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arXiv:2408.06905 · Acta Physica Polonica A

Nonexponential Decay Law of the 2P-1S Transition of the H Atom

Francesco Giacosa, Krzysztof Kyzioł

Published Nov 1, 2024• 2 citations
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Abstract

We evaluate numerically the survival probability P(t) for the unstable 2P excited state of the hydrogen atom that decays into the ground-state 1S by emitting one photon (τ ≈ 1.595 ns), thus extending the analytic study of Facchi and Pascazio, Phys. Lett. A 241, 139 (1998). To this end, we first determine the analytic expression of the spectral function of the unstable state, which allows for an accurate evaluation of P(t). As expected, for short and long time scales, P(t) shows deviations from the exponential law: a 'Zeno' region occurs at extremely short times (up to ≈ 0.3 attosec), followed by a longer 'anti-Zeno' domain (up to ≈ 50 attosec); at long times above 125 τ the decay law scales as t-4.

DOI
10.12693/aphyspola.146.704

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We evaluate numerically the survival probability P(t) for the unstable 2P excited state of the hydrogen atom that decays into the ground-state 1S by emitting one photon (τ ≈ 1.595 ns), thus extending the analytic study of Facchi and Pascazio, Phys. Lett. A 241, 139 (1998). To this end, we first determine the analytic expression of the spectral function of the unstable state, which allows for an accurate evaluation of P(t). As expected, for short and long time scales, P(t) shows deviations from the exponential law: a 'Zeno' region occurs at extremely short times (up to ≈ 0.3 attosec), followed by a longer 'anti-Zeno' domain (up to ≈ 50 attosec); at long times above 125 τ the decay law scales as t-4.
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.
The decaying behavior of both the survival S(t) and total P(t) probabilities for unstable multilevel systems at long times is investigated by using the N-level Friedrichs model. The long-time asymptotic-forms of both S(t) and P(t) are obtained for an arbitrary initial-state extending over the unstable levels. It is then clarified how the asymptotic forms depend on the initial population in unstable levels. In particular, a special initial state that maximizes the asymptotic form of both S(t) and P(t) is found. On the other hand, the initial states eliminating the first term of their asymptotic expnasions also exist, which implies that a faster decay rather than expected can be realized. This faster decay for S(t) is numerically confirmed by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field. It is demonstrated that the t-4-decay and a faster decay are realized depending on the initial states, where the latter is estimated as t-8.
The probabilities of two-photon decay for hydrogen (H) and antihydrogen ( Hˉ \bar H Hˉ ) atoms in the presence and absence of an external electric field are analytically calculated. In particular, the probabilities of the E1E2 and E1M1 transitions between the 2p and 1s levels are calculated for the case when emitted photons are characterized by polarization vectors and wavevectors. It is shown that, in an external electric field, the decay probabilities for 2s and 2p levels differ for H and Hˉ \bar H Hˉ atoms because of interference terms linear in field. Coulomb Green’s function method is used for summing over intermediate states.
The long-time behavior of the survival probability for unstable multilevel systems that follows the power-decay law is studied based on the NNN-level Friedrichs model, and is shown to depend on the initial population in unstable states. A special initial state maximizing the asymptote of the survival probability at long times is found and examined by considering the spontaneous-emission process for the hydrogen atom interacting with the electromagnetic field.
One- and two-photon transitions in the hydrogen atom are analytically evaluated in the absence and in the presence of an external electric field. The emission probabilities are different for the hydrogen and anti-hydrogen atoms due to the existence of contributions, linear in electric field. The magnitude of these contributions is evaluated within the nonrelativistic limit. The Coulomb Green function method is applied. Different nonrelativistic "forms" for the decay probabilities in combination with different gauge choices are considered. The three-photon E1E1E1 2p-1s transition probability is also evaluated and possible applications of the results are discussed.