Consider an atomic two-level system with ground state \(|0\rangle\) and excited state \(|1\rangle\). Recall that in the interaction picture, after making the rotating wave approximation and boosting into a steady-state rotating frame, one had the resultant time-independent steady-state Hamiltonian:
\[H_{\infty}=\frac{\hbar}{2}\tilde{\boldsymbol{\Omega}}\cdot\boldsymbol{\sigma}\]
Invoking the identity of Pauli matrices \((\tilde{\boldsymbol{\Omega}}\cdot\boldsymbol{\sigma})^2=-|\tilde{\boldsymbol{\Omega}}|^21\), it is clear that the eigenvalues of this Hamiltonian are thus \(E_{\pm}=\pm\frac{\hbar|\tilde{\boldsymbol{\Omega}}|}{2}=\frac{\hbar\sqrt{\Omega^2+\delta^2}}{2}\), and this is known as the light shift resulting from the AC Stark effect (also called the Autler-Townes effect). In particular, if \(\Omega=0\) then \(E_{\pm}=\).
\[E_{\pm}=\pm\left(\frac{\hbar\delta}{2}+\frac{\hbar\Omega^2}{4\delta}\right)\]
It is not a coincidence that this light shift calculated from time-independent perturbation theory, after a first-order binomial expansion, to the result of first-order nondegenerate time-independent perturbation theory applied to … turns out in the framework of QED that these correspond to so-called dressed states of the atom-photon system.