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arXiv:0904.2504 · Physical Review A

Theoretical description of two ultracold atoms in a single site of a three-dimensional optical lattice using realistic interatomic interaction potentials

Sergey Grishkevich, Alejandro Sáenz

Published Jul 6, 2009• 26 citations
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Abstract

A theoretical approach was developed for an exact numerical description of a pair of ultracold atoms interacting via a central potential, which is trapped in a three-dimensional optical lattice. The coupling of center-of-mass and relative-motion coordinates is explicitly considered using a configuration-interaction (exact-diagonalization) technique. Deviations from the harmonic approximation are discussed for several heteronuclear alkali-metal atom pairs trapped in a single site of an optical lattice. The consequences are discussed for the analysis of a recent experiment [C. Ospelkaus et al., Phys. Rev. Lett. 97, 120402 (2006)] in which radio-frequency association was used to create diatomic molecules from a fermionic and a bosonic atom and to measure their binding energies close to a magnetic Feshbach resonance.

DOI
10.1103/physreva.80.013403

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A theoretical approach was developed for an exact numerical description of a pair of ultracold atoms interacting via a central potential, which is trapped in a three-dimensional optical lattice. The coupling of center-of-mass and relative-motion coordinates is explicitly considered using a configuration-interaction (exact-diagonalization) technique. Deviations from the harmonic approximation are discussed for several heteronuclear alkali-metal atom pairs trapped in a single site of an optical lattice. The consequences are discussed for the analysis of a recent experiment [C. Ospelkaus et al., Phys. Rev. Lett. 97, 120402 (2006)] in which radio-frequency association was used to create diatomic molecules from a fermionic and a bosonic atom and to measure their binding energies close to a magnetic Feshbach resonance.
A theoretical approach is described for an exact numerical treatment of a pair of ultracold atoms interacting via a central potential and that are trapped in a finite three-dimensional optical lattice. The coupling of center-of-mass and relative-motion coordinates is treated using an exact diagonalization (configuration-interaction) approach. The orthorhombic symmetry of an optical lattice with three different but orthogonal lattice vectors is explicitly considered as is the fermionic or bosonic symmetry in the case of indistinguishable particles.
A theoretical approach was developed that allows for a full numerical description of a pair of ultracold atoms trapped in a three-dimensional optical lattice. This approach includes the possible coupling between centre-of-mass and relative motion coordinates in a configuration-interaction type formulation. The atoms are allowed to interact by their full interaction potential that is, presently, only limited to be central. With the aid of the newly developed method deviations from the harmonic approximation are discussed for the heteronuclear 87 Rb- 40 K pair.
We study properties of two different atoms at a single optical lattice site at a heteronuclear atomic Feshbach resonance. We calculate the energy spectrum, the efficiency of rf association, and the lifetime as a function of magnetic field and compare the results with the experimental data obtained for 40K^{40}\text{K}40K and 87Rb^{87}\text{R}\text{b}87Rb [C. Ospelkaus et al., Phys. Rev. Lett. 97, 120402 (2006)]. We treat the interaction in terms of a regularized $\ensuremath{\delta}$ function pseudopotential and consider the general case of particles with different trap frequencies, where the usual approach of separating center-of-mass and relative motion fails. We develop an exact diagonalization approach to the coupling between center-of-mass and relative motion and numerically determine the spectrum of the system. At the same time, our approach allows us to treat the anharmonicity of the lattice potential exactly. Within the pseudopotential model, the center of the Feshbach resonance can be precisely determined from the experimental data.
In this dissertation, we develop simple single-band effective Hamiltonians for a fermionic gas across a Feshbach resonance in both a quasi-low-dimensional optical trap and a three-dimensional optical lattice. The proper theoretical description of these systems is nontrivial due to the multiband nature of the dimers which form in the ground state of the strongly interacting gas. For the quasi-low-dimensional case, we show that the ground state of the gas has a significant population in the excited levels of the transverse trap. Hence, the effective low-dimensional (single-band) Hamiltonian must include the effects of this real population frozen in many transverse modes. We accomplish this through the definition of a "dressed molecule" state which incorporates the excited fraction. For the three-dimensional case, a valid single-band lattice model at first seems impossible for strong interactions. However, we find effective single-band lattice models that remain valid by subtly accounting for the nontrivial structure of the on-site dimer. We perform explicit numerical calculations for two fermions in a double-well to demonstrate the range of validity of the models and their parameter values. In the process of investigation leading to these models we find other interesting results such as a ground state level crossing for three fermions in a single well and a set of Feshbach resonances induced by anharmonicity of the optical potential. The ultracold neutral atom experiments related to the work presented within provide a flexible and well-controlled tool to observe aspects of quantum physics not easily accessible or controllable in other systems. The combination of optical lattice trapping techniques with the ability to manipulate the interactions via a Feshbach resonance allows for the study of novel strongly correlated physics. This work addresses an important and fundamentally interesting issue for these systems and serves as a proper starting point for further theoretical studies in conjunction with experiment.
We discuss interacting and noninteracting one dimensional atomic systems trapped in an optical lattice plus a parabolic potential. We show that, in the tight-binding approximation, the noninteracting problem is exactly solvable in terms of Mathieu functions. We use the analytic solutions to study the collective oscillations of ideal bosonic and fermionic ensembles induced by small displacements of the parabolic potential. We treat the interacting boson problem by numerical diagonalization of the Bose-Hubbard Hamiltonian. From analysis of the dependence upon lattice depth of the low-energy excitation spectrum of the interacting system, we consider the problems of ``fermionization'' of a Bose gas, and the superfluid-Mott insulator transition. The spectrum of the noninteracting system turns out to provide a useful guide to understanding the collective oscillations of the interacting system, throughout a large and experimentally relevant parameter regime.
We report on the creation of ultracold heteronuclear molecules assembled from fermionic 40K and bosonic 87Rb atoms in a 3D optical lattice. Molecules are produced at a heteronuclear Feshbach resonance on both the attractive and the repulsive sides of the resonance. We precisely determine the binding energy of the heteronuclear molecules from rf spectroscopy across the Feshbach resonance. We characterize the lifetime of the molecular sample as a function of magnetic field and measure lifetimes between 20 and 120 ms. The efficiency of molecule creation via rf association is measured and is found to decrease as expected for more deeply bound molecules.