Spinor Dynamics in an Antiferromagnetic Spin-1 Thermal Bose Gas

We present experimental observations of coherent spin-population oscillations in a cold thermal, Bose gas of spin-1 ${}^{23}\mathrm{Na}$ atoms. The population oscillations in a multi-spatial-mode thermal gas have the same behavior as those observed in a single-spatial-mode antiferromagnetic spinor Bose-Einstein condensate. We demonstrate this by showing that the two situations are described by the same dynamical equations, with a factor of 2 change in the spin-dependent interaction coefficient, which results from the change to particles with distinguishable momentum states in the thermal gas. We compare this theory to the measured spin population evolution after times up to a few hundreds of ms, finding quantitative agreement with the amplitude and period. We also measure the damping time of the oscillations as a function of magnetic field.

Figure 1

(a) Spin population oscillations of a spin-1 Na spinor BEC. The fractional $m_F=0$ population is shown as a function of time. The black circles represent a single-mode BEC with initial relative phase $\theta =\pi$ and magnetic field $B=37\mu \mathrm{T}$. Here, $\theta ={\theta }_1+{\theta }_{-1}-2{\theta }_0$, and ${\theta }_i$ is the phase of spin component $i$. The blue triangles correspond to a multi-spatial-mode BEC ($\le 5$ modes) with initial $\theta =\pi$ and $B=42\mu \mathrm{T}$. The solid curve is a sine wave fit to the data. The dashed line is a guide to the eye. (b) Fractional $m_F=0$ spin population of a cloud of thermal spin-1 Na atoms as a function of time with sinusoidal fits shown as solid lines. The solid (open) triangles are for an initial condition of $\theta =\pi$ (0) and $B=11\mu \mathrm{T}$ ($21\mu \mathrm{T}$). Note the very different time scales.

Figure 2

(a) Amplitude, (b) period (black points), and damping time (blue triangles) of the spin-population oscillations of thermal atoms that are prepared in the $m_F=-1$ state (initial $\theta =0$) as functions of the magnetic field. The solid curves are predictions from the single-mode theory (see text) with the initial fraction in $m_F=0$ set to the measured mean of ${\rho }_0=0.45$. The theory curve would go to zero at zero field in panel (a) if ${\rho }_0=0.5$.

Figure 3

(a) Amplitude, (b) period (black points), and damping time (blue triangles) of spin-population oscillations of thermal atoms that are prepared in the $m_F=0$ state (initial $\theta =\pi$) as functions of the magnetic field. Solid curves are predictions of the amplitude and period from the single-mode theory (see text) with the initial $m_F=0$ fraction set to the measured mean ${\rho }_0=0.52$.