Long-time-scale dynamics of spin textures in a degenerate $F=1$ ${}^{87}$Rb spinor Bose gas

We investigate the long-term dynamics of spin textures prepared by cooling unmagnetized spinor gases of $F=1$ ${}^{87}$Rb to quantum degeneracy, observing domain coarsening and a strong dependence of the equilibration dynamics on the quadratic Zeeman shift $q$. For small values of $\left|q\right|$, the textures arrive at a configuration independent of the initial spin-state composition, characterized by large length-scale spin domains and the establishment of easy-axis (negative $q$) or easy-plane (positive $q$) magnetic anisotropy. For larger $\left|q\right|$, equilibration is delayed as the spin-state composition of the degenerate spinor gas remains close to its initial value. These observations support the mean-field equilibrium phase diagram predicted for a ferromagnetic spinor Bose-Einstein condensate and also illustrate that equilibration is achieved under a narrow range of experimental settings, making the $F=1$ ${}^{87}$Rb gas more suitable for studies of nonequilibrium quantum dynamics.

Figure 1

Condensate Zeeman populations ${\zeta }_{\pm 1}^{\left(c\right)}$ during equilibration of unmagnetized spinor gases at variable $q$. Measurements at various evolution times $t=$ 200 ms (purple circles), 1 s (red squares), and 3 s (black diamonds) are shown with initial ${\zeta }_{\pm 1}^{\left(\mathrm{th}\right)}=$ (a) 1/3 and (b) $1/4$. (c) Zeeman populations at $t=2$ s are compared for initial ${\zeta }_{\pm 1}^{\left(\mathrm{th}\right)}=1/3$ (green triangles), $1/4$ (blue circles), and 0 (yellow squares). For a narrow range $\left|q\right|/h\lesssim 5$ Hz (shaded), Zeeman populations evolve toward a common steady state within about 2 s, toward values qualitatively in accord with mean-field predictions (solid line) based on the spin-dependent contact interaction. Outside that range, initial population differences persist throughout the experimentally accessed equilibration times. Data are averages of three experimental runs.

Figure 2

(top) Transverse and (bottom) longitudinal magnetization after a variable evolution time at a quadratic shift of $q/h=$ (a) $-5$, (b) 0, and (c) 5 Hz for an initial spin mixture of ${\zeta }_{\pm 1}^{\left(\mathrm{th}\right)}=1/3$. The transverse magnetization is represented using the color scheme shown, where the hue indicates the magnetization orientation and the brightness its magnitude. The maximum brightness is set by a fully polarized condensate at 2-s evolution time. The longitudinal magnetization is represented by the color bar. Textures form from initially small domains and then coarsen to where only a few domains span the entire condensate. The late-time images show a spin-space anisotropy; for positive $q$ the transverse magnetization appears brighter, while for negative $q$ the opposite trend is observed.

Figure 3

The magnetization variance vs $q$ for ${\zeta }_{\pm 1}^{\left(\mathrm{th}\right)}=1/3$ at $t=2$ s for transverse magnetization $G_T\left(0\right)$ (red circles) and longitudinal magnetization $G_L\left(0\right)$ (green triangles). For $q=0$ the longitudinal and transverse magnetization variances are equal, showing no preference in spin orientation (isotropic), while for $q>0$ ($q<0$), the data show a preference for transverse (longitudinal) magnetization, seen in the larger transverse (longitudinal) magnetization variance (easy plane or easy axis). The inset shows the temporal evolution of the transverse [$G_T\left(0\right)$, red circles] and longitudinal [$G_L\left(0\right)$, green triangles] magnetization variance, evaluated within the central regions of spin textures produced from ${\zeta }_{\pm 1}^{\left(\mathrm{th}\right)}=1/3$ spin mixtures at $q/h=$ $-5$ Hz (left), 0 Hz (middle), and 5 Hz (right). At early equilibration times, $t\lesssim 700$ ms, the magnetization has equal variance in the longitudinal and transverse orientations. Within 1–2 s, a clear preference is seen for longitudinal orientation at $q>0$ (middle), while for $q=0$ the spin distribution remains isotropic. Data are the average of six experimental repetitions, and error bars are the standard deviation of the data.

Figure 4

Temporal evolution of domain length after variable evolution time at a quadratic shift of 0 Hz for an ${\zeta }_{\pm 1}^{\mathrm{th}}=1/3$ mixture. The domain length, as defined in the text, grows smoothly as a function of the evolution time, with domains of about 10 $\mu$m to final values of 40 $\mu$m within 2 s.