Observation of Spin Superfluidity in a Bose Gas Mixture

The spin dynamics of a harmonically trapped Bose-Einstein condensed binary mixture of sodium atoms is experimentally investigated at finite temperature. In the collisional regime the motion of the thermal component is shown to be damped because of spin drag, while the two condensates exhibit a counterflow oscillation without friction, thereby providing direct evidence for spin superfluidity. Results are also reported in the collisionless regime where the spin components of both the condensate and thermal part oscillate without damping, their relative motion being driven by a mean-field effect. We also measure the static polarizability of the condensed and thermal parts and we find a large increase of the condensate polarizability with respect to the $T=0$ value, in agreement with the predictions of theory.

Figure 1

Computed atomic density distribution $n_{\uparrow ,\downarrow }(x,0,0)$ of the binary mixture at finite temperature showing the component $\uparrow$ (violet) and $\downarrow$ (green), each one of these being composed of a superfluid (top) and a thermal part (bottom). (a) In the absence of any external force the centers of mass of all four components overlap. (b) In the presence of a differential force $F_{\uparrow ,\downarrow }$, the condensed part shows a large positive polarization, while the thermal component interacting with the condensate is polarized in the opposite direction. The thermal part lying outside the BEC region has a small positive polarization.

Figure 2

(a) Spin oscillations for the thermal $S_T$ (red) and condensed $S_0$ (blue) parts of the mixture with $N_0/N=0.3$ ($T/T_c\simeq 0.85$) for configuration (A). After a small transient period, $S_T$ oscillates at ${\omega }_T=0.207(2){\omega }_x$ which turns out to be equal, within error bars, to the oscillation frequency of $S_0$, ${\omega }_{\mathrm{SD}}=0.205(2){\omega }_x$. The ratio of the oscillation amplitude of $S_T$ and $S_0$ is 0.18(2). (b) Spin oscillations for the condensed and the thermal $\{S_0,S_T\}$ parts for a mixture with $N_0/N=0.4$ ($T/T_c\simeq 0.75$) in configuration (B). The condensed component oscillates at ${\omega }_{\mathrm{SD}}=0.233(5){\omega }_x$, while the thermal relative motion is quickly damped. We measure an exponential decay of $S_T$ corresponding to ${\omega }_x\tau =1.5(6)$. (c) Thermal spin current $S_T$ for a nonsuperfluid mixture (above $T_c$) in configuration (A) where we observe a few damped oscillations at the trap frequency ${\omega }_x$ with an exponentially decaying envelope from which we extract the decay lifetime, and obtain ${\omega }_x\tau =11(2)$. (d) Same measurement for configuration (B) where we observe a purely exponential decay and extract ${\omega }_x\tau =1.2(4)$, compatible with the measurement of $\tau$ below $T_c$. To maintain a roughly constant condensed fraction during the measurement, we limit the observation time to the first 500 ms after excitation. This explains why, due to the different trapping frequency ${\omega }_x$, more oscillations are shown for configuration (A) than for (B).

Figure 3

(a) Static spin-dipole polarizability as a function of temperature showing, respectively, the different contributions from the superfluid (blue), the thermal component (red), and the total one (black). (b) The thermal part lying in the region occupied by the superfluid has a negative polarization (green) whereas the outer part has a small positive polarization. The calculation has been performed for the two different configurations (A) (solid) and (B) (dashed). The static polarizabilities measured for $N_0/N=0.4$ are also shown and well agree with the predictions of theory.

Figure 4

Measured spin displacements $\{S_0,S_T\}$ for the thermal (red) and condensed components (blue) of the mixture as a function of $x_0$ for $N_0/N=0.4$ for configuration (B). From such data, we extract $\{{\mathcal{P}}_0,{\mathcal{P}}_T\}$ using a linear fit to the data in the linear region around the origin. We obtain $\{{\mathcal{P}}_0=56(8),{\mathcal{P}}_T=-3(3)\}$.