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 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.
A relativistic and non-Markovian study is carried out on the magnetic-dipole emission of a hydrogen-like atom of large atomic number Z without perturbation approximation. The correlation spectra are derived analytically and the corresponding correlation functions are calculated numerically. The Markovian approximation is seen to be satisfied with more accuracy as compared with the case of electric-dipole emission while the relativistic corrections are quite larger. In transition 2S1/2 to 1S1/2, the relativistic values even become several times of the non-relativistic values. As a comparison with the magnetic-dipole emission, the electric-quadrupole emission in the case of the 2D3/2 to 1S1/2 transition is also studied.
The non-Markovian correlation spectra of spontaneous emission of an excited two-level atom are derived including the effect of finite size of the atom and all the possible contribution of allowed multipole radiations. The emission process is then analyzed by the quantum stochastic trajectory approach. The non-Markovian effect is counted in by expanding the original system to an enlarged system with Markovian reservoirs. In the case of a hydrogenlike atom with large atomic number Z, the deviation from the Weisskopf-Wigner result is quite evident.
In this paper, we extend to three-photon radiative transitions in hydrogenic ions our fully relativistic multipole approach [Z. Bona et al., Phys. Rev. A 89, 022514 (2014)], which has proven so far to yield extremely accurate results for two-photon emission processes. Closed-form formulas of double and single differential frequency distributions as well as total emission probabilities are derived for arbitrary multipole channels by using the Dirac-Coulomb Sturmian functions of the first order. Two cases of atomic systems are considered in the study, namely, ions with a spinless nucleus and those with a nonzero nuclear spin to check the validity of the Bose-Einstein statistics for three-photon emission investigated by Zalialiutdinov et al. [J. Phys. B: At. Mol. Opt. Phys. 49, 055001 (2016)]. In an effort to assess relativistic effects and the influence of the negative spectrum of the set of Dirac-Coulomb Sturmians, we formulate two nonrelativistic schemes that involve Schr\"odinger-Coulomb Sturmians, a transition operator with retardation, on one hand, and a transition operator in the long wavelength approximation, on the other hand. An application of these theories is made for the relativistic and nonrelativistic atomic transitions $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}, 2{p}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}, 2{p}_{3/2}\ensuremath{\rightarrow}2{s}_{1/2}$ and $2s\ensuremath{\rightarrow}1s, 2p\ensuremath{\rightarrow}1s$, respectively, with nuclear charge ranging from 1 to 100. Some of our numerical values are compared with the scarce data available in the literature and they enable us to draw inferences as to the improvements obtained with our approach, while the others are additional results that we provide as a further step for more investigations of three-photon decays.
For the first time to the author's knowledge, a rigorous mathematical treatment of the spontaneous 2P 1/2 to 1S 1/2 transition, going beyond the usual Markov approximation, is presented in the case of the Dirac hydrogen atom. The relevant transition matrix elements of the Dirac Hamiltonian are calculated without the dipole approximation. Further, a non-Markovian equation of motion for the non-decay probability of the unstable 2P 1/2 2 state is derived, by using a self-consistent projection-operator method recently developed by the present author. By a rigorous analytical treatment of the non-Markovian equation, deviations from exponential decay for asymptotic times are obtained. Moreover, by careful error estimations deviations from exponential decay for finite times with definite validity range are also calculated. Finally, for the radiative lineshape a new quasi-Lorentzian distribution is obtained.
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.
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.