A simple quantum-defect model gives analytic expressions for the complex scattering length and threshold collision rates of ultracold molecules. If the probability of reaction in the short-range part of the collision is high, the model gives universal rate constants for s- and p-wave collisions that are independent of short-range dynamics. This model explains the magnitudes of the recently measured rate constants for collisions of two ultracold 40K87Rb molecules, or an ultracold 40K atom with the 40K87Rb molecule [S. Ospelkaus et al., Science 327, 853 (2010).

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. We provide a broad overview of the theoretical description of ion-atom mixtures and their applications, and report 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. We start with microscopic models describing the electronic structure, interactions, and collisional physics of ion-atom systems at low and ultralow temperatures, including radiative and non-radiative charge transfer processes and their control with magnetically tunable Feshbach resonances. Then we describe the relevant experimental techniques and the intrinsic properties of hybrid systems. In particular, we discuss the impact of the micromotion of ions in Paul traps on ion-atom hybrid systems. Next, we review recent proposals for using ions immersed in ultracold gases for studying cold collisions, chemistry, many-body physics, quantum simulation, and quantum computation and their experimental realizations. In the last part we focus on the formation of molecular ions via spontaneous radiative association, photoassociation, magnetoassociation, and sympathetic cooling. We discuss applications and prospects of cold molecular ions for cold controlled chemistry and precision spectroscopy.

We study atom-ion scattering in the ultracold regime. To this aim, an analytical model based on the multichannel quantum defect formalism is developed and compared to close-coupled numerical calculations. We investigate the occurrence of magnetic Feshbach resonances focusing on the specific 40 Ca + + Na system. The presence of several resonances at experimentally accessible magnetic fields should allow the atom-ion interaction to be precisely tuned. A fully quantum-mechanical study of charge exchange processes shows that charge-exchange rates should remain small even in the presence of resonance effects. Most of our results can be cast in a system-independent form and are important for the realization of the charge-neutral ultracold systems.Advances in trapping, cooling, manipulation and readout of single atoms and ions have led over recent years to a range of fundamental as well as applied investigations on the quantum properties of such systems. Nowadays, an increasing number of experimental groups worldwide are starting experiments with combined charged-neutral systems in various configurations [1]. While the theory of atom-ion collisions is well established for high collision energies [2,3], a theoretical description in the ultracold domain is still largely missing.This letter presents the first study of magnetic Feshbach resonances and the first fully quantum study of the radiative charge exchange process for ultracold atomion systems that includes effects of Feshbach and shape resonances. Here we consider only two-body collisions in free space, a necessary prelude to further studies incorporating effects of ion micromotion or trap confinement. We develop a reliable yet manageable effective model of atom-ion collisions by applying multichannel quantum defect theory (MQDT) [4,5,6] based on the long range ion-induced-dipole potential that varies as r −4 at large ion-atom distance r [7,8]. This powerful tool has proven effective as a few-parameter approach for describing scattering and bound states in electron-ion core [4], electron-atom [9] and neutral atom systems [10]. Although the literature on the subject is rich, here we discuss some details of MQDT illustrating how it works in the ultracold domain, so we can reveal the new and interesting ultracold ion-atom physics. We adapt MQDT to the atom-ion realm, utilizing the analytical solutions for the r −4 asymptotic potential [9,11] and applying the frame transformation [10,12] at short distances to reduce the number of quantum defect parameters in the model. We verify the model predictions by comparing to our own numerical close-coupled calculations, taking 40 Ca + − 23 Na [13] as a reference system.We describe the S-state atom and S-state ion collisions with the close-coupled radial Schrödinger equationHere, µ = m i m a /(m i + m a ) denotes the reduced mass, W(r) is the interaction matrix, and F(r) is the matrix of radial solutions. The wave function for N scattering channels reads Ψ i (r) = N j=1 A j Y j (r)F ij (r)/r where Y j (r) denotes the angular par...

We discuss exact solutions of the Schrödinger equation for the system of two ultracold atoms confined in an axially symmetric harmonic potential. We investigate different geometries of the trapping potential, in particular we study the properties of eigenenergies and eigenfunctions for quasi-one-and quasi-two-dimensional traps. We show that the quasi-one-and the quasi-two-dimensional regimes for two atoms can be already realized in the traps with moderately large (or small) ratios of the trapping frequencies in the axial and the transverse directions. Finally, we apply our theory to Feshbach resonances for trapped atoms. Introducing in our description an energy-dependent scattering length we calculate analytically the eigenenergies for two trapped atoms in the presence of a Feshbach resonance.

We propose and theoretically investigate a hybrid system composed of a crystal of trapped ions coupled to a cloud of ultracold fermions. The ions form a periodic lattice and induce a band structure in the atoms. This system combines the advantages of high fidelity operations and detection offered by trapped ion systems with ultracold atomic systems. It also features close analogies to natural solid-state systems, as the atomic degrees of freedom couple to phonons of the ion lattice, thereby emulating a solid-state system. Starting from the microscopic many-body Hamiltonian, we derive the low energy Hamiltonian, including the atomic band structure, and give an expression for the atom-phonon coupling. We discuss possible experimental implementations such as a Peierls-like transition into a period-doubled dimerized state.

We study the Bose-Einstein condensate (BEC) with a fixed number of particles. On the basis of conventional statistical mechanics we introduce the, so called, Maxwell's demon ensemble, where only particle transfer (without energy exchange) is allowed. We show that this new ensemble can be used for the microcanonical description of the system. We apply our formalism to the case of BEC in a harmonic trap and give the analytic expressions for the ground state fluctuations. We compare these expressions with the exact numerical results obtained for relatively small condensates. [S0031-9007(97)03974-4]

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We consider a system composed of a trapped atom and a trapped ion. The ion charge induces in the atom an electric dipole moment, which attracts it with an r −4 dependence at large distances. In the regime considered here, the characteristic range of the atom-ion interaction is comparable or larger than the characteristic size of the trapping potential, which excludes the application of the contact pseudopotential. The short-range part of the interaction is described in the framework of quantum-defect theory, by introducing some short-range parameters, which can be related to the s-wave scattering length. When the separation between traps is changed we observe trap-induced shape resonances between molecular bound states and vibrational states of the external trapping potential. Our analysis is extended to quasi-one-dimensional geometries, when the scattering exhibit confinement-induced resonances, similar to the ones studied before for short-range interactions. For quasi-one-dimensional systems we investigate the effects of coupling between the center of mass and relative motion, which occurs for different trapping frequencies of atom and ion traps. Finally, we show how the two types of resonances can be employed for quantum state control and spectroscopy of atom-ion molecules.