Trapping of Bose-Einstein condensates in a three-dimensional dark focus generated by conical refraction

We present an efficient three-dimensional dark-focus optical trapping potential for neutral atoms and Bose-Einstein condensates. This “optical bottle” is created by a single blue-detuned light field exploiting the phenomenon of conical refraction occurring in biaxial crystals. The conversion of a Gaussian input beam to the bottle beam has an efficiency of close to 100% and the optical setup requires the addition of the biaxial crystal and a circular polarizer only. Based on the conical-refraction theory, we derive the general form of the potential, the trapping frequencies, and the potential barrier heights. We present experiments on confining a ${}^{87}\mathrm{Rb}$ Bose-Einstein condensate in three dimensions. We determine the trap shape, the vibrational frequencies along the weak axis, as well as the lifetime of ultracold atoms in this type of potential.

Figure 1

(a) Schematic setup for a red-detuned attractive potential for neutral atoms provided by a focused Gaussian beam and density plots of relative intensity for the focal plane $(x,y,z=0)$ and along the propagation of the beam $(x=0,y,z)$. (b) Schematic setup for a blue-detuned repulsive potential provided by the conical-refraction bottle beam and the respective density plots of relative intensity.

Figure 2

Intensity distribution of the dark-focus bottle beam: (a) Two-dimensional density plot of the experimental intensity distribution in the $xz$ plane. (b) Corresponding calculated intensity distribution.

Figure 3

(a) Maximum (red) and minimum (blue) values of the relative intensity of the CR pattern at different planes along the beam propagation axis $z$. Calculated values are shown as dashed lines. (b) Relative intensity along the $x$ and $y$ axis in the focal plane at $z=0$. The dashed line shows the calculated values.

Figure 4

Experimental setup for the creation of a dark-focus potential based on conical refraction. The axis of the bottle beam is oriented along the direction of gravity. The inset shows the transverse intensity distributions in the focal plane. Two detection paths can be used to image the atom distribution via absorption imaging. The primary path is aligned with gravity. The secondary path is in the plane of the crossed dipole trap oriented perpendicular to gravity. (BS: beam splitter; PBS: polarizing beam splitter; PD: photodiode; CCD: detection camera; BC: biaxial crystal; L1–L5: achromatic lenses).

Figure 5

Axial oscillations of the center of the atom distribution in the bottle-beam potential modified by gravity. Oscillations are recorded for varying initial displacements of the atoms in the CDT relative to the equilibrium position in the modified potential. Every data set (colored dots) is fitted with a damped sinusoidal function (colored lines) to extract the oscillation frequency.

Figure 6

(a) Density plots for six different planes along $z$. The white scale bar at left has a length of $250\mu \mathrm{m}$ for reference. The selected planes correspond to the positions $-1218\mu \mathrm{m}, -696\mu \mathrm{m}, -174\mu \mathrm{m}, 348\mu \mathrm{m}, 870\mu \mathrm{m}$, and $1392\mu \mathrm{m}$. Atoms that are not captured in the bottle beam are pushed outside by the blue-detuned light and can be seen as an annular distribution. (b) Summed line density of the atoms trapped radially by the bottle beam. The step size of $174\mu \mathrm{m}$ along $z$ between recorded planes is indicated as the horizontal scale bar at the lower right. The white vertical scale bar at left has a length of $50\mu \mathrm{m}$. (c) Number of radially trapped atoms in the bottle beam as a function of axial position. From this we estimate the length of the bottle as $l_{\mathrm{BB}}=1570\left(174\right)\mu \mathrm{m}$, taking the background of $500\mathrm{atoms}$ into account.

Figure 7

Atom number as a function of holding time in the bottle-beam potential showing an exponential decay for trapping times larger than $100\mathrm{m}\mathrm{s}$. For trapping times smaller than $100\mathrm{m}\mathrm{s}$, a deviation from the exponential behavior is visible. Therefore, the exponential fit is carried out for $t\ge 100\mathrm{m}\mathrm{s}$. From the fit (solid red line), a lifetime of $\tau =205\left(3\right)\mathrm{m}\mathrm{s}$ is extracted. The inset shows the density plot of the atom distribution for $t=100\mathrm{m}\mathrm{s}$ after $t_{\mathrm{TOF}}=2\mathrm{m}\mathrm{s}$.