Resonant $d$-wave scattering in harmonic waveguides

We observe and analyze $d$-wave resonant scattering of bosons in tightly confining harmonic waveguides. It is shown that the $d$-wave resonance emerges in the quasi-1D regime as an imprint of a 3D $d$-wave shape resonance. A scaling relation for the position of the $d$-wave resonance is provided. By changing the trap frequency, ultracold scattering can be continuously tuned from $s$-wave to $d$-wave resonant behavior. The effect can be utilized for the realization of ultracold atomic gases interacting via higher partial waves and opens a possibility for studying strongly correlated atomic systems beyond $s$-wave physics.

Figure 1

(a) Partial cross sections for $s$- and $d$-wave scattering in free space, (b) transmission coefficient $T$ in the waveguide, and (c) interaction strength $g_{1\text{D}}$ as a function of the parameter $C_{12}$ ($C_6=28.82$ for Cs).

Figure 2

(a) Partial cross section ${\sigma }_{\ell =2}$ for $d$-wave scattering in free space and (b) transmission coefficient for several values of the confinement frequency ${\omega }_{\perp }$, as a function of the parameter $C_{12}$ around the $d$-wave resonance.

Figure 3

$s$- and $d$-wave scattering lengths $a_s$ and $a_d$ in free space as a function of the $C_{12}$ parameter, where $a_d$ is defined through $a_{\ell }^{2\ell +1}=-\tan {\delta }_{\ell }/k^{2\ell +1}$ [28] for $\ell =2$. $a_s^{\left(d\right)}({\omega }_{\perp }=0)\equiv a_{s,0}^{\left(d\right)}$ is the value of the $s$-wave scattering length at the point of divergence of the $d$-wave scattering length $a_d$.

Figure 4

Quotient $p=\Delta a_s^{\left(d\right)}/C_6$ as a function of $1/a_{\perp }^2$ (a) for different values of $C_6$ corresponding to the atomic species Cs, Rb, and Sr and (b) for Rb atoms with different interatomic potentials (see text), for confinement frequencies $4\pi \text{KHz}⩽{\omega }_{\perp }⩽400\pi \text{KHz}$. Dashed lines correspond to the resonance condition given in Eq. (5).

Figure 5

Probability density ${\left|\Psi (r,\theta )\right|}^2$ for Cs-Cs collisions (a) for the off-resonant case, (b) at the $s$-wave CIR, and (c) at the DWR, for ${\omega }_{\perp }=2\times {10}^{-4}$.

Figure 6

Transmission coefficient for $C_{12}=0.9989621$ as a function of the confinement frequency ${\omega }_{\perp }$ near the DWR.

Figure 7

Probability density ${\left|\Psi (r,\theta )\right|}^2$ for Cs-Cs collisions (a) ${\omega }_{\perp }=5.7\times {10}^{-4}$ (left side of the DWR), (b) ${\omega }_{\perp }=5.96\times {10}^{-4}$ (position of the DWR), and (c) ${\omega }_{\perp }=6.36\times {10}^{-4}$(right side of the DWR).

Figure 8

Ratio of the energy $E_r^{\text{(3D)}}$ of the resonant state in free space and the confining frequency ${\omega }_{\perp }$ as a function of the $C_{12}$ parameter. ${\omega }_{\perp }$ is varied in the range $4\pi \text{KHz}⩽{\omega }_{\perp }⩽400\pi \text{KHz}$.

Figure 9

A schematic illustration of the scattering potential in free space (blue solid line) and in the presence of confinement (black dashed line).