Figure 1

(Color online). (a) The time dependence of the $B$ fields. In the first $30\mathrm{ms}$, $B_z$ decreases to the opposite of the initial value. The gradient increases from 0 to the maximum in $15\mathrm{ms}$, then decreases to zero. Next, the $B$ field is rotated back, while a uniform transverse bias in the $x$-axis direction is pulsed on for the latter $10\mathrm{ms}$, keeping the $B_z^2+{\left(B_x^r\right)}^2$ a constant. (b) The time-dependent fractional population $n_{M_F}=N_{M_F}∕N$ in the Zeeman state.Reuse & Permissions

Figure 2

The temporal development of the condensate phase structures in a hexapole field ${\vec{B}}_h=B_h^{\prime }[(x^2-y^2)\widehat{x}-2xy\widehat{y}]$ plus a transverse bias field $B_x^r\widehat{x}$. The upper row shows the $M_F=1$ component at the end of the first flips, or in the middle of each cycle at 30, 70, 110, 150, and $190\mathrm{ms}$, respectively; the lower row is for the $M_F=-1$ state at $10\mathrm{ms}$ later with respect to the first row or at the end of each cycle at 40, 80, 120, 160, and $200\mathrm{ms}$. The temporal dependence of the $B$ fields and other parameters are as in Fig. 1 of the supplementary material. The maxima for $B_h^{\prime }$, $B_z$, and $B_x^r$ are $1250\mathrm{G}∕{\mathrm{cm}}^2$, 0.1, and $0.1\mathrm{G}$, respectively. White (black) color denotes phase $-\pi$ $\left(\pi \right)$.Reuse & Permissions

Figure 3

The condensate phase distribution for the $M_F=1$ state after flipping the $z$ bias in an IPT. (a) The ideal case of an IPT with perfect alignment with the optical trap and the plug. The flip times are 30, 60, 90, and $120\mathrm{ms}$ from the left to the right, accompanied by an increasing level of adiabaticity. (b) The same as in (a) except for a misaligned optical plug $V_p^{2\mathrm{D}}=U\exp \{-[{(x-x_0)}^2+y^2]∕2{\rho }_0^2\}$, $U∕\hslash =2\times {10}^5\mathrm{Hz}$, and ${\rho }_0=5\mu \mathrm{m}$. From the left to the right in the upper row, the results are compared for $x_0∕{\rho }_0=0.05$, 0.1, 0.2, and 0.3, respectively, at a bias flip time of $30\mathrm{ms}$. In the lower row $x_0=0.2\rho$, where results from different flip times of 60, 90, 120, and $150\mathrm{ms}$ are compared from the left to the right. (c) The same as in (a) except for a non-axial-symmetric optical trap $V_o^{2\mathrm{D}}=M{\omega }_{\perp }^2[{(1+\epsilon )}^2{x}^2+{(1-\epsilon )}^2{y}^2]∕2$ parametrized by $\epsilon$. The upper row is for $\epsilon =0.05$, and $\epsilon =0.1$ in the lower row. The $z$-bias flipping times used are 60, 90, 120, and $210\mathrm{ms}$ from left to right, respectively.Reuse & Permissions

Figure 4

(Color online). The motional angular momentum gained with our protocol in an asymmetric optical trap and an IPT. (a) The temporal development of $L_z$ during the flip of the $z$-bias field at an asymmetry parameter $\epsilon =0.05$. The flip time $T$ is 30, 60, 90, 120, and $150\mathrm{ms}$, respectively, and is shown by offset lines from the lower to the higher. The offset shift for each curve is given by $L_z^T=0.25\hslash \times [(T∕T_m)-1]$, and $T_m=30\mathrm{ms}$. (b) The difference between angular momentum gained from our protocol in an asymmetric optical trap with $L_z^p=2\hslash$ predicted for the ideal case. The curves denote different asymmetry parameter $\epsilon$ from 0.05 to 0.15 as shown by the inset labels. (c) The same as in (b), but now for the $\epsilon$ dependence at the same flip time $T$. The five curves are labeled by their respective $T∕T_m$ values from 1 to 5.Reuse & Permissions

Figure 5

Different vortex pump protocols compared, from the top to the bottom. The parallelogram and hexagon denote, respectively, 2D quadrupole and hexapole $B$ fields, and the stacked pyramids denote a 3D quadrupole $B$ field. The initial $z$ bias is denoted by the solid black line arrow that is flipped to the opposite direction denoted by gray line arrow in the first step; the slanted solid black line arrow in the right figure of (c) refers to the transverse bias. (a) The original theoretical protocol of Ref. 25, which has been experimentally demonstrated [34, 35, 36], can pump only once. (b) The continuous pump protocol of Ref. 38. It works, but the condensate vorticity does not change monotonically; (c) and (d) are the optimal protocols we discuss. They are capable of continuous and monotonic changes to the vorticity of a condensate. They are the optimal protocols based on the manipulation of $B$ fields.Reuse & Permissions

Figure 6

(Color online). The continuously increasing vorticity of a condensate for the optimal protocols. (a) With only ${\vec{B}}_l^{\left(\mathrm{I}\right)}\left(\vec{r}\right)$ for a hexapole ($l=3$ the upper blue curve) and a quadrupole ($l=2$ the lower red curve). $T_1=40\mathrm{ms}$ is the time of flipping. In the first $30\mathrm{ms}$, ${\vec{B}}_l^{\left(\mathrm{I}\right)}\left(\vec{r}\right)$ is turned on to gain vorticity, while no vorticity is gained in the following $10\mathrm{ms}$ when a transverse bias along the $x$ axis is turned on to reset the internal states to be as they were initially. (b) With both $B$ fields ${\vec{B}}_l^{\left(\mathrm{I}\right)}\left(\vec{r}\right)$ ($l=3$ for the upper blue curve and $l=2$ for the lower red curve) and ${\vec{B}}^{\left(\mathrm{II}\right)}\left(\vec{r}\right)$ available and used for the first and second steps, respectively, of each cycle. $T_2=60\mathrm{ms}$ is the total time for each cycle. In the first $30\mathrm{ms}$ ${\vec{B}}_l^{\left(\mathrm{I}\right)}\left(\vec{r}\right)$ is present, and in the next $30\mathrm{ms}$ ${\vec{B}}^{\left(\mathrm{II}\right)}\left(\vec{r}\right)$ is turned on while ${\vec{B}}_l^{\left(\mathrm{I}\right)}\left(\vec{r}\right)$ is turned off. The vorticity increases at all times in this case. The dashed lines in the two figures are guides for the eye.Reuse & Permissions