Resonant Feshbach scattering of fermions in one-dimensional optical lattices

We consider Feshbach scattering of fermions in a one-dimensional optical lattice. By formulating the scattering theory in the crystal momentum basis, one can exploit the lattice symmetry and factorize the scattering problem in terms of center-of-mass and relative momentum in the reduced Brillouin zone scheme. Within a single band approximation, we can tune the position of a Feshbach resonance with the center-of-mass momentum due to the non-parabolic form of the energy band.

💡 Research Summary

The paper presents a comprehensive theoretical study of Feshbach scattering of two fermionic atoms confined in a one‑dimensional optical lattice. Building on the well‑established two‑channel description of magnetic Feshbach resonances, the authors reformulate the scattering problem in the crystal‑momentum (Bloch) basis, thereby exploiting the discrete translational symmetry of the lattice. In this basis the two‑particle state can be factorized into a conserved center‑of‑mass quasi‑momentum (K) and a relative quasi‑momentum (q) that lives in a reduced Brillouin zone. This factorization reduces the full many‑body problem to a set of independent one‑dimensional scattering problems, each labeled by a specific value of (K).

Within a single‑band approximation (the lowest Bloch band is assumed to dominate the low‑energy physics), the single‑particle dispersion is taken as the tight‑binding cosine form (E(k)=-2t\cos(ka)). Because this dispersion is non‑parabolic, the total two‑particle energy (E_{\text{tot}}(K,q)=E(K/2+q)+E(K/2-q)) depends non‑trivially on both (K) and (q). Consequently, the resonance condition—where the closed‑channel bound state (the “molecule”) becomes degenerate with the open‑channel scattering continuum—shifts as a function of the center‑of‑mass momentum. In other words, the position of the Feshbach resonance can be tuned by preparing the atom pair with a particular lattice quasi‑momentum, without altering the external magnetic field.

The authors derive the Lippmann‑Schwinger equation projected onto the Bloch basis and obtain an explicit expression for the T‑matrix:

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