Periodic spin textures in a degenerate $F=1$ ${}^{87}\mathrm{Rb}$ spinor Bose gas

We report on the spin textures produced by cooling unmagnetized ${}^{87}\mathrm{Rb}$ $F=1$ spinor gases into the regime of quantum degeneracy. At low temperatures, magnetized textures form that break translational symmetry and display short-range periodic magnetic order characterized by one- or two-dimensional spatial modulations with wavelengths much smaller than the extent of the quasi-two-dimensional degenerate gas. Spin textures produced upon cooling spin mixtures with a nonzero initial magnetic quadrupole moment also show ferromagnetic order that, at low temperature, coexists with the spatially modulated structure.

Figure 1

The number of atoms in the thermal fraction of the $|m_z=0\rangle$ sublevel (green circles) is compared with the average number in the thermal fractions of the $|m_z=\pm 1\rangle$ states (blue squares) at different final temperatures of the $\eta =1/4$ clouds. These data are derived from bimodal fits to time-of-flight density images taken after the three spin states were spatially separated in a magnetic field gradient (such an image is shown). While the relative populations in the Zeeman levels at high temperature are consistent with the initial value of $\eta =1/4$, below $T_{c,N_0}$, the thermal components in all Zeeman components become roughly equal. A record of the total atom number (condensed and noncondensed) in all three sublevels vs. temperature is shown in the inset.

Figure 2

Spin textures of ${}^{87}\mathrm{Rb}$ $F=1$ spinor gases produced by cooling thermal spin mixtures with initial values (a) $\eta =0$ and (b) $\eta =1/4$. The transverse magnetization ${\mathrm{M̃}}_{x,y}$ (color wheel shown) is shown in the $\mathrm{x̂}\mathrm{ẑ}$ plane. Temperatures are scaled by the Bose condensation temperature given the trap parameters and the population of the $|m_z=0\rangle$ state. Textures labeled (i), (ii), and (iii) are analyzed further in Fig. 4.

Figure 3

The magnetization vector components ${\mathrm{M̃}}_{x,y,z}$ (grayscale spanning $\pm 2/3$ of the maximum condensate magnetization) are shown for the central $25\times 100$ $\mu {\mathrm{m}}^2$ regions of two samples produced by cooling initial mixtures with $\eta =0$ to 80 nK. Regions of sample (a) show prominent magnetization modulations along either a single near-vertical wave vector (top portion) or along two orthogonal wave vectors (bottom portion). In contrast, modulations in (b) are predominantly along a near-horizonal wave vector. The magnetization spatial correlation function $G(\delta r)$ evaluated over a $40\times 200$ $\mu {\mathrm{m}}^2$ region reveals patterns of alternating spins characteristic of these samples.

Figure 4

Spatial Fourier power spectra of the magnetization, $|\tilde{\mathbf{M}}(k_x,k_y)|{}^2$, for gases prepared with (i) $\eta =0$ and (ii, iii) $\eta =1/4$. Periodic spin modulation in (i) and (ii) is indicated by broad spectral peaks with modulation wavelength $\lambda \simeq 10$ $\mu$m. Ferromagnetic order in (ii) and (iii) is indicated by the spectral peak at zero wave vector. Gases cooled to temperatures of (ii) 100 and (iii) 130 nK, respectively, indicate the abrupt transition to a modulated spin texture. The power spectrum is evaluated for the entire magnetization texture, corresponding to images shown in Fig. 2. Dashed ellipses show demarcation of Fourier space into the central region, indicative of ferromagnetic order, and an annular region, indicative of short-range spin modulation. Such regions are used for an empirical characterization (Fig. 7) of the spin textures.

Figure 5

The periodic magnetization textures are characterized by a spin-space axis along which the magnetization is modulated. Distributions of spins within the central $30\times 20$ $\mu {\mathrm{m}}^2$ regions of the magnetization texture, for samples at the lowest temperatures are shown. For $\eta =0$, both transverse ($F_{x,y}$) and longitudinal ($F_z$) spin variations are evident, while for $\eta =1/4$ the modulation is solely transverse. The long axis of the prolate spin distribution defines the local spin axis (indicated by double-headed arrows). For $\eta =1/4$, this distribution is offset due to the coexistent ferromagnetic order. Spatial patterns corresponding to the observed spin-space modulation are shown schematically at the bottom. Dashed ellipses are guides to the eye.

Figure 6

The local spin axis, determined within a 30-$\mu$m region centered variably along the $\mathrm{ẑ}$ axis, is not uniform across the extent of the condensate but meanders over length scales of around 50 $\mu$m. The azimuthal and polar angles of this axis, given in radians, with typical statistical error bar shown, are presented for the lowest-temperature sample in Fig. 2b, with initial $\eta =1/4$.

Figure 7

Measures for ferromagnetic order (red rectangles) and for the short-length-scale spin modulation (green circles) are defined as the sums $P_M={\sum }_k|\tilde{\mathbf{M}}(k_x,k_y)|{}^2$ in regions of spatial Fourier space defined in Fig. 4. Insets show these order parameters as fractions of the integrated spectral power expected for a uniformly magnetized condensate. (a) Gases cooled from $\eta =0$ show a single transition to a magnetic texture dominated by short-range spin modulation. (b) Data for $\eta =1/4$ show an additional transition between textures with coexistent ferromagnetic order and spin modulation observed at low temperature, and solely ferromagnetic textures observed at high temperature.