Stability of nonstationary states of spin-2 Bose-Einstein condensates

The dynamical stability of nonstationary states of homogeneous spin-2 rubidium Bose-Einstein condensates is studied. The states considered are such that the spin vector remains parallel to the magnetic field throughout the time evolution, making it possible to study the stability analytically. These states are shown to be stable in the absence of an external magnetic field, but they become unstable when a finite magnetic field is introduced. It is found that the growth rate and wavelength of the instabilities can be controlled by tuning the strength of the magnetic field and the size of the condensate.

Figure 1

The positive imaginary part ${\omega }^{\mathrm{i}}$ related to ${\widehat{U}}_{2;-1}\left(T\right)$ for different values of the magnetic field parameter $q$. The unit of ${\omega }^{\mathrm{i}}$ is $|g_1|n/\hslash$. Note that the scales of ${\epsilon }_k$ and ${\omega }^{\mathrm{i}}$ are not equal in the top and bottom rows. Note also that the scale of the $q=5|g_1|n$ figure is shifted with respect to the scale of the $q=|g_1|n$ case. The solid white line gives the approximate location of the fastest-growing instability, and the dashed white line corresponds to the largest possible size of a stable condensate, see Eq. (31) and Table .

Figure 2

The positive imaginary part ${\omega }^{\mathrm{i}}$ related to the eigenvalues (37) and to ${\widehat{U}}_{2;0;-2}\left(T\right)$ for various values of the quadratic Zeeman term $q$ and population ${\rho }_0$. The unit of ${\omega }^{\mathrm{i}}$ is $|g_1|n/\hslash$. We have chosen $f_z=0$ as this choice gives the fastest-growing instabilities and the smallest size of a stable condensate. In the top row the dashed, dotted, and solid lines correspond to $q=1,2,3$, respectively, while in the bottom row they correspond to $q=-1,-2,-3$, respectively. We have set $\theta =0$ in ${\psi }_{2;0;-2}$ as the stability was found to be independent of $\theta$.

Figure 3

The positive imaginary part ${\omega }^{\mathrm{i}}$ of the eigenvalues (45) and (46) related to ${\psi }_{1;-1}$ for $q=3$ and $q=-6$. The unit of ${\omega }^{\mathrm{i}}$ is $|g_1|n/\hslash$. We have chosen $f_z=0$ as it gives the fastest-growing instabilities and the smallest size of a stable system. The solid and dashed lines correspond to ${\omega }_8^{\mathrm{i}}$ and ${\omega }_5^{\mathrm{i}}$, respectively, while the dotted line gives ${\omega }_7^{\mathrm{i}}$ and ${\omega }_{10}^{\mathrm{i}}$ [see Eqs. (45) and (46)].

Figure 4

The positive imaginary part ${\omega }^{\mathrm{i}}$ of the eigenvalues (52) related to ${\psi }_{2;0}$ for $q=5$ and $q=-2$. The unit of ${\omega }^{\mathrm{i}}$ is $|g_1|n/\hslash$. The solid white line gives the approximate location of the fastest-growing instability, and the dashed white line corresponds to the largest possible size of a stable condensate, see Table .