Phase and micromotion of Bose-Einstein condensates in a time-averaged ring trap

Rapidly scanning magnetic and optical dipole traps have been widely utilized to form time-averaged potentials for ultracold quantum gas experiments. Here we theoretically and experimentally characterize the dynamic properties of Bose-Einstein condensates in ring-shaped potentials that are formed by scanning an optical dipole beam in a circular trajectory. We find that unidirectional scanning leads to a nontrivial phase profile of the condensate that can be approximated analytically using the concept of phase imprinting. While the phase profile is not accessible through in-trap imaging, time-of-flight expansion manifests clear density signatures of an in-trap phase step in the condensate, coincident with the instantaneous position of the scanning beam. The phase step remains significant even when scanning the beam at frequencies 2 orders of magnitude larger than the characteristic frequency of the trap. We map out the phase and density properties of the condensate in the scanning trap, both experimentally and using numerical simulations, and find excellent agreement. Furthermore, we demonstrate that bidirectional scanning flattens the phase profile, rendering the system more suitable for coherent matter-wave interferometry.

Figure 1

Schematic of the experimental setup. The ring potential is formed by combining a red-detuned laser sheet that confines atoms in the $z=0$ plane with a vertically propagating red-detuned beam that is rapidly scanned in the $xy$ plane. (Insets) Experimental absorption images of the ring condensate density following feedforward correction. The images are taken after 1 ms (I) and 20 ms (II) TOF expansion and rescaled to the peak density. The ring has radius $R=82\mu \mathrm{m}$, and the scale bar in (I) has $50\mu \mathrm{m}$ length.

Figure 2

Analytical solutions for the phase of the unidirectionally scanned ring condensate. (a) Snapshots at three equally spaced times of the azimuthal phase profile of the BEC in the scanned ring trap, at $\rho =R$. The phase step ${\delta }_{\varphi }$ advances with the scanning beam location. The black arrow indicates the direction of beam motion. (Inset) Schematic of the unidirectional scan ordering. Black dots represent the discrete scan locations around the ring, commencing at $\theta =0$ (open circle). The red arrow shows the scan order. (b) Density plot of the analytical phase solution [Eq. (8)] for scan frequency $f_s=0.5\mathrm{kHz}$. The phase is shown for $\left|\rho -R\right|<R_{\rho }$. (c) Density plot of the numerical phase solution, for the same parameters as (b) and as discussed in Sec. 4. (d) Particle trajectories which illustrate the amplitude of micromotion within the time-averaged trap for scan frequencies $f_s=0.5\mathrm{kHz}$ (blue dashed) and $f_s=6.25\mathrm{kHz}$ (red solid). The resulting micromotion is plotted over three scan cycles for initial radial positions $R_{\rho }$ (top) and $0.2R_{\rho }$ (bottom). The amplitude of the azimuthal and radial motion reduces at higher scan frequency.

Figure 3

Normalized angular density profiles $\chi \left(\theta \right)$ from GPE simulations for scan frequencies $f_s=6.25\mathrm{kHz}$ (triangle) and $f_s=0.5\mathrm{kHz}$ (circle) before time-of-flight expansion. Experimental values for the vertical trap frequency $f_z=140\mathrm{Hz}$, radius $R=82\mu \mathrm{m}$, and atom number $N_0=2\times {10}^6$ are used. The definition of the density step ${\delta }_{\chi }$ is indicated.

Figure 4

Consequences of the condensate phase profile within time-averaged optical ring potentials. (a) Mean experimental absorption images for scan frequencies $f_s=6.25\mathrm{kHz}$ (triangle) and $f_s=0.5\mathrm{kHz}$ (circle), taken after 1- and 20-ms time-of-flight expansion. Imaging noise at 1-ms expansion obscures the numerically predicted density step, and the images are experimentally indistinguishable. Following 20-ms expansion, the change in the condensate for different scan rates becomes apparent and agrees with the numerical simulation results. The trapping beam was unidirectionally scanned in a clockwise direction, and the scale bar has $50\text{−}\mu \mathrm{m}$ length. (b) Normalized angular density profiles $\chi \left(\theta \right)$ for the 20-ms TOF images in (a). The profile asymmetry motivates the definition of peak ${\delta }_{\alpha }$ and trough ${\delta }_{\beta }$ amplitudes. Experimental (solid) and numerical (dashed) profiles are overlaid. (c) A comparison of experimental density peaks (red circles) and troughs (blue triangles) with simulations (solid lines). The AOD access time limits the scan frequency to $f_s\le 6.25\mathrm{kHz}$ for our ring geometry, while for frequencies $f_s\le 0.5\mathrm{kHz}$ the atoms are not confined.

Figure 5

Numerical data for the phase profile of the raster-scanned ring condensate. (a) Snapshots of the central phase profile, for the raster-type scan, with $f_s=6.25\mathrm{kHz}$. The phase profiles at $\{t_1,t_2,t_3\}$ illustrate fractional times $\{0.25,0.5,0.75\}$ through the scan period showing the azimuthal expansion of a flat phase region. The phase profile is uniform at the end of each full scan period. (Inset) (I) Schematic of the idealized bidirectional time-averaged scan, with two beams counter-rotating. (II) The raster scan ordering approximates the bidirectional scan. Black dots represent the discrete points around the ring and open circles the starting locations. The red arrows show the scan orders. (b) Density plot for the $f_s=0.7\mathrm{kHz}$ phase profile after a complete period. The solution is radially truncated where the density becomes vanishing. Residual phase steps, with amplitude ${\delta }_0$ and located at $\theta =\{0,\pi \}$, result from the temporal offset between the effective scan periods. (c) Density plot as in (b) for $f_s=6.25\mathrm{kHz}$. The residual phase steps are negligible for our fastest experimental scan speeds.

Figure 6

Experimental density images for a bidirectionally scanned trap with frequency of $f_s=0.7\mathrm{kHz}$ following 20-ms TOF. (I–III) Images taken at fractional times $\{0.5,0.75,1\}$ through the raster scan period. White arrows indicate the instantaneous beam locations. After one complete period the phase-induced density features are absent. (IV) Comparative density using unidirectional scan ordering. The scale bar in (I) has $50\text{−}\mu \mathrm{m}$ length. The corresponding normalized density profiles $\chi \left(\theta \right)$ are vertically displaced for visual clarity.

Figure 7

Numerical mapping of three distinct regions of BEC behavior within unidirectionally scanned ring traps for scan frequency $f_s$, radial trap frequency $f_{\rho }$, and scan beam waist ${\sigma }_{\rho }$. The vertical trap frequency $f_z=140\mathrm{Hz}$, radius $R=82\mu \mathrm{m}$, and atom number $N_0=2\times {10}^6$. The regions are as follows: (I) experimentally accessible confined region; (II) transition region where the time-averaged condition begins to fail; and (III) untrapped region where atoms are numerically unconfined. Region boundaries are defined where trapped density profiles demonstrate irregularity (I and II) and phase profiles randomize (II and III); see Supplemental Material [39]. The contours indicate the phase step ${\delta }_{\varphi }$ (dashed) and density step ${\delta }_{\chi }$ (dotted). The white circle and triangle mark the parameters for data presented in Figs. 3 and 4.

Figure 8

Schematic of the unidirectional scanning features which motivate the rotationally accumulated mean density image (RAMDI) approach. (a) Black dots represent the discrete points scanned by the trapping beam. The open circle indicates the beam position prior to TOF imaging. Feedforward is performed after TOF and with beam position (I). For increasing hold times (II–IV), the scan beam position increments around the ring. (b) Phase profiles in trap. (c) Density profiles in trap. Markers indicate the orientation of density features induced through feedforward. These stationary features are stored in the scan beam point powers. (d) Density profiles after TOF. Markers orientate features resulting from the in-trap phase (striped) and density (solid). Feedforward with beam position (I) induces the density features (c) to correct phase features (b); they consequently cancel for this beam position only.

Figure 9

Diagnostic data for the raster scan ordering RAMDI saddle artefact. Normalized density profiles for experimental RAMDI (solid), taken for 20-ms TOF, and scan frequencies $f_s=1.25\mathrm{kHz}$ (I) and $f_s=0.7\mathrm{kHz}$ (II). The numerical curves are overlaid (dashed) and computed according to the description in Appendix pp4. The excellent agreement indicates that saddling observed experimentally does not result from the residual phase steps ${\delta }_0$ in Fig. 5. The depth of the saddle artefact increases with increasing scan frequency, while the residual phase steps decrease in amplitude. The scale bar in (II) has $50\mu \mathrm{m}$ length.