Dynamics of vortex formation in merging Bose-Einstein condensate fragments

We study the formation of vortices in a Bose-Einstein condensate (BEC) that has been prepared by allowing isolated and independent condensed fragments to merge together. We focus on the experimental setup of Scherer et al. [Phys. Rev. Lett. 98, 110402 (2007)], where three BECs are created in a magnetic trap that is segmented into three regions by a repulsive optical potential; the BECs merge together as the optical potential is removed. First, we study the two-dimensional case; in particular, we examine the effects of the relative phases of the different fragments and the removal rate of the optical potential on the vortex formation. We find that many vortices are created by instant removal of the optical potential regardless of relative phases, and that fewer vortices are created if the intensity of the optical potential is gradually ramped down and the condensed fragments gradually merge. In all cases, self-annihilation of vortices of opposite charge is observed. We also find that for sufficiently long barrier ramp times, the initial relative phases between the fragments leave a clear imprint on the resulting topological configuration. Finally, we study the three-dimensional system and the formation of vortex lines and vortex rings due to the merger of the BEC fragments; our results illustrate how the relevant vorticity is manifested for appropriate phase differences, as well as how it may be masked by the planar projections observed experimentally.

Figure 1

(Color online) Top left: intensity profile of the optical potential responsible for segmenting the potential well into three local minima. Top center: Thomas-Fermi approximation used as an initial condition for our relaxation method to obtain the ground state of the system. Top right: regions A, B, and C with respective phases ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$. The bottom row of panels depicts the evolution of the phase during our relaxation (imaginary time relaxation) toward the initial steady state with different phases for the fragments. This example shows the phase for the case ${\varphi }_k=2\pi k/3$ at the times indicated. In all panels, the field of view is approximately $70\mu \text{m}$ per side. The axis numbers indicate $x$ and $y$ coordinates relative to the center of the unsegmented harmonic trap.

Figure 2

(Color online) Evolution of the 2D condensate density (respective top series of panels) and vorticity (respective bottom series of panels) for three different ramp-down times of the optical potential barriers. From top to bottom, the three sequences correspond to $t_b=0\text{ms}$ (first and second row), $t_b=25\text{ms}$ (third and fourth row), and $t_b=50\text{ms}$ (fifth and sixth row). The times are indicated in the panels in ms and the field of view is approximately $70\mu \text{m}$ per side. For the initial conditions in all cases, the different condensed fragments have relative phases of $2\pi /3$, namely, ${\varphi }_k=2\pi k/3$ $\left(k\in \left\{1,2,3\right\}\right)$.

Figure 3

(Color online) Evolution of the vortex structures in the 2D condensate density for $t_b=25\text{ms}$ and ${\varphi }_k=2\pi k/3$ (cf. middle rows in Fig. 2).

Figure 4

(Color online) Vorticity indicators for the cases presented in Fig. 2 (namely, $t_b=0\text{ms}$, $t_b=25\text{ms}$, and $t_b=50\text{ms}$, from top to bottom) with ${\varphi }_k=2\pi k/3$. The left panels correspond to the total angular momentum [Eq. (6)] normalized by $\hslash$, while the middle and right panels correspond to the total fluid velocity [Eq. (8)] for the whole cloud (middle) and the central portion (see text) of the cloud (right).

Figure 5

(Color online) Same as in Fig. 2 for three different relative phases and a ramp down of $t_b=25\text{ms}$. The top two, middle two and bottom two series correspond, respectively, to (a) ${\varphi }_k=0$, (b) ${\varphi }_1=0$, ${\varphi }_2=2\pi /3$, and ${\varphi }_3=4\pi /3$, and (c) ${\varphi }_1=0$, ${\varphi }_2=\pi /3$, and ${\varphi }_3=2\pi /3$.

Figure 6

(Color online) Vorticity indicators for the cases depicted in Fig. 5 (the different panels are presented in the same manner as in Fig. 4).

Figure 7

Phase diagram for vortex formation at the trap center region as a function of the relative phases of the merging fragments (where ${\varphi }_1$ is assigned the value of 0). Top: Ramp down time of $t_b=0\text{ms}$ (i.e., instantaneous removal) of the barrier. The different shades of gray indicate the amount of vorticity contained in the central region after 100 ms. White, light gray, gray, black correspond, respectively, to 0, 1, 2, and 3 vortices. Middle: Same as above for a ramp down time of $t_b=50\text{ms}$. This case only produces no vortices (white) or one vortex (gray). Bottom: difference between top and middle diagrams. This corresponds to the vortices formed by the collision of the fragments and not by the intrinsic vorticity of the initial configuration (which depends on the relative phases of the different fragments).

Figure 8

(Color online) Vortex formation by the merging of three-dimensional BECs. Top row: contour surfaces of constant atomic density. Middle row: contour surfaces of the corresponding absolute value of the vorticity. Bottom row: $z$ projection of the density distribution (i.e., column density along $z$) as it would be observed in the laboratory. The snapshots are taken at the indicated times (in ms) for an initial phase distribution corresponding to ${\varphi }_k=0$ and a ramping down time of 25 ms. For the contour-surface images, the axis on the left side of each plot represents the vertical $\left(z\right)$ direction, expressed in units of microns.

Figure 9

(Color online) Same as in Fig. 8 for an initial phase distribution corresponding to ${\varphi }_k=2\pi k/3$.

Figure 10

(Color online) Same as in Fig. 8 for an initial phase distribution corresponding to ${\varphi }_k=k\pi /3$.

Figure 11

(a) In situ phase-contrast image of three BECs trapped in a triple-well potential. (b)–(d) Absorption images of BECs after 56 ms of ballistic expansion. Each BEC was created by merging three BECs, as described in the text.