Two-photon absorption ($E_f-E_i=\hslash ({\omega }_1+{\omega }_2)$)

absorption of two photons whose summed energy matches an atomic transition. In the weak-field regime its rate scales with the product of the two intensities. Two electric-dipole interactions preserve the initial parity.

An intermediate contribution to a second-order transition amplitude need not be an occupied, on-shell stationary atomic state. Both time orderings must be summed with their energy denominators. Large detuning can suppress real intermediate occupation while preserving a coherent two-photon coupling. Near resonance, finite linewidth, saturation and sequential population transfer must be treated explicitly.

Absorbing one photon from each of two opposing equal-frequency beams cancels the first-order Doppler shift through their nearly zero summed wavevector. Absorbing both photons from the same beam does not share this cancellation and can create a broad background.

For identical equal-frequency electric-dipole photons, exchange symmetry removes the antisymmetric rank-one transition tensor. Ranks zero and two survive. A zero-to-zero angular-momentum transition can occur, while zero-to-one is forbidden in this approximation; the rank-two triangle condition still permits some nonzero-to-nonzero transitions with a change of one.