Figure 1

An ultracold-atom storage ring. Axes are indicated with gravity along $-\widehat{z}$. Top or side-view absorption images of the atoms allow their motion to be studied. Dashed circles indicate the $270\mu \mathrm{m}$ radial extent of top-view images displayed in polar coordinates as “annular views” in Figs. 2, 4. We refer to motion in $r$ as radial, in $z$ as axial, and in $\theta$ as azimuthal or longitudinal. The nominally circular storage ring contains radial $\delta r$ (exaggerated for illustration) and axial $\delta z$ beam-path errors and an azimuthal variation $U(\theta )$ in the potential.Reuse & Permissions

Figure 2

Dispersion management of matter waves in a storage ring. Annular (top) views are shown after 2, 11, and 18 complete revolutions for initial mean axial tunes of (a) ${\nu }_z=4.2$ and (b) ${\nu }_z=4.0$. Slight bends in the cloud result from radial betatron motion excited during the injection sequence. (c) The rms width of all (X’s for ${\nu }_z=4.2$) or just the compact portion (circles for ${\nu }_z=4.0$) of the atomic beam is shown vs propagation time, with data for ${\nu }_z=4.0$ limited to times when the compact and diffuse portions are separated. A linear fit to data for ${\nu }_z=4.2$ (solid line), and a line joining earliest and latest data for ${\nu }_z=4.0$ (dashed line) are shown. Here $({\omega }_r,{\omega }_z)=2\pi (48,60)\mathrm{Hz}$.Reuse & Permissions

Figure 3

Matter wave dispersion at an axial betatron resonance. The distribution of final azimuthal velocities is shown for beams with initial mean angular velocities ${\Omega }_i/2\pi$ evenly spaced between 14.3 (bottom curve) and 15.4 Hz (top curve). Data are offset vertically for clarity. These distributions are obtained from the radially integrated column density of top-view images taken after 640 ms of propagation. Atoms expelled from the stop band (gray shading) accumulate at its low-velocity edge. Since the stop band is narrower than the full initial range of velocities in the beam, only a portion of the beam is affected. Low-velocity atoms for ${\Omega }_i/2\pi =14.3\mathrm{Hz}$ are affected by the ${\nu }_r=5$ radial betatron resonance. Here $({\omega }_r,{\omega }_z)=2\pi (70,74)\mathrm{Hz}$.Reuse & Permissions

Figure 4

Stop bands for the ${\nu }_r=5$ radial (lower ${\Omega }_i$) and the ${\nu }_z=5$ axial (higher ${\Omega }_i$) betatron resonances. Mean final angular velocities, determined after 440 ms of propagation in the ring, are shown vs ${\Omega }_i$. Near the resonances, the atomic beam divides into two portions (Fig. 3), and mean angular velocities of each are given. Simulation results for the axial resonance (solid line), and the relation ${\Omega }_i={\Omega }_f$ (dashed line) are shown. The (a) annular and (b) side views of the atoms launched at ${\nu }_r=5$ (radial resonance), and (c) annular and (d) side views of those at ${\nu }_z=5$ (axial resonance), show which transverse oscillation is enhanced at each resonance. The horizontal scale for (a) and (c) is chosen to give equal cloud lengths in annular- and side-view images. Here, $({\omega }_r,{\omega }_z)/2\pi =(69,73)$ Hz.Reuse & Permissions

Figure 5

Tuning the resonant velocities and strength of betatron resonances. (a) Measured angular frequencies ${\Omega }_{\mathrm{res}}$ at the low-velocity edge of stop bands observed at several radial (open circles) and axial (filled circles) betatron resonances are shown vs measured betatron frequencies. (b) The ${\nu }_r=4$ resonance for $({\omega }_r,{\omega }_z)=2\pi (68,74)\mathrm{Hz}$ is characterized with a deliberately applied $U(\theta )=U_4\sin (4\theta )$ azimuthal potential of strength $U_4/k_B=24$ (circles), 45 (triangles), and 81 nK (squares). The dashed line shows the relation ${\Omega }_i={\Omega }_f$.Reuse & Permissions