Patterns and excitations in antiferromagnetic spinor condensates

We consider the phenomenon of spin-pattern formation reported in recent experiments in spin-2 rubidium condensate [J. Kronjäger, C. Becker, P. Soltan-Panahi, K. Bongs, and K. Sengstock, Phys. Rev. Lett. 105, 090402 (2010)]. To understand the mechanism underlying the process and explain the parameters such as pattern period and growth rate, we employ the Bogoliubov–de Gennes equations and an alternative method based on energy conservation and the uncertainty principle. While the two methods agree in the spin-1 case, only the second method can provide analytical results in the spin-2 case. Using these results, we explain the occurrence of two regimes, the spin-dominated and the Zeeman-dominated, with the prevalence of spin-wave modes and quadrupole modes, respectively. We obtain very good agreement between theory and experiment for the pattern period and growth rate over several orders of magnitude of the magnetic field strength.

Figure 1

(a) The initial state of the condensate is polarized transversely to the magnetic field (TP) and corresponds to an energy maximum. Two modes can destabilize the condensate in the presence of magnetic field: (a) the spin-wave (SW) mode, which tilts the spin vector and gives rise to the pattern with alternating spin directions, and (b) the quadrupole mode (QM), which reduces the spin vector and builds up the nematic order parameter depicted by a disk with the axis determining the nematic direction.

Figure 2

Examples of density profiles of spin components in spontaneous patterns of $F=2$ ${}^{87}$Rb: (a) spin-wave pattern at $B=0.25$G, and (b) quadrupole wave pattern at $B=1.1$G. The figures in the top row show the results of numerical simulations within the truncated Wigner approximation, while the bottom row shows the experimental data. The trap frequencies are ${\omega }_{x,y,z}=2\pi \times (85,133,0.8)$Hz and the atom density in the center of the trap (magnified here) is $n_0=8.1\times {10}^{13}$ per cm${}^3$.

Figure 3

(a) Analytical results for the dependence of the maximum growth rate of SW modes and QMs in the function of magnetic field. The figures correspond to a spin-1 Bose-Einstein condensate with a negative quadratic Zeeman splitting $\delta E<0$. The “low B” lines correspond to the PM eigenstate [25], which is close to the initial TP state in weak magnetic field, and “high B” lines correspond to the exact initial TP state in the limit of strong magnetic field. $B_0$ denotes the magnitude of the magnetic field for which the interaction and Zeeman energies are equal, $\left|\delta E\right|=c_{1}n$. The dashed-dotted line shows the overlap of the PM state and the initial TP state, $A=(1/N){\sum }_j\left|{\psi }_j^{*\left(\mathit{PM}\right)}{\psi }_j^{\left(\mathit{TP}\right)}\right|$. (b) The most unstable wave vector $k_{\max }$ as a function of $B/B_0$.

Figure 4

As in Fig. 3 but for a spin-2 condensate. Here we present both theoretical and experimental results on a bilogarithmic scale. The black dots depict the results of numerical simulations within the truncated Wigner approximation, and the crosses correspond to the experimental data. The lines are given by Eqs. (17). In addition, the dotted line represents a numerical Bogoliubov analysis [4].