Comment on ‘‘Lifetime of excited atomic states’’

Abstract

We show that the two-photon nonresonant contributions to the lifetime of excited 3s and 3d states of hydrogen, as given by a formula recently derived by Cresser et al. [Phys. Rev. A 33, 1677 (19S6)],can be evaluated in an economic way using the analytic expressions of the amplitudes of 3s-1s and 3d-1s two-photon transitions. Our numerical evaluation confirms the numbers given by Cresser et al. In a recent paper Cresser et al. ' have considered the contribution of two-photon emission to the lifetime of an excited state which can also decay by one-photon emission. Using time-dependent perturbation theory up to fourth order and Markov approximation, they have separated in the reciprocal of the lifetime of an initial state I i ) two contributions [Eq. (Ig) of Ref. I], I (1)+ I (2) 0& --E; -%co&, and interchanging the polarization vectors 2 2and e2. m, is the electron mass, co& and co2 the two photon frequencies connected by %co, +fico2 --E, -E, . We refer in the following to an initial state with n; =3 and defined orbital momentum l; (0 or 2). We complete XI and Xz with the contribution of the intermediate 2p and 3p states, obtaining in the case of X & with I ', " the spontaneous decay rate of the state I i ) due to one-photon emission, and I', ' the contribution of nonresonant intermediate transitions to all the states of energy higher than the initial-state energy E;. In Ref. 1

Relativistic calculation of the two-photon decay rate of highly excited ionic states

“…In order to avoid problems with non-integrable singularities, the authors of [9] Although the results presented in Refs. [9,10] are in mutual agreement, they are both based on the interpretation of the two-photon decay rate Γ…”

“…One of these problems, which currently attracts a lot of interest, concerns those two-photon transition from highly excited states to the ground state which pass through an intermediate state with a lower energy than the initial state of the two-photon process [9,10] and can alternatively decay to the groundstate via two (or more) sequential one-photon emissions. Such a transition leads to resonance peaks in the energy spectrum of the coherently emitted photons from the two-photon decays which are located at the energies corresponding to the (real) intermediate states.…”

“…Since the seminal work of Göppert-Mayer [1], two-photon decay rates of excited states in hydrogen-like atoms and ions have been the subject of intense experimental [2,3,4,5] and theoretical [6,7,8,9,10,11,12,13] studies. For many years, the investigations have dealt not only with the total decay rates but also with the energy and even angular distributions of the two emitted photons.…”

Two-photon transitions in hydrogen and cosmological recombination

“…explicitly neglecting the transition via the intermediate 3p-state, we can reproduce their values for the non-resonant contribution to the two-photon decay rates of the 3s → 1s and 3d → 1s transitions. Later Florescu et al (1988) computed these values again within the framework of Cresser et al (1986) and A nr,Cr 3d→1s = 0.131814 s −1 ) up to all given figures. Although the discussion in the previous sections has already shown that these values probably have no direct relation to the total corrections in the lifetime of the level due to the two-photon process, we computed them to check our own computational procedure.…”

Two-photon approximation in the theory of electron recombination in hydrogen

“…At nearly the same time as the arrival of the paper [18], the cascade problem was discussed also for the ns (n > 2), nd transitions in the hydrogen atom [23,24]. In these works the pure two-photon contribution was obtained simply by omitting the resonant (singular) terms responsible for the cascades.…”