Abstract:
We use the Bogoliubov-de Gennes formalism to analyze the effects of rotation on the ground state phases of harmonically trapped Fermi gases, under the assumption that quantized vortices are not excited. We find that the rotation breaks Cooper pairs that are located near the trap edge, and that this leads to a phase separation between the nonrotating superfluid (fully paired) atoms located around the trap center and the rigidly rotating normal (nonpaired) atoms located towards the trap edge, with a coexistence (partially paired) region in between. Furthermore, we show that the superfluid phase that occurs in the coexistence region is characterized by a gapless excitation spectrum, and that it is distinct from the gapped phase that occurs near the trap center.

Abstract:
We investigate theoretically the interaction of polar molecules with optical lattices and microwave fields. We demonstrate the existence of frequency windows in the optical domain where the complex internal structure of the molecule does not influence the trapping potential of the lattice. In such frequency windows the Franck-Condon factors are so small that near-resonant interaction of vibrational levels of the molecule with the lattice fields have a negligible contribution to the polarizability and light-induced decoherences are kept to a minimum. In addition, we show that microwave fields can induce a tunable dipole-dipole interaction between ground-state rotationally symmetric (J=0) molecules. A combination of a carefully chosen lattice frequency and microwave-controlled interaction between molecules will enable trapping of polar molecules in a lattice and possibly realize molecular quantum logic gates. Our results are based on ab initio relativistic electronic structure calculations of the polar KRb and RbCs molecules combined with calculations of their rovibrational motion.

Abstract:
Recent experiments have shown a remarkable number of collapse-and-revival oscillations of the matter-wave coherence of ultracold atoms in optical lattices [Will et al., Nature 465, 197 (2010)]. Using a mean-field approximation to the Bose-Hubbard model, we show that the visibility of collapse-and-revival interference patterns reveal number squeezing of the initial superfluid state. To describe the dynamics, we use an effective Hamiltonian that incorporates the intrinsic two-body and induced three-body interactions, and we analyze in detail the resulting complex pattern of collapse-and-revival frequencies generated by virtual transitions to higher bands, as a function of lattice parameters and mean-atom number. Our work shows that a combined analysis of both the multiband, non-stationary dynamics in the final deep lattice, and the number-squeezing of the initial superfluid state, explains important characteristics of optical lattice collapse-and-revival physics. Finally, by treating the two- and three-body interaction strengths, and the coefficients describing the initial superposition of number states, as free parameters in a fit to the experimental data it should be possible to go beyond some of the limitations of our model and obtain insight into the breakdown of the mean-field theory for the initial state or the role of nonperturbative effects in the final state dynamics.

Abstract:
We study the dynamics of spin-1 atoms in a periodic optical-lattice potential and an external magnetic field in a quantum quench scenario where we start from a superfluid ground state in a shallow lattice potential and suddenly raise the lattice depth. The time evolution of the non-equilibrium state, thus created, shows collective collapse-and-revival oscillations of matter-wave coherence as well as oscillations in the spin populations. We show that the complex pattern of these two types of oscillations reveals details about the superfluid and magnetic properties of the initial many-body ground state. Furthermore, we show that the strengths of the spin-dependent and spin-independent atom-atom interactions can be deduced from the observations. The Hamiltonian that describes the physics of the final deep lattice not only contains two-body interactions but also effective multi-body interactions, which arise due to virtual excitations to higher bands. We derive these effective spin-dependent three-body interaction parameters for spin-1 atoms and describe how spin-mixing is affected. Spinor atoms are unique in the sense that multi-body interactions are directly evident in the in-situ number densities in addition to the momentum distributions. We treat both antiferromagnetic (e.g. $^{23}$Na atoms) and ferromagnetic (e.g. $^{87}$Rb and $^{41}$K) condensates.

Abstract:
We investigate the existence of quantum {\it quasi} phase transitions for an ensemble of ultracold bosons in a one-dimensional optical lattice, performing exact diagonalizations of the Bose-Hubbard Hamiltonian. When an external parabolic potential is added to the system {\it quasi} phase transitions are induced by the competition of on-site mean-field energy, hopping energy, and energy offset among lattice sites due to the external potential and lead to the coexistence of regions of particle localization and delocalization in the lattice. We clarify the microscopic mechanisms responsible for these {\it quasi} phase transitions as a function of the depth of the external potential when the on-site mean-field energy is large compared to the hopping energy. In particular, we show that a model Hamiltonian involving a few Fock states can describe the behavior of energy gap, mean particle numbers per site, and number fluctuations per site almost quantitatively. The role of symmetry on the gap as a function of the depth of the external trapping potential is elucidated. We discuss possible experimental signatures of {\it quasi} phase transitions studying the single particle density matrix and explain microscopically the occurrence of local maxima in the momentum distribution. The role of a thermal population of the excited states on the momentum distribution is discussed.

Abstract:
A method is described for estimating effective scattering lengths via spectroscopy on a trapped pair of atoms. The method relies on the phenomena that the energy levels of two atoms in a harmonic trap are shifted by their collisional interaction. The amount of shift depends on the strength of the interaction (i.e. scattering length). By combining the spectra of the trap state energy levels and a suitable model for the effective scattering length, an estimate for the latter may be inferred. Two practical methods for measuring the trap spectra are proposed and illustrated in examples. The accuracy of the scheme is analyzed and requirements on measurement precision are given.

Abstract:
Pairs of trapped atoms can be associated to make a diatomic molecule using a time dependent magnetic field to ramp the energy of a scattering resonance state from above to below the scattering threshold. A relatively simple model, parameterized in terms of the background scattering length and resonance width and magnetic moment, can be used to predict conversion probabilities from atoms to molecules. The model and its Landau-Zener interpretation are described and illustrated by specific calculations for $^{23}$Na, $^{87}$Rb, and $^{133}$Cs resonances. The model can be readily adapted to Bose-Einstein condensates. Comparison with full many-body calculations for the condensate case show that the model is very useful for making simple estimates of molecule conversion efficiencies.

Abstract:
We have constructed the X 1SIGMAg+ potential for the collision between two ground state Mg atoms and analyzed the effect of uncertainties in the shape of the potential on scattering properties at ultra-cold temperatures. This potential reproduces the experimental term values to 0.2 inverse cm and has a scattering length of +1.4(5) nm where the error is prodominantly due to the uncertainty in the dissociation energy and the C6 dispersion coefficient. A positive sign of the scattering length suggests that a Bose-Einstein condensate of ground state Mg atoms is stable.

Abstract:
The relativistic configuration interaction valence bond method has been used to calculate permanent and transition electric dipole moments of the KRb heteronuclear molecule as a function of internuclear separation. The permanent dipole moment of the ground state $X^1\Sigma^+$ potential is found to be 0.30(2) $ea_0$ at the equilibrium internuclear separation with excess negative charge on the potassium atom. For the $a^3\Sigma^+$ potential the dipole moment is an order of magnitude smaller (1 $ea_0=8.47835 10^{-30}$ Cm) In addition, we calculate transition dipole moments between the two ground-state and excited-state potentials that dissociate to the K(4s)+Rb(5p) limits. Using this data we propose a way to produce singlet $X^1\Sigma^+$ KRb molecules by a two-photon Raman process starting from an ultracold mixture of doubly spin-polarized ground state K and Rb atoms. This Raman process is only allowed due to relativistic spin-orbit couplings and the absence of gerade/ungerade selection rules in heteronuclear dimers.

Abstract:
It is possible to tune the scattering length for the collision of ultra-cold 1S0 ground state alkaline-earth atoms using an optical Feshbach resonance. This is achieved with a laser far detuned from an excited molecular level near the frequency of the atomic intercombination 1S0--3P1 transition. Simple resonant scattering theory, illustrated by the example of 40Ca, allows an estimate of the magnitude of the effect. Unlike alkali metal species, large changes of the scattering length are possible while atom loss remains small, because of the very narrow line width of the molecular photoassociation transition. This raises prospects for control of atomic interactions for a system without magnetically tunable Feshbach resonance levels.