Coupled $\ell$-wave confinement-induced resonances in cylindrically symmetric waveguides

A semianalytical approach to atomic waveguide scattering for harmonic confinement is developed, taking into account all partial waves. As a consequence $\ell$-wave confinement-induced resonances are formed, being coupled to each other due to the confinement. The corresponding resonance condition is obtained analytically using the $K$-matrix formalism. Atomic scattering is described by transition diagrams which depict all relevant processes the atoms undergo during the collision. Our analytical results are compared to corresponding numerical data and show very good agreement.

Figure 1

A schematic illustration of the transitions into $N$ channels which contribute in the scattering process, corresponding to the $K$-matrix ${\underline{K}}_{\mathrm{oo},\ell }^{1\mathrm{D},\mathrm{phys}}$. The open (solid) circles indicate the open (closed) channels, the arrows depict the transitions through the $\ell$th partial wave, and the dots denote the $N-3$ channels and their corresponding transitions.

Figure 2

A schematic illustration of the transitions into $N$ channels which contribute in the scattering process, for the case of $s$- and $d$-wave. The open (solid) circles indicate the open (closed) channels, the solid-black (dashed-dotted) arrows depict the transitions through the $s$- ($d$-) partial wave, while the dots denote the $N-3$ channels and their corresponding transitions.

Figure 3

A comparison plot for (a) analytically calculated transmission coefficient $T$ and (b) numerically calculated transmission coefficient $T_n$ of $d$-wave CIR for several confinement frequencies ${\omega }_{\perp }$.

Figure 4

A high-resolution comparison of analytical results (solid line) with numerical data (red squares) for the $d$-wave CIR for two different confinement frequencies, ${\omega }_{\perp }=4\times {10}^{-4},8\times {10}^{-4}$ in (a),(b), respectively.

Figure 5

(a) Transmission coefficient $T$ as a function of the $a_s/a_{\perp }$ parameter, (b) contour plot of the quantity |det$(\mathbb{I}-i{\underline{K}}_{\mathrm{cc}}^{1D})$|versus the parameters $a_s/a_{\perp }$ and ${(a_d/a_{\perp })}^5$, and (c) transmission coefficient $T$ as a function of the ${(a_d/a_{\perp })}^5$. In (b) the black dotted lines indicate the position of the resonance with respect to the panels (a) and (c), the black dotted circles indicate the positions of the zeros of |det$(\mathbb{I}-i{\underline{K}}_{\mathrm{cc}}^{1D})$|, and the red dashed line shows the positions of the resonances for an arbitrary short-range interatomic potential. The inset of (a) refers to the dashed boxed area and depicts on a logarithmic scale the second zero of the transmission coefficient ($s$-wave CIR).