Crossover between Kelvin-Helmholtz and counter-superflow instabilities in two-component Bose-Einstein condensates

Dynamical instabilities at the interface between two Bose-Einstein condensates that are moving relative to each other are investigated using mean-field and Bogoliubov analyses. Kelvin-Helmholtz instability is dominant when the interface thickness is much smaller than the wavelength of the unstable interface mode, whereas the counter-superflow instability becomes dominant in the opposite case. These instabilities emerge not only in an immiscible system but also in a miscible system where an interface is produced by external potential. Dynamics caused by these instabilities are numerically demonstrated in rotating trapped condensates.

Figure 1

Density distributions of the ground states (top panels), maximum values of the imaginary part in the Bogoliubov spectra (middle panels), and profiles $\delta n_1(x,y)$ and $\delta n_2(x,y)$ of the most unstable modes defined in Eq. (34) (bottom panels). The field gradient $G$ is 0 in (a)–(c) and $\mathrm{G̃}=m{\xi }^{3}G/\hslash {}^2={10}^{-7}$ in (d). In the middle panels, the solid lines indicate the boundaries between dynamically stable and unstable regions and the dashed lines are plots of Eq. (29) in (a) and Eq. (31) in (c). The color bars are normalized by $g{n}_0$. In the bottom panels, the relative velocity is ${\mathrm{Ṽ}}_{\mathrm{r}}=m\xi V_{\mathrm{r}}/\hslash =0.08$ and the field of view is $1600\xi \times 800\xi$. All the results are independent of $V_1+V_2$.

Figure 2

Time evolution of the two-component BEC for (a) $\mathit{\Delta }={10}^{-2}$ with $\mathrm{G̃}=0$, (b) $\mathit{\Delta }={10}^{-5}$ with $\mathrm{G̃}=0$, and (c) $\mathit{\Delta }=-{10}^{-5}$ with $\mathrm{G̃}={10}^{-7}$. The relative velocity is ${\mathrm{Ṽ}}_{\mathrm{r}}=0.1$ with $V_1+V_2=0$. The four upper rows are the density profiles $\left|{\psi }_1\right|{}^2$ and the bottom panels are the phase profiles arg ${\psi }_1$. The arrows in the bottom panel indicate an example of a vortex pair. The density is normalized by $n_0$. Time $t$ is normalized as $\mathrm{t̃}=g{n}_{0}t/\hslash$. The field of view is $1000\xi \times 750\xi$.

Figure 3

Time evolution of the density and phase profiles. In the initial state, component 2 rotates with vorticity $n_2=18$ and component 1 is at rest. The total number of ${}^{23}\mathrm{Na}$ atoms is $N=1.63\times {10}^6$ with an equal population in each component. The field of view is $93\times 93$ $\mu \mathrm{m}$. The unit of density is $N/a_{\perp }^2$ with $a_{\perp }=[\hslash /(m{\omega }_{\perp })]{}^{1/2}$.

Figure 4

Time evolution of the density profile of component 2 for $n_2=12$. Other parameters are the same as those in Fig. 3.

Figure 5

Imaginary part of Bogoliubov frequency $\mathrm{\omega ̃}=\omega /{\omega }_{\perp }$ for $n_1=0$ and $n_2=12$ (circles), $n_2=15$ (squares), and $n_2=18$ (triangles). The excitation mode has $L$-fold symmetry. The total number of atoms is $N=1.63\times {10}^6$. The conditions for triangles and circles are the same as those in Figs. 3 and 4, respectively. The solid arrows indicate the modes relevant to the patterns in Figs. 3 and 4. The inset shows a snapshot of the density profile of component 2 obtained by time evolution for $n_2=15$.

Figure 6

Time evolution of the density profiles for counter-rotating miscible condensates. The vorticities of the initial state are $n_1=-20$ and $n_2=30$. The total number of ${}^{87}\mathrm{Rb}$ atoms is $N=3.19\times {10}^6$ with an equal population in each component. The field of view is $84\times 84$ $\mu \mathrm{m}$.

Figure 7

(a) Time evolution of the cross-section densities $|{\psi }_1(x,y,0)|{}^2$ (left panels) and $|{\psi }_2(x,y,0)|{}^2$ (right panels) for $n_1=0$ and $n_2=7$. The field of view is $20\times 20$ $\mu \mathrm{m}$. The total number of ${}^{87}\mathrm{Rb}$ atoms is $N=1.75\times {10}^5$ with an equal population in each component. The bottom panels show the phase profiles at $t=84$ ms. (b) Isodensity surfaces of component 1 (inside) and component 2 (outside) at $t=84$ ms. For clarity, only the surface of component 2 is transparent.

Figure 8

(a) Column densities $\int \mathit{dz}\left|{\psi }_j\right|{}^2$ (left panels, $40\times 40$ $\mu \mathrm{m}$) and $\int \mathit{dy}\left|{\psi }_j\right|{}^2$ (right panels, $40\times 132$ $\mu \mathrm{m}$) of the initial state for $n_1=2$ and $n_2=-2$. The white arrow indicates the direction of the Stern-Gerlach force produced by the field gradient of $\mathit{dB}/\mathit{dz}=-17.3$ mG/cm. (b) Time evolution of the isodensity surfaces of components 1 (lower) and 2 (upper). The solid arrows at $t=0$ ms indicate the directions of the rotation. The phase profile of component 2 at $t=298$ ms is shown in the rightmost image. The total number of ${}^{23}\mathrm{Na}$ atoms is $N=6.6\times {10}^5$ with an equal population in each component.

Figure 9

(a) Column densities $\int \mathit{dz}\left|{\psi }_j\right|{}^2$ (left panels, $27\times 27$ $\mu \mathrm{m}$) and $\int \mathit{dy}\left|{\psi }_j\right|{}^2$ (right panels, $27\times 202$ $\mu \mathrm{m}$) of the initial state for $n_1=3$ and $n_2=-3$. The white arrow shows the direction of the Stern-Gerlach force produced by a field gradient of $\mathit{dB}/\mathit{dz}=0.34$ mG/cm. (b) Time evolution of the isodensity surface of component 1. At $t=0$, component 2 is also shown as the transparent surface. The total number of ${}^{87}\mathrm{Rb}$ atoms is $N=1.3\times {10}^6$ with an equal population in each component.