Imprinting Vortices in a Bose-Einstein Condensate using Topological Phases

Vortices were imprinted in a Bose-Einstein condensate using topological phases. Sodium condensates held in a Ioffe-Pritchard magnetic trap were transformed from a nonrotating state to one with quantized circulation by adiabatically inverting the magnetic bias field along the trap axis. Using surface wave spectroscopy, the axial angular momentum per particle of the vortex states was found to be consistent with $2\hslash$ or $4\hslash$, depending on the hyperfine state of the condensate.

Figure 1

Geometry of the rotating magnetic field for imprinting topological phases. (a) The unit vectors $\widehat{b}(\varphi )$ point in the direction of the two-dimensional quadrupole field providing the radial confinement of a Ioffe-Pritchard magnetic trap. The atomic angular momenta rotate about the unit vectors $\widehat{n}(\varphi )$ as the axial bias field, $B_z$, is ramped from positive to negative values. (b) For an atom in state $|F,m_F⟩$, its atomic angular momentum, $\overrightarrow{F}$, traverses a path on a sphere of radius $|m_F|\hslash$ as it adiabatically follows its local magnetic field. The primed coordinate system is centered on the atomic position and has axes parallel to those of the unprimed coordinate system in (a). For an atomic position described by the azimuthal angle $\varphi$, $\overrightarrow{F}$ rotates in a half-plane defined by ${\varphi }^{\prime }=-\varphi$ for $m_F>0$ and ${\varphi }^{\prime }=-\varphi +\pi$ for $m_F<0$ as $B_z$ is inverted. After inverting $B_z$, the relative topological phase acquired between atoms located at positions 1 and 2 in (a) is proportional to the solid angle subtended by the shaded surface, bounded by the contour marked with arrowheads (see text).

Figure 2

Observation of vortices formed by imprinting topological phases. Axial absorption images of condensates in the $|1,-1⟩$ state after 18 ms of ballistic expansion (a) prior to inverting $B_z$, after inverting $B_z$, and holding the trapped condensate for (b) 5 ms and (c) 20 ms, and (d) after inverting $B_z$ and then returning it to its original direction. Axial absorption images of condensates in the $|2,+2⟩$ state after 7 ms of ballistic expansion (e) prior to inverting $B_z$, after inverting $B_z$, and holding the trapped condensate for (f) 0 ms and (g) 5 ms, and (h) after inverting $B_z$ and then returning it to its original direction. The field of view is (a)–(d) $570\mu \mathrm{m}\times 570\mu \mathrm{m}$ and (e)–(h) $285\mu \mathrm{m}\times 285\mu \mathrm{m}$.

Figure 3

Surface wave spectroscopy. Axial absorption images after 18 ms of ballistic expansion of $|1,-1⟩$ condensates undergoing a quadrupole oscillation (a)–(d) in the presence of a vortex and (e)–(h) in the absence of a vortex. Successive images were taken during successive half periods of the quadrupole oscillation such that the short and long axes of the elliptical cross section were exchanged. Images (a)–(d) show counterclockwise (positive) precession of the quadrupole axes, while images (e)–(h) show no precession. (i)–(l) Axial absorption images after 7 ms of ballistic expansion of $|2,+2⟩$ condensates undergoing a quadrupole oscillation in the presence of a vortex. The images were taken during a single half period of the quadrupole oscillation. Images (i)–(l) show clockwise (negative) precession of the quadrupole axes. The field of view is (a)–(h) $570\mu \mathrm{m}\times 570\mu \mathrm{m}$ and (i)–(l) $285\mu \mathrm{m}\times 285\mu \mathrm{m}$. (m) Precession angle vs time in the presence of a vortex for $|1,-1⟩$ condensates measured after a delay of 0 ms (open circles), 5 ms (open squares), and 20 ms (open triangles) from the completion of the inversion of the axial bias field, in the absence of a vortex for $|1,-1⟩$ (open diamonds) and $|2,+2⟩$ (filled diamonds) condensates, and in the presence of a vortex for $|2,+2⟩$ condensates measured immediately upon the completion of the inversion of the axial bias field (filled circles).