipolar forces. Because of conservation of a total angular momentum the atoms going to other Zeeman components acquire an orbital angular momentum and circulate around the center of the trap. This is a realization of the Einstein-de Haas effect in a system of cold gases. We show that the transfer of atoms via dipolar interactions is possible only when the energies of the initial and the final sates are equal. This condition can be fulfilled utilizing a resonant external magnetic field, which tunes energies of involved states via the linear Zeeman effect. We found that there are many final states of different spatial density, which can be tuned selectively to the initial state. We show a simple model explaining high selectivity and controllability of weak dipolar interactions in the condensate of ${}^{87}\mathrm{Rb}$ atoms.

Figure 1

Population of $m_F=0$ (main frame) and $m_F=-1$ (inset) components as a function of the magnetic field (note that the magnetic field is pointed toward the negative $z$ axis, and only its values are depicted here). We observe many magnetic resonances corresponding to different values of the magnetic field. The initial number of atoms in the $m_F=1$ state is equal to $N_{+1}=5\times {10}^4$, and the aspect ratio is $\beta ={\omega }_z/{\omega }_r=1/4$. Maximal transfer is reached at $t~0.1$ s for each resonance.

Figure 2

Population of $m_F=0$ (black) and $m_F=-1$ (gray) components as a function of time for the first resonance. The magnetic field is equal to $B=-0.12$ mG.

Figure 3

Phase (upper panel) and density (lower panel) of $m_F=+1,0,-1$ components (left, middle, and right columns, respectively) in the $\mathit{xy}$ plane for $N_{+1}=5\times {10}^4$ and $\beta =1/4$. The data correspond to the first maximum in the resonance structure (Fig. 1). The magnetic field is equal to $B=-0.12$ mG. $N_0=15\hspace{0.5em}000$ and $N_{-1}=6000$ atoms both at their maximal transfers.

Figure 4

Density of $m_F=0$ (upper panel) and $m_F=-1$ (lower panel) components in the $\mathit{xz}$ plane for $N_{+1}=5\times {10}^4$ atoms and $\beta =1/4$. Successive frames show density patterns characteristic for resonances depicted in Fig. 1. Note the increasing number of rings with the order of the resonance as well as the presence of even (odd) number of rings in the $m_F=0$ ($m_F=-1$) component.

Figure 5

Population of the $m_F=0$ (main frame) and $m_F=-1$ (inset) components as a function of the magnetic field for the spherically symmetric trap with $\omega =2\pi \times 100$ Hz. We observe magnetic resonances corresponding to different values of the magnetic field. Initial number of atoms in the $m_F=1$ state is equal to $N_{+1}=5\times {10}^4$. Maximal transfer is reached at $t~0.3$ s for each resonance.

Figure 6

Phase (left column) and density (middle and right columns for $\mathit{xy}$ and $\mathit{xz}$ planes, respectively) in the $m_F=0$ (upper panel) and $m_F=-1$ (lower panel) components corresponding to the second resonance in Fig. 5 at maximal population. Initial number of atoms is $N_{+1}=5\times {10}^4$, $B=-0.16$ mG, and the trap is spherically symmetric with frequency $\omega =2\pi \times 100\hspace{0.5em}\text{Hz}.$

Figure 7

Spatial densities of the three lowest states coupled by dipolar interactions. (a) Ground state of the trap with $m_F=+1$. (b) Lowest excited state with one quanta of angular momentum and symmetric in the $z$ direction. It is accessible when two interacting atoms flip their spins simultaneously. (c) Lowest excited state with one quanta of angular momentum and antisymmetric in the $z$ direction. It is accessible when one ore two interacting atoms flip their spins.

Figure 8

Energy of two-atom quantum states as a function of an external magnetic field for ${\omega }_z/{\omega }_{\perp }=1/4$. For some values of the external magnetic field energies of some states become equal due to the linear Zeeman effect. Note, however, that, according to (11b), the crossing of Zeeman levels happens for magnetic fields oriented along the negative $z$ direction. Two insets show real and avoided crossings. Note that avoided crossings appear only for states coupled by dipolar interactions.