= Lamb-Dicke regime
{c}
{title2=$\eta=|k_{\rm eff}|\sqrt{\hbar/(2m\nu)}$}
The Lamb-Dicke regime means the wave-induced <phase> change across the occupied motional wavepacket is small. For one <normal mode>, let $x_0=\sqrt{\hbar/(2m\nu)}$ be the <ground state>'s position <standard deviation>, and write $x=x_0(a+a^\dagger)$ using <creation and annihilation operators>. Here $m$ is <mass> and $\hbar$ the <Planck constant> divided by $2\pi$. The effective <wave vector> component $k_{\rm eff}$ is the single beam's component for a direct transition, or the difference of two beam components for a <Raman transition>. With dimensionless $\eta=|k_{\rm eff}|x_0$, the useful condition is $\eta\sqrt{2n+1}\ll1$ over the populated <number states>, rather than merely $\eta\ll1$.
Absorb the sign of $k_{\rm eff}$ into the pulse <phase>. Expanding the coupling factor gives
$$
e^{i\eta(a+a^\dagger)}=1+i\eta(a+a^\dagger)+O(\eta^2).
$$
The constant term preserves $n$; the <annihilation operator> gives $a|n\rangle=\sqrt n|n-1\rangle$; the <creation operator> gives $a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$. The different square roots explain why removing a motional quantum becomes impossible at $n=0$, while adding one remains possible. Smallness of $\eta$ controls this spatial expansion; resolving the resulting transitions is a separate <frequency> requirement.
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