Characterizing Feshbach resonances in ultracold scattering calculations

We describe procedures for converging on and characterizing zero-energy Feshbach resonances that appear in scattering lengths as a function of an external field. The elastic procedure is appropriate for purely elastic scattering, where the scattering length is real and displays a true pole. The regularized scattering length (RSL) procedure is appropriate when there is weak background inelasticity, so that the scattering length is complex and displays an oscillation rather than a pole, but the resonant scattering length $a_{\rm res}$ is close to real. The fully complex procedure is appropriate when there is substantial background inelasticity and the real and complex parts of $a_{\rm res}$ are required. We demonstrate these procedures for scattering of ultracold $^{85}$Rb in various initial states. All of them can converge on and provide full characterization of resonances, from initial guesses many thousands of widths away, using scattering calculations at only about 10 values of the external field.