Topological formation of a multiply charged vortex in the Rb Bose-Einstein condensate: Effectiveness of the gravity compensation

In a Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ $(F=2,m_F=2)$ atoms we have topologically created a quantized vortex with a charge of 4 by reversing the magnetic field of the trap. Experimental conditions of reversal time and initial magnetic field strength for the successful vortex creation were restricted within narrower ranges, compared to those in the case of the ${}^{23}\mathrm{Na}$ condensate. The experimental difficulty was explained in terms of a non-negligible gravitational sag arising from its large atomic mass. We have successfully stabilized the vortex formation by compensating gravity with a blue-detuned laser beam.

Figure 1

(Color online) Experimental setup. The magnetic trap consists of 12 coils placed in the cloverleaf configuration. The magnetic field at the trap center is directed along the $z$ direction, and its strength is about $0.4\mathrm{G}$, which is reversed by applying an additional axial magnetic field with an external set of Helmholtz coils for the topological vortex formation. The gravity is in the $-x$ direction.

Figure 2

(Color online) Schematic diagram of the topological vortex formation. (a) The magnetic field direction in the $x\text{−}y$ plane, which remains constant. (b) The axial magnetic field is gradually reversed by applying an additional axial magnetic field. Each arrow represents the magnetic field direction, and the direction of each atomic spin follows the change of the local field direction adiabatically. The needles on the small disks indicate the atomic phases. Initially the atomic phase is identical all over the BEC. During the axial field reversal, because the direction of the rotation of the atomic spin is different depending on the spatial position of the atom, the different Berry phase is imprinted to the atom depending on its position, which results in the quantized vortex.

Figure 3

(Color online) Typical time sequence of the axial magnetic field reversal. The axial magnetic field at the trap center $B_0$ was reversed typically from 0.4 to $-0.2\mathrm{G}$ in a time of $2.2\mathrm{ms}$ by applying an additional axial magnetic field with an external set of Helmholtz coils. A created vortex was observed by the absorption imaging method after releasing the BEC from the trap. The time-of-flight was typically $13\mathrm{ms}$.

Figure 4

(Color online) Images of the BEC with a quantized vortex observed from (a) the axial direction (the $z$ direction) and (b) the radial direction (the $y$ direction). In (a) a created vortex is seen as a hole in the atom density, while in (b) a vortex line is seen as a linear depletion along the horizontal axis (the $z$ axis). The resolution of the image is about $16\mathrm{\mu }\mathrm{m}$. These images were observed after a time-of-flight of $13\mathrm{ms}$ by the absorption imaging.

Figure 5

(Color online) Gravitational sag $d$ (broken line) and BEC radius (solid line) as a function of the magnetic field strength at the trap center $B_0$. The calculation was made for Rb atoms (atom number of $3\times {10}^5$) by using the radial magnetic field gradient of $180\mathrm{G}∕\mathrm{cm}$. The BEC radius was calculated under the Thomas-Fermi approximation.

Figure 6

(Color online) Typical center-of-mass motion during the field reversal. (a) The axial magnetic field is changed from $B_0=0.4$ to $-0.2\mathrm{G}$ in a time of $2.2\mathrm{ms}$. (b) The center-of-mass motion and the potential minimum are plotted as a function of time. The center of mass moves up and down around the potential minimum during the field reversal.

Figure 7

(Color online) The distance $d_{CM}$ between the CM and the symmetric axis at the moment $B_0=0$ as a function of the field reversal time $T_{rev}$. The distance $d_{CM}$ was calculated in the case where the axial field was changed from $B_0=0.4$ to $-0.2\mathrm{G}$. The solid and dotted lines are for the Rb and Na BECs, respectively.

Figure 8

(Color online) Compensation of the gravitational field by using a blue-detuned laser beam ($532\mathrm{nm}$, $0.75\mathrm{W}$). The beam is softly focused $54\mathrm{\mu }\mathrm{m}$ below the BEC, whose waist size is about $53\mathrm{\mu }\mathrm{m}$. The vertical potential gradient generated by the laser beam is set nearly $-mg$ at the position of the BEC, so as to compensate the gravitational field.

Figure 9

(Color online) Probability distribution for the vertical displacement of the vortex center from the BEC center. (a) and (b) were obtained, respectively, with and without the gravity compensation. The axial magnetic field was reversed from $B_0=0.4$ to $-0.4\mathrm{G}$ in a time of $3\mathrm{ms}$. The displacement was measured in the direction of the gravity. The inset presented in (a) is the image of the BEC with a vortex, which was observed along the $z$ direction with the gravity compensation $(\mathrm{TOF}=25\mathrm{ms})$. The solid lines in (a) and (b) are Gaussian curves fitted to the distributions.