Long-range scattering resonances in strong-field-seeking states of polar molecules

We present first steps toward understanding the ultracold scattering properties of polar molecules in strong electric field-seeking states. We have found that the elastic cross section displays a quasiregular set of potential resonances as a function of the electric field, which potentially offers intimate details about the intermolecular interaction. We illustrate these resonances in a “toy” model composed of pure dipoles, and in more physically realistic systems. To analyze these resonances, we use a simple WKB approximation to the eigenphase, which proves both reasonably accurate and meaningful. A general treatment of the Stark effect and dipolar interactions is also presented.

Figure 1

(a) Adiabatic curves of the pure dipole system for two values of the dipole moment $\mu$, $\mu =0.1$ (gray) and $\mu =1$ (black) (b) The cross section of the polarized dipole model versus $\mu$, at a collision energy $E={10}^{-12}\mathrm{K}$. The brackets denote predicted resonant positions using the adiabatic WKB phase (AWP) approximation. The black bracket represents for the phase contribution from the lowest adiabatic curve and the gray is from all remaining contributions.

Figure 2

(a) Eigenphase (solid) and total adiabatic WKB phase (AWP, dashed) for the dipole-scattering model. (b) AWP contributions for the lowest adiabatic curves $\left({\varphi }_{\mathrm{WKB}}^{\left(1\right)}\right)$ with the black bracket indicating when ${\varphi }_{\mathrm{WKB}}^{\left(1\right)}$ passes through a multiple of $\pi$. The two remaining curves are contributions with nonzero partial waves $\left({\varphi }_{\mathrm{WKB}}^{(2,3)}\right)$. The gray bracket indicates where the sum of these contributions pass through a multiple of $\pi$.

Figure 3

(a) Adiabatic curves for the RbCs system at two different field values, $\mathcal{E}=0(\mathrm{V}∕\mathrm{cm})$ (gray) and $\mathcal{E}=5000(\mathrm{V}∕\mathrm{cm})$ (black). (b) Log-log plot of the absolute value of the two lowest adiabatic curves. The blue dashed line is proportional to $1∕R^3$.

Figure 4

(a) Cross section for RbCs in strong-field-seeking states, including the 10 potential resonances that the AWP predicts (solid bracket). The AWP phases are shown in (b). The black bracket corresponds to the AWP predicted resonances for the lowest adiabatic curve after a defect, ${\delta }_{\text{defect}}=0.14$, has been added according to Eq. (13). The gray triangles are the AWP predicted resonances without the defect.

Figure 5

Cross section for SrO with a bracket indicating resonance positions predicted by the phase shifted AWP for the lowest adiabatic curve. There are 33 PR shown in this range of electric field, ${\delta }_{\text{defect}}=0.215$.

Figure 6

The vertical curves represent electric field values where the AWP predicts potential resonances for a given defect, ${\mathcal{E}}_{\mathrm{WKB}}\left({\delta }_{\text{defect}}\right)$. The points represent resonant field values, ${\mathcal{E}}^{\left(b\right)}$, in the full calculations with different initial boundary conditions. The values of $b$ are given by $\mathrm{cotan}\left(\pi \beta \right)$ and $\beta$ for the different full numerical calculations are 0 (filled circle), 0.11 (filled square), 0.22 (filled triangle), 0.56 (hollow circle), 0.78 (hollow square), and 0.89 (hollow triangle). Each set of values ${\mathcal{E}}^{\left(b\right)}$ is plotted at a height corresponding to its best-fit value of ${\delta }_{\text{defect}}$.