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arXiv:1011.0599 · Physical Review A

Bound states of two bosons in an optical lattice near an association resonance

Jerome Sanders, Otim Odong, Juha Javanainen, Matt Mackie

Published Mar 21, 2011• 13 citations
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Abstract

We model two bosons in an optical lattice near a Feshbach or photoassociation resonance, focusing on the Bose-Hubbard model in one dimension. Whereas the usual atoms-only theory with a tunable scattering length yields one bound state for a molecular dimer for either an attractive or repulsive atom-atom interaction, for a sufficiently small direct background interaction between the atoms a two-channel atom-molecule theory may give two bound states that represent attractively and repulsively bound dimers occurring simultaneously. Such unusual molecular physics may be observable for an atom-molecule coupling strength comparable to the width of the dissociation continuum of the lattice dimer, which is the case, for instance, with narrow Feshbach resonances in Na, $^{87}\mathrm{Rb}$, and $^{133}\mathrm{Cs}$ or low-intensity photoassociation in $^{174}\mathrm{Yb}$.

DOI
10.1103/physreva.83.031607

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We theoretically and numerically study the Bose-Hubbard model for two bosons in a one-dimensional optical lattice. We begin with a single-channel model describing two interacting atoms, known as a lattice dimer, and then examine the more nuanced two-channel model. The two-channel model accounts for the presence of an association resonance that facilitates the conversion of an atom pair into a molecule and adds a separate molecular degree of freedom. Lattice dimers in the single-channel case have been extensively investigated, but we resolve the conundrum between these dimers and molecules by employing a two-channel model. The stationary states for the time-independent Schrödinger equation are determined and these states reveal some unusual features, including the existence of the celebrated repulsively bound state. We examine the similarities and differences between the single- and two-channel models and find a peculiar qualitative difference between the two. Whereas the usual atoms-only theory yields one bound state for a molecular dimer for either an attractive or repulsive atom-atom interaction, a two-channel atom-molecule theory may give two bound states that represent attractively and repulsively bound dimers occurring simultaneously. Possible methods for experimentally detecting both the dimers and the simultaneous bound states are discussed and we identify several systems for observing this novel aspect of molecular physics. In addition to our theoretical work, we also present the results and verification of our quantum Monte Carlo simulations of the two-channel model. ^
We analyze the scattering and bound state physics of a pair of atoms in a one-dimensional optical lattice interacting via a narrow Feshbach resonance. The lattice provides a structured continuum allowing for the existence of bound dimer states both below and above the continuum bands, with pairs above the continuum stabilized by either repulsive interactions or their center-of-mass motion. Inside the band the Feshbach coupling to a closed channel bound state leads to a Fano resonance profile for the transmission, which may be mapped out by rf or photodissociative spectroscopy. We generalize the scattering length concept to the one-dimensional lattice, where a scattering length may be defined at both the lower and the upper continuum thresholds. As a function of the applied magnetic field the scattering length at either band edge exhibits the usual Feshbach divergence when a bound state enters or exits the continuum. Near the scattering length divergences the binding energy and wave function of the weakly bound dimer state acquires a universal form reminiscent of those of free-space Feshbach molecules. We give numerical examples of our analytic results for a specific Feshbach resonance, which has been studied experimentally.
We study the resonant effects produced when a Feshbach dimer crosses a scattering continuum band of atoms in an optical lattice. We numerically obtain the exact spectrum of two particles in a one-dimensional lattice and develop an effective atom-dimer Hamiltonian that accurately captures resonant effects. The lattice-induced resonances lead to the formation of bound states simultaneously above and below the scattering continuum and significantly modify the curvature of the dimer dispersion relation. The nature of the atom-dimer coupling depends strongly on the parity of the dimer state leading to a novel coupling in the case of negative parity dimers.
We numerically study the problem of two fermions in a three-dimensional optical lattice interacting via a zero-range Feshbach resonance and display the dispersions of the bound states as a two-particle band structure with unique features compared to typical single-particle band structures. We show that the exact two-particle solutions of a projected Hamiltonian may be used to define an effective two-channel, few-band model for the low-energy, low-density physics of many fermions at arbitrary s-wave scattering length. Our method applies to resonances of any width and can be adapted to multichannel situations or higher-ℓ pairing. In strong contrast to usual Hubbard physics, we find that pair hopping is significantly altered by strong interactions and the presence of the lattice, and the lattice induces multiple molecular bound states.
We present the theory of an atomic gas in an optical lattice near a Feshbach resonance. We derive from first principles a generalized Hubbard model, that incorporates all the relevant two-body physics exactly, except for the background atom-atom scattering. For most atoms the background interactions are negligible, but this is not true for 6Li, which has an exceptionally large background scattering length near the experimentally relevant Feshbach resonance at 834 G. Therefore, we show how to include background atom-atom scattering by solving the on-site two-body Feshbach problem exactly. We apply the obtained solution to 6Li and find that the background interactions indeed have a significant effect in this case.
We present the theory of an atomic gas in an optical lattice near a Feshbach resonance. We derive from first principles a generalized Hubbard model that incorporates all the relevant two-body physics exactly, except for the background atom–atom scattering. For most atoms, the background interactions are negligible, but this is not true for 6 Li, which has an exceptionally large background scattering length near the experimentally relevant Feshbach resonance at 834 G. Therefore, we show how to include background atom–atom scattering by solving the on-site two-body Feshbach problem exactly. We apply the obtained solution to 6 Li and find that the background interactions indeed have a significant effect in this case.