Energy-dependent $\ell$-wave confinement-induced resonances

The universal aspects of two-body collisions in the presence of a harmonic confinement are investigated for both bosons and fermions. The main focus of this study are the confinement-induced resonances (CIRs) which are attributed to different angular momentum states $\ell$, and we explicitly show that in alkaline collisions only four universal $\ell$-wave CIRs emerge given that the interatomic potential is deep enough. Going beyond the single mode regime the energy dependence of $\ell$-wave CIRs is studied. In particular we show that all the $\ell$-wave CIRs may emerge even when the underlying two-body potential cannot support any bound state. We observe that the intricate dependence on the energy yields resonant features where the colliding system within the confining potential experiences an effective free-space scattering. Our analysis is done within the framework of the generalized $K$-matrix theory, and the relevant analytical calculations are in very good agreement with the corresponding ab initio numerical scattering simulations.

Figure 1

The $s$-wave scattering length $a_0$ (red dashed line) and the $g$-wave scattering length $a_4$ (blue solid line) are shown versus the $C_{10}$ parameter. It is observed that both scattering lengths are diverging at the same values of $C_{10}$, meaning that the $s$- and $g$-wave bound and quasibound states appear simultaneously at the threshold.

Figure 2

The coefficient ${\mathsf{RC}}_{\ell }\left(\epsilon \right)$ (black solid lines) from Eq. (12) versus the dimensionless energy $\epsilon$ is shown for the case of $\ell =0$. (a) to (c) refer to the values of energy for which the transmission coefficient $T$ is shown versus the scattering length in Fig. 4. The shading indicates the magnitude of the scaled scattering length (${\overline{a}}_0$) that has to be met, fulfilling the equality of the resonance condition Eq. (11), while the white line represents the pole in the resonance condition after which the CIR occurs with a negative sign in scattering length.

Figure 3

Analogous to Fig. 2, here ${\mathsf{RC}}_1\left(\epsilon \right)$ is shown. (a) to (d) correspond to the values of energy for which the transmission coefficient $T$ is shown in Fig. 5. Note that this time the resonance condition has two poles, resulting in a twofold sign change in the scattering length.

Figure 4

The analytically calculated transmission coefficient (solid lines) is plotted versus the scaled $s$-wave scattering length ${\overline{a}}_0$ for energies $\epsilon =0.1,0.69,0.8$ from left to right, corresponding to the energy values indicated by the red dots in Fig. 2. The black dots indicate numerical values for the transmission coefficient included for comparison.

Figure 5

Same as Fig. 4, but this time for $\ell =1$ and energies $\epsilon =0.2,0.7,0.93,0.97$, from left to right, as indicated in Fig. 3.

Figure 6

The transmission coefficient $T_{\mathrm{unitary}}$ at unitarity is shown versus the channel normalized energy $\epsilon$ for $\ell =0$ and $\ell =1$, solid and dashed curves, respectively. Note that the minima indeed appear at ${\epsilon }_{\infty }^0$ for $\ell =0$ and at ${\epsilon }_{\infty }^1$ and ${\epsilon }_{\infty }^2$ for $\ell =1$, as these denote the locations of the poles in Figs. 2 and 3, respectively.

Figure 7

The solid lines in panels (a) and (b) depict ${\mathsf{RC}}_0\left(\epsilon \right)$ and ${\mathsf{RC}}_1\left(\epsilon \right)$, respectively, while both dashed lines show the condition for $T=1$, i.e., total transparency.

Figure 8

For the case of $d$-wave CIR, panel (a) shows the resonance coefficients ${\mathsf{RC}}_{\mathsf{2},\mathsf{0}}\left(\epsilon \right)$ versus the total colliding energy $\epsilon$. Panel (b) shows, also versus $\epsilon$, the transparency coefficients ${\mathsf{TC}}_{2,0}\left(\epsilon \right)$, which are seen to depend strongly on energy.

Figure 9

For the case of $f$-wave CIR, analogous to Fig. 8. Panel (a) depicts ${\mathsf{RC}}_{3,1}\left(\epsilon \right)$ and panel (b) ${\mathsf{TC}}_{3,1}\left(\epsilon \right)$. Contrary to the bosonic counterpart shown in Fig. 8, we observe from panel (b), that there is now $f$-wave dual CIR for negative values of ${\overline{a}}_4$.

Figure 10

The eigenenergies of the closed channel bound states of the $\ell$-wave CIRs as a function of the corresponding inverse scattering length, namely ${\overline{a}}_{\ell }^{-1}$. Panel (a) shows the bound states of $s$- (red solid line) and $p$-wave (blue dashed line) CIRs and (b) depicts the bound-states of $d$- (red solid line) and $f$-wave CIRs (blue dashed line). The horizontal dashed line illustrates the total colliding energy ${\epsilon }_{\mathrm{tot}}$.