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arXiv:1109.1510 · Physical Review Letters

Microscopic Model for Feshbach Interacting Fermions in an Optical Lattice with Arbitrary Scattering Length and Resonance Width

Michael L. Wall, Lincoln D. Carr

Published Jul 31, 2012• 15 citations
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Abstract

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.

DOI
10.1103/physrevlett.109.055302

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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 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 study the effective Bloch-wave scattering of a spinless Fermi gas in one-dimensional (1D) optical lattices. By tuning the odd-wave scattering length, we find multiple resonances of Bloch waves scattering at the bottom (and the top) of the lowest band, beyond which an attractive (and a repulsive) two-body bound state starts to emerge. These resonances exhibit comparable widths in the deep lattice limit, and the finite interaction range plays an essential role in determining their locations. Based on exact two-body solutions, we construct an effective microscopic model for the low-energy scattering of fermions. The model can reproduce not only the scattering amplitudes of Bloch waves at the lowest-band bottom or top, but also the attractive or repulsive bound states within a reasonably large energy range below or above the band. These results lay the foundation for quantum simulating topological states in cold Fermi gases confined in 1D optical lattices.
To test effective Hamiltonians for strongly interacting fermions in an optical lattice, we numerically find the energy spectrum for two fermions interacting across a Feshbach resonance in a double-well potential. From the spectrum, we determine the range of detunings for which the system can be described by an effective lattice model, and how the model parameters are related to the experimental parameters. We find that for a range of strong interactions the system is well described by an effective ttt-JJJ model, and the effective superexchange term, JJJ, can be smoothly tuned through zero on either side of unitarity. Right at and around unitarity, an effective one-band general Hubbard model is appropriate, with a finite and small onsite energy, due to a lattice-induced anharmonic coupling between atoms at the scattering threshold and a weakly bound Feshbach molecule in an excited center-of-mass state.
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.
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 study the exact solution for two atomic particles in an optical lattice interacting via a Feshbach resonance. The analysis includes the influence of all higher bands, as well as the proper renormalization of molecular energy in the closed channel. This exact solution allows for the precise determination of the parameters in the Hubbard model and the two-particle bound state energy. We identify the regime, where a single band Hubbard model fails to describe the scattering properties of the atoms as well as the bound states energies.