Vortex generation in stirred binary Bose-Einstein condensates

Binary Bose-Einstein condensates of two distinct atomic species are studied, with interesting effects being verified due to the imbalance of atomic masses. We study the formation of vortices and associated turbulence flows created by stirring two mass-imbalanced coupled Bose-Einstein condensates in the miscible regime. We consider the mixture in the laboratory frame, confined in a pancake-like harmonic trap slightly perturbed elliptically by a time-dependent periodic potential, which introduces turbulent dynamics with final stable patterns of vortices, induced by the periodicity of the perturbation. Derived rotational frequencies are shown to be associated with the observed vortex patterns in the asymptotic regime, with a correspondence between the stirring procedure and a direct time-independent approach being indicated. It was also confirmed that a larger pattern of vortices is supported by the higher mass condensed species at the same frequency. For the experimentally accessible mixtures ${}^{85}\mathrm{Rb}\text{−}{}^{133}\mathrm{Cs}$ and ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$, we numerically solve the equations of motion and analyze the system dynamics using a range of measures including energy spectra. In the transient turbulent regime, spectral analysis shows evidence for the characteristic $k^{-5/3}$ Kolmogorov classical law associated with turbulence, modified by the universal $k^{-3}$ vortex core scaling in the ultraviolet regime.

Figure 1

Set of panels indicating the time evolution of the densities for the condensed mixture ${}^{85}\mathrm{Rb}\text{−}{}^{133}\mathrm{Cs}$ in a miscible configuration with $\delta =0.5$. The upper panels are for ${}^{85}\mathrm{Rb}$ (component 1), with the lower panels for ${}^{133}\mathrm{Cs}$ (component 2). By starting with a ground-state solution, the system is evolved in real-time with ${\mathrm{\Omega }}_E=1.25$ and $\epsilon =0.025$. The density levels are indicated at the top of the panels. Within given units, all quantities are dimensionless.

Figure 2

For each component of the mixture, ${}^{85}\mathrm{Rb}$ ($i=1$, solid blue lines) and ${}^{133}\mathrm{Cs}$ ($i=2$, dashed red lines), the time evolutions are presented (in the laboratory frame) for (a) the kinetic energies, $E_i^K\equiv E_i^K\left(t\right)$, and (b) their corresponding compressible ($E_i^c$, indicated with bullets) and incompressible ($E_i^{nc}$, indicated with triangles) parts. With the initial configuration being ellipsoidal, the evolution is shown until stable vortex patterns are verified. Within given units, all quantities are dimensionless.

Figure 3

For the ${}^{85}\mathrm{Rb}$ ($i=1$) and ${}^{133}\mathrm{Cs}$ ($i=2$) mixture, the respective time evolutions of the current densities [(a), given by (8)] and torques [(b), given by (12)] are shown until stable vortex patterns are verified. Within given units, all quantities are dimensionless.

Figure 4

Time evolutions of the rotational frequencies [Eq. (13)]: ${\mathrm{\Omega }}_1\left(t\right)$, for ${}^{85}\mathrm{Rb}$ (solid red line); ${\mathrm{\Omega }}_2\left(t\right)$, for ${}^{133}\mathrm{Cs}$ (dot-dashed green line), and with the corresponding averaging, ${\mathrm{\Omega }}_{av}\left(t\right)\equiv [{\mathrm{\Omega }}_1\left(t\right)+{\mathrm{\Omega }}_2\left(t\right)]/2$ (dashed blue line). The vertical lines are approximately separating three time intervals: (I) shape deformations; (II) turbulent regime, with starting vortex nucleations; and (III) when vortex patterns are settled. Within given units, all quantities are dimensionless.

Figure 5

Vortex-pattern solutions, obtained at $t=27000$ with the stirring potential [(${\mathrm{a}}_i$) = (${\mathrm{d}}_i$) of Fig. 1], are compared with ground-state solutions of Eq. (3), using (6) with ${\mathrm{\Omega }}_0=0.63$ [shown in panels (${\mathrm{b}}_i$)]. The density levels are indicated at the top of each panel. Within given units, the plotted quantities are dimensionless.

Figure 6

Set of panels indicating the time evolution of the densities for the mixture ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$. The upper panels are for the ${}^{85}\mathrm{Rb}$ (component 1), with the lower panels for ${}^{87}\mathrm{Rb}$ (component 2). As in the case of Fig. 1, by starting with a ground-state solution, the system is evolved in real time with ${\mathrm{\Omega }}_E=1.25$ and $\epsilon =0.025$. The density levels are indicated at the top of the panels. Within given units, the plotted quantities are dimensionless.

Figure 7

For both components, ${}^{85}\mathrm{Rb}$ ($i=1$) and ${}^{87}\mathrm{Rb}$ ($i=2$), (a) shows the almost complete overlap of the total kinetic energies (solid blue for $E_1^K$, with dashed red for $E_2^K$). The compressible (bullets) and incompressible (triangles) parts of $E_i^K$ are shown in (b). Within given units, all quantities are dimensionless.

Figure 8

The current densities (a) with the torques (b), respectively given by Eqs. (8) and (12) for both components, are shown for the ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ mixture, during the dynamical process that occurs until the vortex generation. Within given units, all quantities are dimensionless.

Figure 9

Time evolution of ${\mathrm{\Omega }}_i\left(t\right)$ [Eq. (13)] of both components, ${}^{85}\mathrm{Rb}$ (solid red curve) and ${}^{87}\mathrm{Rb}$ (dot-dashed green curve), with the corresponding averaged result, ${\mathrm{\Omega }}_c\left(t\right)=[{\mathrm{\Omega }}_1\left(t\right)+{\mathrm{\Omega }}_2\left(t\right)]/2$ (dashed blue curve). Vertical lines are indicating three time intervals in the evolution: (I) shape deformations; (II) transition period with nucleation of vortices at the surface; and (III) with vortex lattices being formed. Within given units, all quantities are dimensionless.

Figure 10

Vortex-pattern results, at $t=35000$ with the stirring potential [(${\mathrm{a}}_i$) = (${\mathrm{d}}_i$) of Fig. 6], are compared with ground-state solutions of Eq. (3), using (6) with ${\mathrm{\Omega }}_0=0.6$ [panels (${\mathrm{b}}_i$)], for the coupled system ${}^{85}\mathrm{Rb}$ (species 1) and ${}^{87}\mathrm{Rb}$ (species 2) in miscible configuration $a_{12}/a_{11}=0.5$. The density levels are indicated at the top of each panel. Within given units, all quantities are dimensionless.

Figure 11

Incompressible kinetic energy spectra, $E^{nc}\equiv E^{nc}(k,t)$, for the ${}^{85}\mathrm{Rb}\text{−}{}^{133}\mathrm{Cs}$ mixture, obtained by averaging over 50 samples in the turbulent time interval (II) indicated in Fig. 4, which approximately agrees with the classical Kolmogorov $k^{-5/3}$ power-law behavior in the shown interval $0.1<k\xi <1$ (red dashed lines), being modified to the $k^{-3}$ in the ultraviolet region. Within given units, all quantities are dimensionless.

Figure 12

Incompressible kinetic energy spectra, $E^{nc}\equiv E^{nc}(k,t)$, for the ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ mixture, by averaging over 50 samples in the turbulent interval (II) indicated in Fig. 9. With behavior similar to the one observed for the stronger mass-imbalanced case in Fig. 11, the change in the power-law behavior is verified for larger values of $k\xi$. Within given units, all quantities are dimensionless.