Imaging of $s$ and $d$ Partial-Wave Interference in Quantum Scattering of Identical Bosonic Atoms

We report on the direct imaging of $s$ and $d$ partial-wave interference in cold collisions of atoms. Two ultracold clouds of ${}^{87}\mathrm{R}\mathrm{b}$ atoms were accelerated by magnetic fields to collide at energies near a $d$-wave shape resonance. The resulting halos of scattered particles were imaged using laser absorption. By scanning across the resonance we observed a marked evolution of the scattering patterns due to the energy dependent phase shifts for the interfering $s$ and $d$ waves. Since only two partial-wave states are involved in the collision process the scattering yield and angular distributions have a simple interpretation in terms of a theoretical model.

Figure 1

Illustration of the process of using absorption imaging for the detection of scattered particles. We present the case of pure $d$-wave scattering occurring at the origin for particles coming in along the $z$ axis. Scattered particles will be situated on an expanding sphere and distributed according to the $d$-wave angular emission pattern $|f(\theta ){|}^2$. Absorption imaging along the $x$ axis projects this distribution onto the $yz$ plane.

Figure 2

Absorption images acquired at a quarter of a radial trap period after the collision of two doubly spin-polarized Rb clouds (visible as dark ellipses) for various collision energies. The halos of scattered particles have elliptical envelopes since they are evolving in an anisotropic harmonic trap which is weakest in the horizontal direction ($z$ direction). At the selected time of acquisition the scattering halos have the maximum radial excursion in the trap.

Figure 3

Dependence on collision energy. (a) The $s$ (dotted line) and $d$ (dashed line) partial-wave phase shifts from the theoretical model. (b) The $s$-wave (dotted line), $d$-wave (dashed line), and total (solid line) cross sections calculated from the model partial-wave phase shifts. (c) The measured scattered fraction of atoms $N_{\mathrm{s}\mathrm{c}}/N_{\mathrm{t}\mathrm{o}\mathrm{t}}$ (filled circles). The black curve shows the fraction as given by the model cross section when depletion of the colliding atom clouds is accounted for.

Figure 4

Polar plots of the normalized angular scattering probability density for different collision energies in $\mu \mathrm{K}$. (a) Experimental results from the absorption images of Fig. 2 after Abel inversion and a transformation from trap to free space. (b) Characteristic patterns as predicted by Eq. (1) using the partial-wave shifts from our theoretical model.