Longitudinal character of atom-chip-based rf-dressed potentials

We experimentally investigate the properties of radio-frequency-dressed potentials for Bose-Einstein condensates on atom chips. The three-dimensional potential forms a connected pair of parallel waveguides. We show that rf-dressed potentials are robust against the effect of small magnetic-field variations on the trap potential. Long-lived dipole oscillations of condensates induced in the rf-dressed potentials can be tuned to a remarkably low damping rate. We study a beam splitter for Bose-Einstein condensates and show that a propagating condensate can be dynamically split in two vertically separated parts and guided along two paths. The effect of gravity on the potential can be tuned and compensated for using a rf-field gradient.

Figure 1

Resonant absorption images of ultracold ${}^{87}\text{R}\text{b}$ atom clouds trapped in a rf-dressed potential. The final temperature of the cloud is varied by varying the final trap depth during evaporative cooling before transferring the atoms to the rf-dressed potential. The final trap depths are (a) $48\mu \text{K}$, (b) $29\mu \text{K}$, (c) $9.6\mu \text{K}$, and (d) $2.4\mu \text{K}$. The observed elliptical distribution of atoms results from the longitudinal variation of the character of the dressed potential. Near the center, $x=0$, the dressed potential forms two parallel waveguides which merge toward the edges of the image at $\left|x\right|\approx 250\mu \text{m}$.

Figure 2

(Color online) Schematic of the atom chip used to produce rf-dressed potentials. The central $Z$-shaped wire carries a dc current and is used together with an external bias field along $y$ to produce an Ioffe-Pritchard magnetic microtrap. Current through a macroscopic copper wire beneath the chip surface produces a dip in the longitudinal magnetic field (along $x$) for additional confinement. Positioned next to the $Z$-shaped wire are two wires which carry rf currents. The amplitudes of the rf currents are controlled to tune the linear polarization angle in the transverse $y\text{−}z$ plane, producing either horizontally split (along $y$) or vertically split (along $z$) rf-dressed potentials. Potential-energy cross-sections for vertical splitting are depicted on the back planes of the image. A sketch of the trapped atom cloud is shown in red.

Figure 3

The suppressed effect of small magnetic field variations as a function of rf-dressing frequency $\omega$ measured using rf spectroscopy of dressed Bose-Einstein condensates. Measurements have been performed for both horizontally split (a) and vertically split (b) potentials. The inset of (a) shows typical rf spectra for two dressing frequencies: $\omega =2\pi \times 3.5\text{MHz}$ (triangles) and $\omega =2\pi \times 1.3\text{MHz}$ (squares) for two values of $B_I$ differing by 0.186 G. The measured suppression ratio $dU/\mu d{B}_I$ is in excellent agreement with the prediction of Eq. (7) (solid lines) and the approximation of Eq. (8) (dashed lines) in the limit of large dressing frequency. In (b) we also consider the finite rf-field gradient which accounts for the small difference between the two vertically split traps (solid and open circles).

Figure 4

Measured longitudinal dipole oscillation frequencies $(●)$ of Bose-Einstein condensates in radio-frequency-dressed potentials. The data is obtained for horizontal splitting and $b_{\text{rf}}=1.7\text{G}$. The dashed line is the calculated decrease in potential curvature at the trap minimum $\left({\omega }_x^{\prime }/{\omega }_x\approx \sqrt{dU/\mu d{B}_I}\right)$ based on the parameters obtained from the suppression measurements of Sec. 4A. The solid line includes corrections due to anharmonicity of the longitudinal potential which slightly increases the BEC oscillation frequency below resonance. The inset shows the two transverse trapping frequencies (solid line, ${\omega }_y/2\pi$; dashed line, ${\omega }_z/2\pi$) calculated as a function of rf-dressing frequency $\omega$.

Figure 5

(Color online) Damping of longitudinal dipole oscillations in the rf-dressed potentials. The data is obtained for horizontal splitting. The center-of-mass position of a BEC as a function of time is recorded after 14 ms expansion for (a) the bare magnetic trap and for (b) a rf-dressed potential with dressing frequency $2\pi \times 2.02\text{MHz}$. The inset of (a) shows a zoomed-in section of the oscillation in the bare magnetic trap. In (c) we show the measured $e^{-1}$-damping time for various traps as a function of the rf-dressing frequency. The two data sets (circles and triangles) correspond to measurements obtained on two different days.

Figure 6

(Color online) Effect of potential gradients on the time-dependent splitting of rf-dressed Bose-Einstein condensates. A BEC is first prepared in a single well and then the dressing radio frequency is linearly increased by 0.25 MHz to split the BEC in the (vertical) $z$ direction. The rf current is varied to change the rf-field strength and gradient thereby tuning the potential symmetry for fixed rf polarization. The rf-field gradient in the $z$ direction is determined by the wire geometry and is calibrated to approximately $0.25\text{G}/\text{mA}\text{cm}$. Solid symbols correspond to the population in the lower well, open symbols to the upper well. We observe equal splitting for a rf current of 73 mA $\left(b_{\text{rf}}^{\prime }\approx 18\text{G}/\text{cm}\right)$ indicating that the effect of gravity can be compensated. Data is provided for three rf-ramp durations $t_s$ of 250 ms (triangles), 75 ms (circles), and 2.5 ms (squares). The complex behavior observed in the 2.5 ms data is most likely due to collective excitations induced during rapid splitting.

Figure 7

A spatial beam splitter for a propagating Bose-Einstein condensate. In (a) a single BEC is prepared near the edge of the potential by exciting a large-amplitude longitudinal oscillation. At the turning point the parameters of the rf-dressed potential are switched to create a pair of waveguides. The BEC propagates at a velocity of 20 mm/s and is split at a position of $x=-160\mu \text{m}$ (b). Two parts of the initial BEC propagate in two vertically separated waveguides. About 2 ms after splitting the two BECs are destroyed (see text for details). The two clouds continue moving through the waveguides (c)–(e) and recombine after 16 ms at a position of $x=160\mu \text{m}$ (f).