Quantum Defect Theory for Orbital Feshbach Resonance

In the ultracold gases of alkali-earth (like) atoms, a new type of Feshbach resonance, i.e., the orbital Feshbach resonance (OFR), has been proposed and experimentally observed in ultracold $^{173}$Yb atoms. When the OFR of the $^{173}$Yb atoms occurs, the energy gap between the open and closed channels is smaller by two orders of magnitudes than the van der Waals energy. As a result, quantitative accurate results for the low-energy two-body problems can be obtained via multi-channel quantum defect theory (MQDT), which is based on the exact solution of the Schr$\ddot{{\rm o}}$dinger equation with the van der Waals potential. In this paper we use the MQDT to calculate the two-atom scattering length, effective range, and the binding energy of two-body bound states for the systems with OFR. With these results we further study the clock-transition spectrum for the two-body bound states, which can be used to experimentally measure the binding energy. Our results are helpful for the quantitative theoretical and experimental researches for the ultracold gases of alkali-earth (like) atoms with OFR.