Enhanced Sensitivity to Fundamental Constants In Ultracold Atomic and Molecular Systems near Feshbach Resonances

Scattering length, which can be measured in Bose-Einstein condensate and Feshbach molecule experiments, is extremely sensitive to the variation of fundamental constants, in particular, the electron-to-proton mass ratio ($m_e/m_p$ or $m_e/{\Lambda }_{\mathrm{QCD}}$, where ${\Lambda }_{\mathrm{QCD}}$ is the QCD scale). Based on single- and two-channel scattering models, we show how the variation of the mass ratio propagates to the scattering length. Our results suggest that variation of $m_e/m_p$ on the level of ${10}^{-11}–{10}^{-14}$ can be detected near a narrow magnetic or an optical Feshbach resonance by monitoring the scattering length on the 1% level. Derived formulas may also be used to estimate the isotopic shift of the scattering length.