= Motional sideband
{title2=$\omega_{\rm drive}=\omega_0,\ \omega_0-\nu,\ \omega_0+\nu$}
A motional sideband is a transition that changes internal excitation and a motional <quantum number> together. Define the internal <energy> gap as $\hbar\omega_0>0$. A carrier drives $|g,n\rangle\leftrightarrow|e,n\rangle$; the first red sideband drives $|g,n\rangle\leftrightarrow|e,n-1\rangle$; the first blue sideband drives $|g,n\rangle\leftrightarrow|e,n+1\rangle$. Their resonant drive <angular frequencies> follow by subtracting these <energy levels>. For a <Raman transition>, the drive is a beam-difference <frequency> with a sign fixed by the selected internal transition.
Choose a real carrier <Rabi angular frequency> $\Omega$ defined so that resonant carrier excitation is $\sin^2(\Omega t/2)$. In the <Lamb-Dicke regime>, isolated first-sideband <Hamiltonians> in a rotating frame can be written, after choosing the pulse <phase>, as
$$
H_r=\frac{\hbar\eta\Omega}{2}(\sigma_+a+\sigma_-a^\dagger),\qquad
H_b=\frac{\hbar\eta\Omega}{2}(\sigma_+a^\dagger+\sigma_-a),\qquad
\sigma_+=|e\rangle\langle g|,\quad\sigma_-=\sigma_+^\dagger.
$$
These formulas use a <rotating-wave approximation>: neglected carrier and other-sideband terms oscillate rapidly compared with the retained coupling. The drive's effective width and sideband coupling must be small compared with $\nu$; sufficient suppression of off-resonant carrier excitation must also be checked. The red interaction is a <Jaynes-Cummings model> with a motional quantum replacing a cavity <photon>. The blue interaction creates internal excitation and a motional quantum together; the external <laser> supplies <energy>. The driven-system rotating-frame <Hamiltonian> alone is not the total laboratory <energy>.
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