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arXiv:cond-mat/0605056 · New Journal of Physics

Dressed molecules in an optical lattice

K. B. Gubbels, D. B. M. Dickerscheid, Henk T. C. Stoof

Published Aug 30, 2006• 11 citations
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Abstract

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.

DOI
10.1088/1367-2630/8/8/151

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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.
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.
A Feshbach resonance in the s-wave scattering length occurs if the energy of the two atoms in the incoming open channel is close to the energy of a bound state in a coupled closed channel. Starting from the microscopic Hamiltonian that describes this situation, we derive the effective atom–molecule theory for a Bose gas near a Feshbach resonance. In order to take into account all two-body processes, we have to dress the bare couplings of the atom–molecule model with ladder diagrams. This results in a quantum field theory that exactly reproduces the scattering amplitude of the atoms and the bound-state energy of the molecules. Since these properties are incorporated at the quantum level, the theory can be applied both above and below the critical temperature of the gas. Moreover, making use of the true interatomic potentials ensures that no divergences are encountered at any stage of the calculation. We also present the mean-field theory for the Bose–Einstein condensed phase of the gas.
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, 87Rb^{87}\mathrm{Rb}87Rb, and 133Cs^{133}\mathrm{Cs}133Cs or low-intensity photoassociation in 174Yb^{174}\mathrm{Yb}174Yb.
We present a theory for ultracold atomic gases in an optical lattice near a Feshbach resonance. In the single-band approximation the theory describes atoms and molecules that can both tunnel through the lattice. Moreover, an avoided crossing between the two-atom and the molecular states occurs at every site. We determine the microscopic parameters of the generalized Hubbard model that describes this physics, using the experimentally known parameters of the Feshbach resonance in the absence of the optical lattice. As an application we also calculate the zero-temperature phase diagram of an atomic Bose gas in an optical lattice.
We analyze a system of two-component fermions, which interact via a Feshbach resonance in the presence of a three-dimensional lattice potential. By expressing a two-channel model of the resonance in the basis of Bloch states appropriate for the lattice, we derive an eigenvalue equation for the two-particle bound states, which is nonlinear in the energy eigenvalue. Compact expressions for the interchannel matrix elements, numerical methods for the solution of the nonlinear eigenvalue problem, and a renormalization procedure to remove ultraviolet divergences are presented. From the structure of the two-body solutions we identify the relevant degrees of freedom, which describe the resonance behavior in the lowest Bloch band. These degrees of freedom, which we call dressed molecules, form an effective closed channel in a many-body model of the resonance, the Fermi resonance Hamiltonian (FRH). It is shown how the properties of the FRH can be determined numerically by solving a projected lattice two-channel model at the two-particle level. As opposed to single-channel lattice models such as the Hubbard model, the FRH is valid for general sss-wave scattering length and resonance width. Hence, the FRH provides an accurate description of the BEC-BCS crossover for ultracold fermions on an optical lattice.
We present a detailed analysis of the two-channel atom-molecule effective Hamiltonian for an ultracold two-component homogeneous Fermi gas interacting near a Feshbach resonance. We particularly focus on the two-body and many-body properties of the dressed molecules in such a gas. An exact result for the many-body T matrix of the two-channel theory is derived by both considering coupled vertex equations and functional-integral methods. Making use of this result allows us to incorporate exactly into the many-body theory the two-body physics of the Feshbach scattering by means of simple analytical formulas without any fitting parameters. New interesting many-body effects are discussed in the case of narrow resonances. We give also a description of the BEC-BCS crossover above and below TC{T}_{C}TC​. The effects of different approximations for the self-energy of the dressed molecules are discussed. The single-channel results are derived as a special limit for broad resonances. Moreover, through an analytic analysis of the BEC limit, the relation between the composite boson of the single-channel model and the dressed-molecule of the two-channel model is established.