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Physical Review A

Radiative decay of hydrogenlike atoms in an electric field

Mark Hilley, Peter J. Mohr

Published Jan 1, 1980• 50 citations
Publisher

Abstract

Photon emission by an atom in the metastable $2S$ state in a weak static electric field is examined within the context of bound-state quantum electrodynamics. Low's method is applied to obtain the expansion in terms of Coulomb wave functions of the propagation function for an electron interacting with the nuclear Coulomb field, the applied electric field, and the quantized radiation field. For a weak applied field, and to lowest relative order in $\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}$, the $2S$-state resonant contribution to the propagation function can be factorized to provide the basis for a calculation of the angular distribution of single-photon radiation and lifetime of the excited state. A numerical evaluation is given, which takes into account effects of electric dipole, magnetic dipole, and magnetic quadrupole emission, relativistic corrections, and finite level widths.

DOI
10.1103/physreva.21.24

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Photon emission by an atom in the metastable 2S2S2S state in a weak static electric field is examined within the context of bound-state quantum electrodynamics. Low's method is applied to obtain the expansion in terms of Coulomb wave functions of the propagation function for an electron interacting with the nuclear Coulomb field, the applied electric field, and the quantized radiation field. For a weak applied field, and to lowest relative order in \ensuremathα\ensuremathπ\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}\ensuremathπ\ensuremathα​, the 2S2S2S-state resonant contribution to the propagation function can be factorized to provide the basis for a calculation of the angular distribution of single-photon radiation and lifetime of the excited state. A numerical evaluation is given, which takes into account effects of electric dipole, magnetic dipole, and magnetic quadrupole emission, relativistic corrections, and finite level widths.
This paper presents a quantum-field-theoretic formulation of the hydrogen 2p → 1s Lyman-α transition, replacing the semiclassical Bohr–Maxwell description with the modern framework of quantum electrodynamics. The initial and final bound-electron states are described by the Dirac equation in the proton’s Coulomb field, with the Schrödinger equation recovered as the leading nonrelativistic limit. The electromagnetic radiation field is quantized, and photon emission is generated by the QED interaction Hamiltonian. The transition amplitude is evaluated from the atomic current and quantized electromagnetic field, leading through Fermi’s golden rule to the spontaneous-emission rate and Einstein coefficient A₂₁. Energy and momentum conservation require the emitted photon to carry the atomic transition energy, including the small correction associated with proton recoil. At leading order, the Lyman-α photon has ΔE ≈ 10.2 eV, ν ≈ 2.47 × 10¹⁵ Hz, and λ ≈ 121.6 nm. In the electric-dipole approximation, the field-theoretic calculation reduces to the standard matrix element of the interaction Hamiltonian H_int = −μ·E, thereby connecting QED directly to the transition dipole moment, oscillator strength, Einstein A coefficient, and radiative lifetime. The calculation reproduces the accepted numerical results for hydrogen while identifying their origin in the coupling between the quantized electron field and the quantized electromagnetic field. The paper also explains how classical electrodynamics is recovered from QED. Maxwell’s equations govern the expectation values of the electromagnetic-field operators in the appropriate classical-field limit, particularly for coherent states containing many photons. A single spontaneously emitted Lyman-α photon, however, remains a genuinely quantum excitation rather than a classical electromagnetic wave. Leading corrections beyond the nonrelativistic Coulomb approximation are then considered, including relativistic fine structure, the Lamb shift, and atomic recoil. The resulting treatment provides a rigorous bridge from the historical Bohr model and Schrödinger wave mechanics to Dirac theory and full QED, verifying the Lyman-α energy, frequency, wavelength, transition probability, and radiative decay structure within a unified quantum-field framework.
The angular correlations in the two-photon decay of hydrogenlike ions are studied within the framework of second-order perturbation theory, based on Dirac's equation. Particular attention has been paid to the effects which arise from the higher (nondipole) terms in the expansion of the electron-photon interaction. It is shown that the photon-photon angular correlation function, which is found symmetric with respect to the angle $\ensuremath{\theta}=90\ifmmode^\circ\else\textdegree\fi{}$ in the electric dipole approximation, becomes asymmetric because of the nondipole contributions, and that this effect is enhanced as the nuclear charge ZZZ increases. Detailed computations on the photon-photon angular distribution have been carried out for the $2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ and $3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ transitions in neutral hydrogen (H) as well as for hydrogenlike xenon (Xe53+)({\mathrm{Xe}}^{53+})(Xe53+) and uranium (U91+)({\mathrm{U}}^{91+})(U91+) ions, and are compared with previous nonrelativistic results by Au [Phys. Rev. A 14, 531 (1976)]).
The interaction of a low-intensity laser field with a nonrelativistic one-electron atom is considered to the first order of perturbation theory and in the electric dipole approximation. The radiation field is turned on adiabatically modifying the initial unperturbed atomic state, which is either an angular-momentum eigenstate ∣nlm〉|\mathrm{nlm}〉∣nlm〉 or a Stark state ∣nnem〉|{\mathrm{nn}}_{e}m〉∣nne​m〉. The first-order correction to the wave function is expressed both in the length and velocity gauges in terms of a vector function called the linear-response vector and depending on the field-free energy eigenstate. We derive in a unifying manner the linear-response vectors in the position representation, as closed-form contour integrals, starting from a unique generating function built up with the Coulomb Green function. The linear-response vectors are then evaluated in momentum space via a Fourier transformation: they are obtained as integral representations and also in explicit form, as generalized hypergeometric functions. With reference to the static limit, we complement earlier results and find the reduced linear-response vectors in momentum space, as Fourier transforms of their coordinate-representation counterparts. The low-frequency behavior of the length-gauge first-order correction to the ground-state wave function is established in coordinate as well as in momentum space. We finally point out the high-frequency limit of the linear response in the velocity gauge.
The mean life of the 2S state of hydrogen in the absence of collisions is primarily determined by the simultaneous emission of two photons. Owing to this cause the mean life is `-*4 sec. It is expected that the mean life of is 2S `S of He i is of the same order of magnitude and is much shorter than that of is 2S 3S of He i. The non-adiabatic collisions with electrons are found to be more important than the static effects at small electron densities, leading to a transition probability `-*-`~h sec ` for an electron temper- ature `-.--`io,ooo° C and an electron density 30 cm3. The effect of static electric fields on the 2S state of hydrogen is worked out in more detail than previously by taking into account the hyperfine structure. The mean life in static electric fields which are strong enough to make double emission of little importance and yet weak enough to give a mean life > io7 sec is found to be the sum of two lives: (i) the mean life calculated neglecting hyperfine structure and (2) the mean life calculated neglecting radiation coupling. In both (i) and (2) the mixing of 2~ into 2S by the Stark effect is taken into account. Radiations due to quadrupole and magnetic dipole effects are considered qualita- tively for is 2S 3S of helium and quantitatively for 2S of hydrogen. For the latter there exists a small radiative magnetic dipole effect on Dirac's theory of the electron which is absent in the ordinary spin model
The non-Markovian corrections to electric-dipole emission of hydrogen-like atoms of large atomic number Z are studied by multipole photon field formulation. The correlation functions are derived by the corresponding spectra, which show how the δ-function type correlation is approached. The decay of upper-level population is calculated by the relevant integro-differential equation. The numerical results show that the relativistic and finite atom-size corrections are evident; while the non-Markovian corrections are negligible in the main period of decay, they can manifest themselves in the far remote tail.
The thermal two-photon level broadening of the excited energy levels in the hydrogen and H-like helium is evaluated via the imaginary part of thermal two-loop self-energy correction for bound electrons. All the derivations are presented in the framework of rigorous quantum electrodynamic theory at finite temperatures and are applicable for the H-like ions. On this basis, we found a contribution to the level broadening induced by the blackbody radiation which is fundamentally different from the usual line broadening caused by the stimulated two-photon decay and the Raman scattering of thermal photons. Numerical calculations of the two-loop thermal correction to the two-photon width for the 2s2s2s state in hydrogen and singly ionized helium atoms show that the effect could significantly exceed the higher-order relativistic and radiative QED corrections commonly included in the calculations. In addition, the thermal two-loop self-energy correction significantly exceeds the ``ordinary'' stimulated one-photon depopulation rate at the relevant laboratory temperatures. In this work, detailed analysis and the corresponding comparison of the effect with the existing laboratory measurements in H-like ions are carried out.