An electric dipole transition is a transition driven, in the long-wavelength approximation, by the electric-dipole interaction . Its leading amplitude is proportional to a matrix element of the electric dipole moment between the initial and final states. A vanishing element gives an electric-dipole selection rule; a nonzero element alone does not ensure a large probability, since the drive frequency, duration and strength also matter. For a charged quantum harmonic oscillator driven along its axis, gives the single-quantum rule .
A coherent transition involving two atom-field interactions. two-photon absorption uses a sum of photon energies; stimulated absorption-emission uses their difference. A second-order intermediate contribution need not represent real population in a stationary atomic state.
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.
The matrix element of the electric dipole moment operator between two quantum states. Electric-dipole transition strength is proportional to its squared magnitude; molecular symmetry can force every Cartesian component to vanish.
