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Journal of Physics B Atomic Molecular and Optical Physics

Non-Markovian study of the relativistic magnetic-dipole spontaneous emission process of hydrogen-like atoms

Chang-qi Cao, Xiaowei Fu, Hui W. Cao

Published Apr 10, 2006• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/39/8/022

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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 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 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.
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.
We show how the radial position–momentum uncertainty product can be obtained analytically for the Dirac hydrogen-like atoms. Some interesting features for this system are found. First, for the same principal quantum number n , as the azimuthal quantum number l increases, the uncertainties Δ p r and Δ r Δ p r decrease. However, the uncertainty Δ r does not always decrease, which is different from the non-relativistic hydrogen-like atoms case, where it always decreases. Second, for the same l both Δ r and Δ r Δ p r increase as n increases. Third, all uncertainties for the same n and l = n − 1 are smallest in comparison with those for the same n but . Fourth, the uncertainty Δ r Δ p r is independent of the value of charge Z in the non-relativistic case, while it is related with the value of charge Z in the relativistic case. Fifth, the relativistic corrections to the non-relativistic values of uncertainties are very small when the values of charge Z are not too big, while the relativistic corrections to them will appear explicitly for a large value of charge Z .
We show how the radial position–momentum uncertainty product can be obtained analytically for Klein–Gordon hydrogen-like atoms. Some interesting features of this quantum system are found. First, for the same principal quantum number n, when the angular momentum quantum number l increases, the uncertainties Δpr and ΔrΔpr decrease. However, Δr does not always decrease, which is different from the non-relativistic hydrogen-like atoms case, where it always decreases. Second, for same l both Δr and ΔrΔpr increase as n increases. Third, all uncertainties for same n and l = n − 1 are smallest in comparison with those for same n but l ≠ n − 1. Fourth, ΔrΔpr is independent of the value of charge Z in the non-relativistic case, while it is related to the value of charge Z in the relativistic case. Fifth, the relativistic corrections to the non-relativistic uncertainties are very small when the values of the charge Z are not too big. However, relativistic corrections to them will appear explicitly for large Z.