Analytical treatment of bosonic $d$-wave scattering in isotropic harmonic waveguides

We analyze $d$-wave resonances in atom-atom scattering in the presence of harmonic confinement by employing a higher-partial-wave pseudopotential. Analytical results for the scattering amplitude and transmission are obtained and compared to corresponding numerical ones, which employ the Lennard-Jones potential. Qualitative agreement is observed for weak confinement. For strong confinement the pseudopotential does not capture the $s$- and $d$-wave interference phenomena, yielding an asymmetric Fano profile for the transmission resonance.

Figure 1

(a) Numerically calculated transmission coefficient $T_n$, which shows the DWR on the LHS and the $s$-wave CIR on the RHS. The analytically calculated (b) $s$-wave transmission coefficient $T_s$ and (c) $d$-wave transmission coefficient $T_d$. The transverse confinement is ${\omega }_{\perp }={10}^{-5}$.

Figure 2

(a) Left panel: The numerically calculated transmission coefficient $T_n$ versus $C_{12}$ parameter for several confinement frequencies ${\omega }_{\perp }$; right panel: high-resolution graph of the DWR for ${\omega }_{\perp }={10}^{-5}$ where the inward-pointing arrows denote the width of the resonance. (b) Left panel: The analytically calculated transmission coefficient $T_d$ as a function of the $C_{12}$ parameter calculated with the $d$-wave pseudopotential for the corresponding values of ${\omega }_{\perp }$; right panel: high-resolution graph of the DWR for ${\omega }_{\perp }={10}^{-5}$. The yellow dashed line in the right panels indicates the minimum of the numerical calculated transmission coefficient ($T_n=0$) in comparison with the corresponding analytical result.