= Solution
Write $g=\hbar\Omega/2$, with <Rabi frequency> $\Omega$, for the internal coupling and use normalized oscillator states. The <motional carrier transition> <matrix element> is the Gaussian <characteristic function>,
$$
M_{00}=g\int_{-\infty}^{\infty}\frac{dx}{\sqrt\pi l_0}
\exp(-x^2/l_0^2-ikx)
=\boxed{g\exp(-k^2l_0^2/4)=g e^{-\eta^2/2}.}
$$
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 $g\sqrt2\exp(-k^2l_0^2/2)$. This is not a <matrix element> between normalized <eigenstates> of the stated trap; even its zero-recoil value is incorrectly $\sqrt2g$.
Back to article page
