Relativistic recoil in radiative atomic transitions

Abstract

A modified time-dependent perturbation theory for single-photon emission is proposed, which yields the exact photon energy for a relativistically recoiling atom. The relevant unperturbed "Hamiltonian" has stationary eigenvalues E 2 . Its form is generalized here to hydrogenic atoms of arbitrary nuclear spins.When an atom of mass M emits or absorbs a single photon of momentum បk, it suffers a recoil, which is used for example in laser cooling. In emission from an atom at rest, the nonrelativistic recoil energy is P 2 /2M. From momentum conservation P + បk = 0 and with k = / c, the corresponding photon energy ប is ⌬E − ͑⌬E͒ 2 /2Mc 2 , where ⌬E = E − EЈ is the difference of the initial and final atomic energy levels. More precisely, EЈ is the total energy of the final atomic state in its own rest frame, which for the ground state is just Mc 2 (c 2 times the sum of the masses of its constituents, minus the total binding energy). Another nonrelativistic expression for ប is thus ⌬E − ͑⌬E͒ 2 /2EЈ. However, relativistic energy conservation requires E = ប + ͑EЈ 2 + ប 2 2 ͒ 1/2 , which leads to