 Solution

Write $g=\hslash \Omega /2$, with Rabi frequency $\Omega$, for the internal coupling and use normalized oscillator states. The motional carrier transition matrix element is the Gaussian characteristic function,In the Lamb-Dicke regime, $M_{00}=g[1-{\eta }^2/2+O({\eta }^4)]$. The diagonal internal-energy terms cannot connect $g$ to $e$.

If the two unnormalized printed functions are used verbatim, the literal ground-state integral is instead $g\sqrt{2}\exp (-k^2{l}_0^2/2)$. This is not a matrix element between normalized eigenstates of the stated trap; even its zero-recoil value is incorrectly $\sqrt{2}g$.