Magnetic field dependence of the dynamics of ${}^{87}\mathrm{Rb}$ spin-2 Bose-Einstein condensates

We experimentally studied the spin-dependent collision dynamics of ${}^{87}\mathrm{Rb}$ spin-2 Bose-Einstein condensates confined in an optical trap. The condensed atoms were initially populated in the $\mid F=2,m_F=0⟩$ state, and their time evolutions in the trap were measured in the presence of external magnetic field strengths ranging from 0.1 to $3.0\mathrm{G}$. The atom loss rate due to inelastic two-body collisions was found to be $1.4\left(2\right)\times {10}^{-13}{\text{cm}}^3{\mathrm{s}}^{-1}$. Spin mixing in the $F=2$ manifold developed dramatically for the first few tens of milliseconds, and the oscillations in the population distribution between different magnetic components were observed over a limited range of magnetic field strengths. The antiferromagnetic property of this system was deduced from the magnetic field dependence on the evolution of relative populations for each $m_F$ component.

Figure 1

Reduction in the number of condensed atoms in an optical trap at a magnetic field strength of $3.0\mathrm{G}$. (a) Absorption images of atomic clouds after free expansions of $22\text{ms}$ (upper two) and $18\text{ms}$ (lowest one). Storage times are also shown on the right-hand side. A Stern-Gerlach separation was applied to distinguish the different $m_F$ components. $\mid 2,0⟩$ was the predominant component observed. The size of the field of view for each image is $1.1\times 0.45\text{mm}$. (b) Number of condensed atoms in the $\mid 2,0⟩$ state remaining in the trap as a function of trap time. Each data point represents the average of three measurements. Error bars indicate the standard deviation. The curve was fitted to Eq. (3), after omitting the $K_1$ and $K_3$ terms.

Figure 2

Number of atoms in the $\mid 2,0⟩$ (solid circles), $\mid 2,\pm 1⟩$ (open squares), and $\mid 2,\pm 2⟩$ (solid triangles) states, as a function of trap time at a magnetic field strength of $1.5\mathrm{G}$. The total number of atoms (solid diamonds) is also shown. Each point represents the average of five measurements. Error bars indicate the standard deviation.

Figure 3

Relative populations in the $\mid 2,0⟩$ (solid circles), $\mid 2,\pm 1⟩$ (open squares), and $\mid 2,\pm 2⟩$ (solid triangle) states, as a function of trap time at magnetic field strengths of $1.5\mathrm{G}$ (a), $0.75\mathrm{G}$ (b), $0.3\mathrm{G}$ (c), and $0.1\mathrm{G}$ (d). Each point represents the average of five measurements. Error bars indicate the statistical error.

Figure 4

Images taken after evolutions of $7\text{ms}$ (a) and $10\text{ms}$ (b) in a magnetic field strength of $0.75\mathrm{G}$. The dips in the density distributions of the condensed atoms are shown in the center of the condensates in a vertical direction for the $\mid 2,0⟩$ component (a) and the $\mid 2,\pm 1⟩$ components (b). The size of the field of view for each image is $1.0\times 0.4\text{mm}$. (c) Horizontal profiles of each component in (a) were cut at the center in the vertical direction.

Figure 5

Relative populations of each spin component after $70\text{ms}$ storage time in the optical trap represented as a function of the various magnetic field strengths. Solid circles indicate the $\mid 2,0⟩$ component, solid squares the $\mid 2,\pm 1⟩$ component, and solid triangles the $\mid 2,\pm 2⟩$ component.

Figure 6

Loss rates for an inelastic two-body collision at various magnetic field strengths. The values were determined by fitting Eq. (3) and omitting the $K_1$ and $K_3$ terms for the evolution of the total number of atoms in each data set. At a magnetic field strength of $1.0\mathrm{G}$, the data points were plotted separately because the two values were very close. The magnitude of the error for each point was dependent on the error in the measured trap frequency, which was estimated to be $8\text{Hz}$.