Feshbach Resonance in a Tight-Binding Model

TL;DR

This paper provides an analytic analysis of Feshbach resonance within a two-channel tight-binding model, revealing new conditions for resonance and deriving key scattering parameters relevant for ultra-cold atomic systems.

Contribution

It introduces a unified analytic framework for bound states and resonances, showing Feshbach resonance can occur without a bound state in the closed channel, and derives explicit expressions for scattering parameters.

Findings

Feshbach resonance can occur without a bound state in the closed channel for strong coupling.

Analytic expressions for scattering length, resonance magnetic field, and energy shifts are derived.

The sign of the scattering length depends on coupling strength in the unitary limit.

Abstract

The physics of Feshbach resonance is analyzed using an analytic expression for the $s$-wave scattering phase-shift and the scattering length $a$ which we derive within a two-channel tight-binding model. Employing a unified treatment of bound states and resonances in terms of the Jost function, it is shown that for strong inter-channel coupling, Feshbach resonance can occur even when the closed channel does not have a bound state. This may extend the range of ultra-cold atomic systems that can be manipulated by Feshbach resonance. The dependence of the sign of $a$ on the coupling strength in the unitary limit is elucidated. As a by-product, analytic expressions are derived for the background scattering length, the external magnetic field at which resonance occurs, and the energy shift $\varepsilon-\varepsilon_B$, where $\varepsilon$ is the scattering energy and $\varepsilon_B$ is the bound state energy in the closed channel (when there is one).