Spontaneous Pattern Formation in an Antiferromagnetic Quantum Gas

In this Letter we report on the spontaneous formation of surprisingly regular periodic magnetic patterns in an antiferromagnetic Bose-Einstein condensate (BEC). The structures evolve within a quasi-one-dimensional BEC of ${}^{87}\mathrm{Rb}$ atoms on length scales of a millimeter with typical periodicities of $20\dots 30\mu \mathrm{m}$, given by the spin healing length. We observe two sets of characteristic patterns which can be controlled by an external magnetic field. We identify these patterns as linearly unstable modes within a mean-field approach and calculate their mode structure as well as time and energy scales, which we find to be in good agreement with observations. These investigations open new prospects for controlled studies of symmetry breaking and complex quantum magnetism in bulk BEC.

Figure 1

Initial state (a) and saturated spin patterns arising in spinor condensates for increasing magnetic field (b)–(f). The individual 3D rendered false color absorption profiles in each image visualize the spatial density distribution for each of the five magnetic projections $m_F=-2\dots +2$ of the $F=2$ hyperfine manifold. Imperfections in the preparation and detection artifacts (e.g., optical interference fringes), which appear as fluctuations visible in the initial state (a), are distinctly different from the patterns of (b)–(f). The cases marked “$*$” are chosen to represent the initial state, the interaction dominated regime, and the Zeeman-dominated regime, respectively, and are referred to in the following figures.

Figure 2

Simplified illustration of the local spin vector in Figs. 1a, 1c, 1e (marked “$*$”). (a) Initial state, the spin vector is fully stretched and points in the transverse $x$ direction. (b) Interaction dominated regime, the spin vector is fully stretched but rotates from axial to transverse and back over one spatial period. (c) Zeeman-dominated regime, the local spin vector has zero length but still a defined orientation ($m_F=0$ state) that rotates from axial to transverse and back.

Figure 3

Bogoliubov spectra resulting from a linear stability analysis of the 1D GPE, (a) Interaction regime at 0.25 G, (b) Zeeman regime at $\infty \mathrm{G}$. The interaction parameter $g_1⟨n⟩=2\pi \times 5\mathrm{Hz}$ is obtained from the observation of homogeneous spin oscillations. For $F=2$, five pairs of complex conjugate branches exist (of which only the positive real frequencies are plotted). Branches with positive imaginary frequency indicate unstable modes. Compare the most unstable modes (squares) to the observed modes (crosses).

Figure 4

Comparison of simulated Stern-Gerlach images with experimental data in the linear growth regime, (a) interaction dominated regime 0.25 G, (b) Zeeman-dominated regime $\infty \mathrm{G}$. The mode pattern of the calculated most unstable modes coincide with the experimentally observed modes during linear growth.

Figure 5

Cross correlations of spin densities. (a) Interaction regime at 0.25 G, (b) Zeeman regime at 1.1 G. Color-coded plots (left and right) of the normalized correlation as a function of relative axial displacement and time of evolution allow to determine the symmetry and spatial periodicity of the evolving modes—(a) $28\mu \mathrm{m}$ and (b) $22\mu \mathrm{m}$. The correlation contrast (center) can be tracked over 2 orders of magnitude and allows us to extract the growth rate of the mode (half the growth rate of the correlation function)—(a) $20{\mathrm{s}}^{-1}$ and (b) $28{\mathrm{s}}^{-1}$.