Collective Spin Modes of a Trapped Quantum Ferrofluid

We report on the observation of a collective spin mode in a spinor Bose-Einstein condensate. Initially, all spins point perpendicular to the external magnetic field. The lowest energy mode consists of a sinusoidal oscillation of the local spin around its original axis, with an oscillation amplitude that linearly depends on the spatial coordinates. The frequency of the oscillation is set by the zero-point kinetic energy of the BEC. The observations are in excellent agreement with hydrodynamic equations. The observed spin mode has a universal character, independent of the atomic spin and spin-dependent contact interactions.

Figure 1

(a) A BEC is initially polarized perpendicular to the external magnetic field. Spin collective modes are excited by a magnetic field inhomogeneities (generated by several coils, two of them being sketched). In the rotating frame, these modes correspond to spins oscillating around their initial direction (and not around the inhomogeneous field) with an amplitude which linearly depends (in this figure) on the position $x$. The mode is analyzed after separation of the spin components by a Stern-Gerlach procedure [which produces absorption images such as the one shown in (b)]. From the absorption pictures, we extract (c) the dynamics of the population in the different Zeeman states $p_{m_s}$ (for clarity, only $m_s=0$ and $m_s=-3$ are shown), as well as (d) the separation $\delta$ between the Zeeman states, which is associated to the in situ spin flux. Solid lines are fits by a damped sinusoid (top), and fits by a sum of two sinusoids (bottom). Here $g{\mu }_{B}b/h=30.1\mathrm{MHz}/\mathrm{m}$. In (c) and (d) data points are averages over 10 realizations, while error bars are standard deviations, dominated by technical noise in the imaging procedure.

Figure 2

Experimental data as a function of the magnetic field gradient. Left: fractional populations in $m_s=-3$ as a function of time, for various magnetic field gradients $b$, showing oscillations for each $b$. Right: Fitted oscillation frequency associated to population dynamics and spin motion. The solid line results from solving Eq. (1) beyond the small oscillation approximation [23], and converges to the result of Eq. (3) when $b\to 0$.

Figure 3

Spin dynamics for a magnetic field gradient of $g{\mu }_{B}b/h=5.5\mathrm{MHz}{\mathrm{m}}^{-1}$. (a) Fractional spin populations; from top to bottom $p_0$, $p_{-1}$, $p_{-2}$, $p_{-3}$, the solid lines result from the analytical expression detailed in Ref. [23]. (b) Spatial spin separation $\delta (t)$ is shown, together with a sinusoidal fit. In both cases, error bar correspond to statistical fluctuations.