Vortex dynamics in coherently coupled Bose-Einstein condensates

In classical hydrodynamics with uniform density, vortices move with the local fluid velocity. This description is rewritten in terms of forces arising from the interaction with other vortices. Two such positive straight vortices experience a repulsive interaction and precess in a positive (anticlockwise) sense around their common centroid. A similar picture applies to vortices in a two-component, two-dimensional uniform Bose-Einstein condensate (BEC) coherently coupled through rf Rabi fields. Unlike the classical case, however, the rf Rabi coupling induces an attractive interaction and two such vortices with positive signs now rotate in the negative (clockwise) sense. Pairs of counter-rotating vortices are instead found to translate with uniform velocity perpendicular to the line joining their cores. This picture is extended to a single vortex in a two-component trapped BEC. Although two uniform vortex-free components experience familiar Rabi oscillations of particle-number difference, such behavior is absent for a vortex in one component because of the nonuniform vortex phase. Instead the coherent Rabi coupling induces a periodic vorticity transfer between the two components.

Figure 1

Precession frequency of two corotating vortices, one per component. The lines indicate results obtained for different radii $R$ of the circular container (from top to bottom, $R/r_{12}=2,3.5,5$); solid lines are results with $\xi =r_{12}/10$, and dashed ones results with $\xi =r_{12}/40$. The dash-dotted line is the strong-coupling limit, Eq. (36). The diamonds are instead results for counter-rotating vortices: for large Rabi coupling vortex-antivortex pairs move uniformly, without precessing. (Right) The same data are plotted with different axes, to highlight the behavior at weak coupling. Here ${\omega }_{12}\equiv \hslash /M{r}_{12}^2$, and the dots are the result expected for a single vortex in a single-component BEC inside a cylinder, Eq. (38).

Figure 2

Translational velocity of two oppositely charged vortices, one in each component, in a circular container of radius $R$. The simulation result, shown with diamonds, is compared to the analytical formula Eq. (37), valid for strong Rabi coupling.

Figure 3

Contours of equal dimensionless energy $\tilde{E}$, for $N_1=N_2, \overline{\varphi }=0$, and $R/\xi =5$. From left to right, $M\mathrm{\Omega }{R}^2/\hslash =0,1,10$. For $\overline{\varphi }\ne 0$, the contours would be rotated in the plane by an angle $\overline{\varphi }$.

Figure 4

Transfer of vorticity in a harmonically trapped two-component BEC, in the presence of a Rabi frequency $\mathrm{\Omega }=2{\omega }_{\perp }$. In this simulation we used a relatively strong coupling, $gN=g_{12}N=40{\hslash }^2/M$, so that $R^2/l_{\mathrm{\Omega }}^2\approx 10$ and ${(R/d_{\perp })}^4\approx 50$. From left to right, the columns show, respectively, the particle densities $n_1$ and $n_2$ (arbitrary units, with brighter colors indicating higher densities), the phase $S_1$ of ${\psi }_1$, and the phase difference $S_1-S_2$ at given times: from top to bottom, $t/T=\{0,0.25,0.5,0.75,1\}$, with $T=2\pi /\mathrm{\Omega }$ the Rabi period. Markers indicate the trajectory of the vortex core, up to the time at which the screenshot is taken (circles and squares for vortex core in first and second components respectively). The color of the markers indicate at what (past) time the core was at that specific position. Top row: initial condition, with a vortex at the center of the first component. Second row: the system imaged after one quarter of a Rabi period ($t=T/4$). The vortex core follows the trajectory marked by the symbols: from black ($t=0$) to gray ($t=T/4$), it travels until the edge of the first component; just before $t=T/4$, while the first vortex is still inside the first cloud, a second vortex enters the second cloud, and a domain wall in the relative phase is clearly visible in the fourth column. Third row: after half a Rabi period ($t=T/2$), the first vortex has completely disappeared from the first component, while the new one gradually migrates to the center of the second component. Fourth row: at $t=3T/4$, two vortices are again visible, one in each component, with a domain wall in-between them. Bottom row: after a full Rabi period ($t=T$), the vortex leaves the second component, reappears inside the first, and returns back to its center. Continuous lines show the predicted trajectory given by Eqs. (46) and (47), for $\kappa =0.485{\omega }_{\perp }$. A video of the complete simulation, running over five full Rabi cycles, is available in the Supplemental Material [24].

Figure 5

Dynamics at weak coupling ($gN=g_{12}N=2{\hslash }^2/M$): From left to right, we show the particle densities (arbitrary units) and phases of first and second components of the BEC, after an evolution of half a Rabi period. Markers depict the numerical vortex trajectory, while continuous lines show the predicted trajectory as given by Eqs. (57) and (58). The Rabi frequency is set to $\mathrm{\Omega }=2{\omega }_{\perp }$, as in Fig. 4.

Figure 6

Dependence of $\kappa$ on the coupling $gN$. Dots represent the values of $\kappa$ that minimize the difference between the simulated trajectories, and the ones predicted by Eqs. (57) and (58) for the weak coupling regime ($gN<20{\hslash }^2/M$), or Eqs. (46) and (47) for the strong coupling regime ($gN>20{\hslash }^2/M$). For $gN<20{\hslash }^2/M$ we used $\mathrm{\Omega }=2{\omega }_{\perp }$, whereas for $gN>20{\hslash }^2/M$ we used $\mathrm{\Omega }=2{\omega }_{\perp }/5$. In all simulations we used $g_{12}=g$. We do not see variation of $\kappa$ for different values of $\mathrm{\Omega }$, and our results are in close agreement with Eq. (60), plotted here with $C=4$ as a continuous line.

Figure 7

The vortex energy $\tilde{E}$, given by Eq. (45), is evaluated at the vortex core as a function of time for the simulation shown in Fig. 4. The blue (dark gray) and green (light gray) lines depict, respectively, the vortex energy $\tilde{E}$ in components 1 and 2, evaluated at the solutions of Eqs. (46) and (47), using $\kappa =0.485{\omega }_{\perp }$. Dots represent $\tilde{E}$ computed using the position of the vortex given by the GPE simulation.

Figure 8

Pseudospin oscillation of Rabi-coupled BECs, obtained by the numerical solution of the coupled GP equations in a square box of area $L^2$. As initial conditions we take the ground state of the uncoupled BECs with $N_1=N_2=N/2, gN=40{\hslash }^2/M$, and $T=2\pi /\mathrm{\Omega }=\pi M{L}^2/\hslash$. (Top) Dynamics of the population imbalance $\eta$ with $g=g_{12}$, for various values of the initial phase difference $S\left(0\right)$. (Bottom) Evolution of the population imbalance $\eta$ at fixed phase difference $S\left(0\right)=\pi /6$, for different values of the interspecies interaction $g_{12}$; the pseudospin oscillations become perfectly harmonic in the SU(2)-invariant case $g_{12}=g$.

Figure 9

Evolution of population imbalance $\eta$ and relative phase $S$ for two Rabi-coupled BECs in a harmonic trap. The results of our simulations, shown with symbols, are compared with equations (62), shown as lines. As initial conditions we took the ground state of the uncoupled BECs with $N_1=N_2=N/2$ and $S\left(0\right)=\pi /3$. Here $gN=g_{12}N=40{\hslash }^2/M$ and $T=\pi /\omega$.

Figure 10

Pseudospin oscillations in the presence of both trapping and interactions. The dotted line displays the population imbalance $\eta$ for the simulation shown in Fig. 4: The transfer of vorticity we observed there happens in complete absence of population transfer. For comparison, the lines show the evolution obtained starting from the same initial conditions, but without vortices in either component: Solid green and dashed blue lines represent, respectively, the cases where the relative phase at $t=0$ is 0, and $\pi /2$.