Hybrid systems of laser-cooled trapped ions and ultracold atoms combined in a single experimental setup have recently emerged as a new platform for fundamental research in quantum physics. This paper reviews the theoretical and experimental progress in research on cold hybrid ion-atom systems which aim to combine the best features of the two well-established fields. A broad overview is provided of the theoretical description of ion-atom mixtures and their applications, and a report is given on advances in experiments with ions trapped in Paul or dipole traps overlapped with a cloud of cold atoms, and with ions directly produced in a Bose-Einstein condensate. This review begins with microscopic models describing the electronic structure, interactions, and collisional physics of ion-atom systems at low and ultralow temperatures, including radiative and nonradiative charge-transfer processes and their control with magnetically tunable Feshbach resonances. Then the relevant experimental techniques and the intrinsic properties of hybrid systems are described. In particular, the impact is discussed of the micromotion of ions in Paul traps on ion-atom hybrid systems. Next, a review of recent proposals is given for using ions immersed in ultracold gases for studying cold collisions, chemistry, many-body physics, quantum simulation, and quantum computation and their experimental realizations. The last part focuses on the formation of molecular ions via spontaneous radiative association, photoassociation, magnetoassociation, and sympathetic cooling. Applications and prospects are discussed of cold molecular ions for cold controlled chemistry and precision spectroscopy.

We formulate the theory for a diatomic molecule in a spatially degenerate electronic state interacting with a non-resonant laser field and investigate its rovibrational structure in the presence of the field. We report on \textit{ab initio} calculations employing the double electron attachment intermediate Hamiltonian Fock space coupled cluster method restricted to single and double excitations for all electronic states of the Rb$_2$ molecule up to $5s+5d$ dissociation limit of about 26.000$\,$cm$^{-1}$. In order to correctly predict the spectroscopic behavior of Rb$_2$, we have also calculated the electric transition dipole moments, non-adiabatic coupling and spin-orbit coupling matrix elements, and static dipole polarizabilities, using the multireference configuration interaction method. When a molecule is exposed to strong non-resonant light, its rovibrational levels get hybridized. We study the spectroscopic signatures of this effect for transitions between the X$^1\Sigma_g^+$ electronic ground state and the A$^1\Sigma_u^+$ and b$^3\Pi_u$ excited state manifold. The latter is characterized by strong perturbations due to the spin-orbit interaction. We find that for non-resonant field strengths of the order $10^9$W/cm$^2$, the spin-orbit interaction and coupling to the non-resonant field become comparable. The non-resonant field can then be used to control the singlet-triplet character of a rovibrational level.Comment: 28 pages, 14 figures, 5 table

The electronic structure of the $(\mathrm{LiYb}{)}^{+}$ molecular ion is investigated with two variants of the coupled cluster method restricted to single, double, and noniterative or linear triple excitations. Potential-energy curves for the ground and excited states, permanent and transition electric dipole moments, and long-range interaction coefficients $C_4$ and $C_6$ are reported. The data are subsequently employed in scattering calculations and photoassociation studies. Feshbach resonances are shown to be measurable, despite the ion's micromotion in the Paul trap. Molecular ions can be formed in their singlet electronic ground state by one-photon photoassociation and in triplet states by two-photon photoassociation; and control of cold atom-ion chemistry based on Feshbach resonances should be feasible. Conditions for sympathetic cooling of an ${\mathrm{Yb}}^{+}$ ion by an ultracold gas of Li atoms are found to be favorable in the temperature range $10\mathrm{nK}–10\mathrm{mK}$, and further improvements using Feshbach resonances should be possible. Overall, these results suggest excellent prospects for building a quantum simulator with ultracold ${\mathrm{Yb}}^{+}$ ions and Li atoms.