Sympathetic cooling of the Ba${}^{+}$ ion by collisions with ultracold Rb atoms: Theoretical prospects

State-of-the-art ab initio techniques have been applied to compute the potential energy curves of the (BaRb)${}^{+}$ molecular ion in the Born-Oppenheimer approximation for the singlet and triplet states dissociating into the ground-state ${}^1$$S$ Rb${}^{+}$ ion and the Ba atom in the ground ${}^1$$S$ state or the lowest singlet or triplet $d$ excited states, and for the singlet and triplet states dissociating into the ground-state ${}^2$$S$ Rb atom and the ground-state ${}^2$$S$ Ba${}^{+}$ ion. The ground-state potential energy was obtained with the coupled-cluster method restricted to single, double, and nonperturbative triple excitations. The first triplet states in the $\Sigma$, $\Pi$, and $\Delta$ symmetries were computed with the restricted open-shell coupled-cluster method restricted to single, double, and nonperturbative triple excitations. All other excited-state potential energy curves were computed using the equation of motion approach within the coupled-cluster singles, doubles, and linear triples framework. The long-range coefficients describing the electrostatic, induction, and dispersion interactions at large interatomic distances are also reported. The electric transition dipole moments governing the $x$ ${}^1\Sigma \to {}^1\Sigma ,{}^1\Pi$ transitions have been obtained as the first residue of the polarization propagator computed with the linear response coupled-cluster method restricted to single and double excitations. Nonadiabatic radial and angular coupling matrix elements as well as the spin-orbit coupling matrix elements have been evaluated using the multireference configuration-interaction method restricted to single and double excitations with a large active space. With these couplings, the spin-orbit-coupled (relativistic) potential energy curves for the 0${}^{+}$ and 1 states relevant for the running experiments have been obtained. Finally, relativistic transition moments and nonadiabatic coupling matrix elements were obtained from the nonrelativistic results and spin-orbit eigenvectors. The electronic structure input has been employed in the single-channel scattering calculations of the collisional cross sections between the Ba${}^{+}$ ion and Rb atom. Both nonrelativistic and relativistic potentials were used in these calculations. Our results show that the inelastic cross section corresponding to the charge transfer from the Rb atom to the Ba${}^{+}$ ion is much smaller than the elastic one over a wide range of energies up to 1 mK. This suggests that sympathetic cooling of the Ba${}^{+}$ ion by collisions with ultracold Rb atoms should be possible.

Figure 1

Nonrelativistic potential energy curves for the excited states of the (BaRb)${}^{+}$ molecular ion.

Figure 2

Comparison of the LRCC3 with CCSD, and RCCSD(T) with LRCCSD potential energy curves for selected states of the (BaRb)${}^{+}$ molecular ion.

Figure 3

Nonadiabatic coupling matrix elements of the nonrelativistic electronic states of the (BaRb)${}^{+}$ molecular ion. The left and right panels correspond to the radial and angular couplings, respectively.

Figure 4

Electric transition dipole moments from the ground $x$$\hspace{0.5em}{}^1\Sigma$ state to the ${}^1\Sigma$ and ${}^1\Pi$ states of (BaRb)${}^{+}$.

Figure 5

Matrix elements of the spin-orbit coupling for the electronic states of (BaRb)${}^{+}$. The left and right panels correspond to $0^{+}$ and 1 states, respectively.

Figure 6

Relativistic potential energy curves for the excited states of the (BaRb)${}^{+}$ molecular ion. Red dashed lines correspond to the $0^{+}$ states and the blue dotted lines to the $1$ states.

Figure 7

Nonadiabatic coupling matrix elements of the relativistic electronic states and relativistic electric transition dipole moments of the (BaRb)${}^{+}$ molecular ion. The left and right panels correspond to the nonadiabatic couplings and transition moments, respectively.

Figure 8

Long-range nonrelativistic (left panel) and relativistic (right panel) potential energy curves of the (BaRb)${}^{+}$ molecular ion. Stars are based on long-range coefficients like in Eq. (3).

Figure 9

Elastic, spin-flip, and charge-transfer cross sections for collisions of ${}^{138}\mathrm{Ba}$${}^{+}$(${}^2$$S$) and ${}^{87}\mathrm{Rb}$(${}^2$$S$) as functions of the collision energy from nonrelativistic potentials.

Figure 10

Elastic, spin-flip, and charge-transfer cross sections for collisions of ${}^{138}\mathrm{Ba}$${}^{+}$(${}^2$$S$) and ${}^{87}\mathrm{Rb}$(${}^2$$S$) as functions of the collision energy from relativistic potentials.