Shielding ${}^2\mathrm{\Sigma }$ ultracold dipolar molecular collisions with electric fields

The prospects for shielding ultracold, paramagnetic, dipolar molecules from inelastic and chemical collisions are investigated. Molecules placed in their first rotationally excited states are found to exhibit effective long-range repulsion for applied electric fields above a certain critical value, as previously shown for nonparamagnetic molecules. This repulsion can safely allow the molecules to scatter while reducing the risk of inelastic or chemically reactive collisions. Several molecular species of ${}^2\mathrm{\Sigma }$ molecules of experimental interest—RbSr, SrF, BaF, and YO—are considered, and all are shown to exhibit orders of magnitude suppression in quenching rates in a sufficiently strong laboratory electric field. It is further shown that, for these molecules described by Hund's coupling case (b), electronic and nuclear spins play the role of spectator with respect to the shielding.

Figure 1

Energy spectrum of a single RbSr molecule in an electric field. The dressed states are noted $|\tilde{n},{\tilde{m}}_n\rangle$ (see text). The initial state considered for scattering is $|\tilde{1},\tilde{0}\rangle$.

Figure 2

Energy of the combined RbSr + RbSr molecular states. The initial colliding state is indicated with an arrow as well as the “crossing” state, at the crossing field $E^{*}=2.27$ kV/cm indicated inside the blue box. The combined molecular states are denoted $|{\tilde{n}}_1,{\tilde{m}}_{n_1}\rangle +|{\tilde{n}}_2,{\tilde{m}}_{n_2}\rangle$.

Figure 3

Elastic (red), inelastic (green), reactive (black), and quenching (blue) rate coefficients as a function of the electric field for a fixed collision energy of $E_c=500$ nK, for bosonic ${}^{87}{\mathrm{Rb}}^{84}\mathrm{Sr}+{}^{87}{\mathrm{Rb}}^{84}\mathrm{Sr}$ collisions initially in the state indicated in Fig. 2.

Figure 4

Adiabatic energies as a function of the molecule-molecule separation $r$ at a fixed field of $E=2.5$ kV/cm for $l=0\text{–}2$. The black curves represent a full calculation including all rotational states as well as the fine and hyperfine structure. The blue arrow indicates where the combined molecular state above the initial state starts to play a role. The red curves correspond to a simplified calculation involving $m_{n_i}=m_{s_i}=m_{i_i}=0,i=1,2$. To compare with the full structure calculation, all the red curves have been translated so that the two combined molecular states have the same threshold energies at large $r$.

Figure 5

Quenching (solid lines) and elastic (dashed lines) rate coefficients of ${}^{87}{\mathrm{Rb}}^{84}\mathrm{Sr}+{}^{87}{\mathrm{Rb}}^{84}\mathrm{Sr}$ collisions as a function of the electric field for the partial waves $l=0\text{–}2$ for a fixed collision energy of $E_c=500$ nK. Two simplifications are used: $m_{f_i}=m_{n_i}+m_{s_i}+m_{i_i}=0$ (blue lines) and $m_{n_i}=m_{s_i}=m_{i_i}=0$ (red curves). The thin black lines recall the same calculation with the full rotational and spin structure from Fig. 3. Finally, the thick black lines are the converged result with the partial waves $l=0\text{–}20$ for the $m_{n_i}=m_{s_i}=m_{i_i}=0$ simplification.

Figure 6

Same as Fig. 5 but as a function of the collision energy for a fixed electric field of $E=2.5$ kV/cm, for the $m_{n_i}=m_{s_i}=m_{i_i}=0$ simplification. The range of partial waves is $l=0\text{–}20$ and $M=m_l=[-8,8]$. The ratio $\gamma ={\beta }^{\text{el}}/{\beta }^{\text{qu}}$ is indicated for every decade of the collision energy on top of the elastic rate curve.

Figure 7

Quenching (solid lines) and elastic (dashed lines) rate coefficients of ${}^{84}{\mathrm{Sr}}^{19}\mathrm{F}+{}^{84}{\mathrm{Sr}}^{19}\mathrm{F}$ (red), ${}^{138}{\mathrm{Ba}}^{19}\mathrm{F}+{}^{138}{\mathrm{Ba}}^{19}\mathrm{F}$ (blue), and ${}^{89}{\mathrm{Y}}^{16}\mathrm{O}+{}^{89}{\mathrm{Y}}^{16}\mathrm{O}$ (green) collisions as a function of the electric field, for a fixed collisions energy of $E_c=500$ nK. The range of partial waves is, respectively, $l=0\text{–}30,l=0\text{–}40$, and $l=0\text{–}50$. The curve from Fig. 5 for RbSr has been added for comparison.

Figure 8

Same as Fig. 7 but as a function of the collision energy for a fixed electric field of $E=15$ kV/cm for the SrF and BaF molecules, and $E=18$ kV/cm for the YO molecules. The curve from Fig. 6 for RbSr has been added for comparison.