Nonresonant corrections for the optical resonance frequency measurements in the hydrogen atom

The deviation of the natural spectral line profile from the Lorentz shape for the optical resonant frequency measurements is considered. This deviation leads to an asymmetry, which is mainly due to nonresonant correction to the resonant Lorentz profile. The nonresonant corrections are studied for the different types of the atomic resonant experiments. The most accurate recent optical resonance experiments are analyzed, i.e., the two-photon $1s\text{−}2s$ resonance excitation of the hydrogen atom with the delayed decay in the external electric field. The description of the nonresonant correction in the latter case requires the employment of QED with different in and out Hamiltonians. The nonresonant corrections for this experiment are investigated and found to be about ${10}^{-5}\text{Hz}$, while the recent experimental uncertainty is 34 Hz and in the near feature is expected to be a few hertz. The projected $1s\text{−}2s$ resonance excitation experiment with the three-photon ionization detection (which is now in progress) is also considered.

Figure 1

The Feynman graphs describing the elastic photon scattering on atomic electron. The solid line describes the electron in the field of the nucleus; the wavy line with an arrow denotes the absorption or the emission of a photon. Symbol $a$ corresponds to the certain electron level $\left(njl\right)$ in an atom; $\omega$ and ${\omega }^{\prime }$ denote the absorbed and emitted photon frequencies. The right-hand side of both graphs corresponds to the initial state and the left-hand side corresponds the final state. In case of the elastic scattering (the initial and final atomic states are the same) $\omega ={\omega }^{\prime }$.

Figure 2

The diagrammatic representation of the FGS electron propagator [Eq. (33)] in the coordinate space. See explanations in the text.

Figure 3

The process of the two-photon excitation $1s\text{−}2s$ followed by the delayed decay in an external electric field. The picture is understood with respect to the frame of reference of the hydrogen atom. The ordinary external and internal solid lines describe the electron wave function and electron propagator for the bound electron in the absence of external electric field. The double internal electron line represents the FGS electron propagator as described in Fig. 2. The external double solid line corresponds to the electron wave function in the external electric field. The two absorbed photons are the laser photons with frequencies $\omega =\frac{1}{2}\left(E_{2s}-E_{1s}\right)$, where $E_i$ are the energies of the atomic electron states in the absence of the electric field. The notations $n,\tilde{n}$ for the wave functions were introduced in Sec. 5.

Figure 4

Scheme of the levels for the two-photon $1s\text{−}2s$ transition in hydrogen. The vertical double lines denote the two-photon transitions. The $K$ numbers denote the total angular momentum for a two-photon system, possible for the different two-photon transitions.

Figure 5

The Feynman graph corresponding to the process of three-photon ionization of hydrogen atom. Here $\epsilon jl$ are the quantum numbers of the electron in continuous spectrum with the energy $\epsilon =3\omega -\left|E_{1s}\right|=\frac{1}{16}\text{a}\text{.}\text{u}\text{.}$