Persistent Currents in Spinor Condensates

We create and study persistent currents in a toroidal two-component Bose gas, consisting of ${}^{87}\mathrm{Rb}$ atoms in two different spin states. For a large spin-population imbalance we observe supercurrents persisting for over two minutes. However, we find that the supercurrent is unstable for spin polarization below a well-defined critical value. We also investigate the role of phase coherence between the two spin components and show that only the magnitude of the spin-polarization vector, rather than its orientation in spin space, is relevant for supercurrent stability.

Figure 1

Preparation and detection of supercurrent in a two-component gas. (a) The ring trap is formed by a horizontal “sheet” beam and a vertical Laguerre-Gauss (LG) “tube” beam. $B$ is the external magnetic field. (b) Supercurrent is induced by a Raman transfer of atoms between two spin states, $|\uparrow ⟩$ and $|\downarrow ⟩$, using the LG beam and an auxiliary Gaussian (G) beam. During the transfer each atom absorbs $3\hslash$ of angular momentum from the LG beam. Two-component gas is created by coupling $|\uparrow ⟩$ and $|\downarrow ⟩$ states with an rf field. The characteristic rotational energy is $E_r/h\approx 0.4\mathrm{Hz}$. (c) Time-of-flight image of the atoms, with spin states separated using a Stern-Gerlach gradient. The rotational state $q$ is deduced from the radius $R$ characterizing the central hole in the density distribution. The image shown was taken after $t=4\mathrm{s}$ of rotation; the longitudinal spin polarization is $P_z=0.44$ and $q=3$ for both spin states. (d) Histogram of $\approx 900$ measurements of $R$ at various $P_z$ and $t$.

Figure 2

Single- versus two-component supercurrent. (a) In a pure $|\uparrow ⟩$ state ($P_z=1$) supercurrent persists for over 2 min, with no phase slips occurring for $\approx 90\mathrm{s}$. (b) At $P_z=0$ the first phase slip occurs within 5 s and we observe no rotation beyond 20 s. (c) Total atom number decay for $P_z=1$ (open symbols) and $P_z=0$ (solid symbols). Dashed lines are double-exponential fits.

Figure 3

Supercurrent stability in a partially spin-polarized gas. The statistically averaged supercurrent state, $⟨q⟩$, of the majority spin component is shown as a function of $P_z$ and the evolution time $t$. The contour plot is based on $\approx 1600$ measurements of $q(P_z,t)$. The transition between stable- and unstable-current regimes occurs at $0.6<P_z<0.7$. In the stable regime the current eventually decays due to the atom-number decay.

Figure 4

The role of spin-coherence in superflow stability. (a) An rf pulse at $t=0$ creates a spin-superposition state in which $P_z<1$ but $P=1$. As the superposition decoheres, transverse spin-polarization decays and $P\to P_z$. (b) Transverse-polarization decay (see text). Double-exponential fit (solid red line) gives the decay function $f(t)$. (c) Adiabatic dressing of the spin state. In presence of a resonant rf field ${\overrightarrow{B}}_{\mathrm{eff}}\propto \widehat{y}$ and the dressed $P_z=0$ state $|y⟩$ is stable against decoherence. (d) In the dressed $|y⟩$ state the supercurrent is also stable.

Figure 5

Critical spin polarization $P_c$. (a) Characteristic time of the first phase slip, $\tau$, versus $P_z$. Vertical blue line marks $P_c$, accurately determined in (b). (b) $|\overrightarrow{P}|$ at the onset of the supercurrent decay. Fit to the points with $P(\tau )>P_z$ (horizontal blue line) gives $P_c=0.64(1)$. (c) Stability diagram on the Bloch sphere. The blue-shaded region in Fig. 3 maps into the outer shell $|\overrightarrow{P}|>P_c$.