Energy dependence of scattering ground-state polar molecules

We explore the total collisional cross section of ground-state polar molecules in an electric field at various energies, focusing on $\mathrm{RbCs}$ and $\mathrm{RbK}$. An external electric field polarizes the molecules and induces strong dipolar interactions, leading to nonzero partial waves contributing to the scattering even as the collision energy goes to zero. This results in the need to compute scattering problems with many different values of total $M$ to converge the total cross section. An accurate and efficient approximate total cross section is introduced and used to study the low-field temperature dependence. To understand the scattering of the polar molecules we compare a semiclassical cross section with the quantum unitarity limit. This comparison leads to the ability to characterize the scattering based on the value of the electric field and the collision energy. General and simple forms of the characteristic electric field and energy are given, enabling characterization of the scattering.

Figure 1

(Color online) $\mathrm{RbCs}$ molecular energies shown as a function of electric field. The energies are normalized by the rotational constant. Different color curves represent values of $M$: $M=0$ (black), $\mid M\mid =1$ (red), and $\mid M\mid =2$ (blue). The black dashed line is the projection of the dressed molecular ground state onto the field axis, $⟨00\mid \widehat{z}\mid 00⟩$, and its value is given on the right vertical axis. The red dotted curve is a simple and approximate form of $⟨00\mid \widehat{z}\mid 00⟩$ given in the text. The top horizontal axis shows the electric field normalized by the critical field value, $\mathcal{E}∕{\mathcal{E}}_0$.

Figure 2

(Color online) The energy dependence of $\sigma$ (solid lines) and $\tilde{\sigma }$ (dashed lines) for various electric fields. In ascending order the fields are 0 (black), ${\mathcal{E}}_0$ (blue), and $3{\mathcal{E}}_0$ (brown) for (a) $\mathrm{RbCs}$ and (b) $\mathrm{RbK}$ where ${\mathcal{E}}_0^{\mathrm{RbCs}}\simeq 780\mathrm{V}∕\mathrm{cm}$ and ${\mathcal{E}}_0^{\mathrm{RbK}}\simeq 3000\mathrm{V}∕\mathrm{cm}$. The dotted lines are ${\sigma }_{SC}$ from Eq. (9) for different electric fields. In ascending order they are ${\mathcal{E}}_0$ (red) and $3{\mathcal{E}}_0$ (purple).

Figure 3

(Color online) The total cross section (solid circles) for $\mathrm{RbK}$ decomposed into three ${\sigma }^{\left(M\right)}$’s for different field values. The fields are (a) $\mathcal{E}$=0 and (b) $\mathcal{E}=3{\mathcal{E}}_0$, and their comparison clearly shows the change in threshold behavior induced by the electric field. The different ${\sigma }^{\left(M\right)}$’s are for $M$ equal to 0 (solid lines), 2 (dashed lines), and 4 (dash-dotted lines). The total cross section is also shown as circles in Fig. 2a.

Figure 4

(Color online) Electric field dependence of the total cross section (solid lines) and approximate total cross section (dashed lines) at various energies for $\mathrm{RbCs}$ in (a) and $\mathrm{RbK}$ in (b). In (a) and (b) the energies of the curves in descending order are 0.1 (brown), 1 (red), 10 (blue), and 100 (black) $\mathrm{\mu }\mathrm{K}$. Electric field dependence of the approximate thermally averaged cross at various temperatures for $\mathrm{RbCs}$ (c) and $\mathrm{RbK}$ (d). The temperatures of the curves in descending order are 1 (circle), 5 (open circle), 10 (square), 25 (open square), 50 (open triangle), and 100 (triangle) $\mathrm{\mu }\mathrm{K}$.

Figure 5

(Color online) Comparisons of $\sigma$ (solid lines), ${\sigma }_{SC}$ (dashed lines), and ${\sigma }_Q$ (dotted lines) show the change of character in scattering for both $\mathrm{RbCs}$ (a) and $\mathrm{RbK}$ (b). Sets of curves ($\sigma$, ${\sigma }_{SC}$, and ${\sigma }_Q$) are given for different energies, and in descending order the sets are for 1 (red), 10 (blue), and 100 (black) $\mathrm{\mu }\mathrm{K}$. The critical electric field, when ${\sigma }_{SC}={\sigma }_Q$, is indicated by a symbol for each energy, and these symbols are also shown in (c). (c) The critical electric field in units of ${\mathcal{E}}_0$ is plotted as a function of energy for various molecules. For a particular molecule, the scattering is semiclassical in character when the electric field is greater than ${\mathcal{E}}_X$ and the collision energy is greater than $E_Q$. The scattering is quantum mechanical when either the electric field is less than ${\mathcal{E}}_X$ or the collision energy is less than $E_Q$. The parameters of the molecules are given in the text.