# Natural line profile asymmetry

A. Anikin1,2, T. Zalialiutdinov1,3, D. Solovyev1 1 Department of Physics,
St.Petersburg State University, St.Petersburg, 198504, Russia
2 D. I. Mendeleev Institute for Metrology, St. Petersburg, 190005, Russia
3 National Research Center, Kurchatov Institute, B.P. Konstantinov Petersburg
Nuclear Physics Institute, Gatchina, Leningrad District 188300, Russia

###### Abstract

The paper discusses the line profile asymmetry of the photon scattering
process that arises naturally in quantum electrodynamics (QED). Based on
precision spectroscopic experiments conducted on hydrogen atoms, we focus our
attention on the two-photon $1s-2s$ transition. As one of the most precisely
determined transition frequencies, it is a key pillar of optical frequency
standards and is used in determining fundamental physical constants, testing
physical principles, and searching constraints on new fundamental
interactions. The results obtained in this work show the need to take into
account the natural line profile asymmetry in precision spectroscopic
experiments.

##  I Introduction

Precise measurements of transition frequencies in various atomic systems are
of paramount importance in modern physics. By providing a detailed
interpretation of a theory, spectroscopic experiments are able to identify
effects that can be used to enhance our understanding of physical processes,
test theoretical hypotheses, or verify fundamental physical principles. For
such attractive astrophysical applications as the search for Dark Matter
Kennedy  or the variation in time of physical constants DIRAC1937 ; Webb , the
laboratory measurements of transition frequencies with increasing accuracy are
required. Another important aspect of improving the accuracy of the experiment
is the determination of the fundamental physical constants and the definition
of frequency standards. The leaders for the latter are atomic clocks, which
have an unprecedented relative error of $2.1\times 10^{-18}$, see, e.g., AtCl-
Cs ; AtCl-Sr . Laboratory experiments with simple atomic systems such as
hydrogen or helium atoms have also demonstrated fast progress Parthey ; Mat ;
van2011frequency , where relative uncertainties were reported at the order of
$4.5\times 10^{-15}$ and $8\times 10^{-12}$, respectively.

Requiring increasingly subtle effects to be taken into account, experiments
with hydrogen have reached a level where the spectral line profile has become
an observable quantity H-exp . Theoretical derivation of the line shape can be
obtained within the framework of quantum mechanics (QM) or more rigorously
within the framework of QED (quantum electrodynamics) theory. According to F.
Low Low , the Lorentz line profile in QED arises in a natural way within the
resonance approximation. Until recently, the resonant approximation was
sufficient for the above purposes, although theoretical studies were also
carried out for effects lying beyond its limits, see Andr ; ZSLP-report ;
AZSL-2022  and references therein.

In particular, it was shown that the nonresonant contributions arising from
the differential or total cross section of the photon scattering process lead
to a significant line profile distortion. This, in turn, affects the
determination of the transition frequency. Being close to the experimental
scheme, the quantum interference effect (QIE), responsible for nonresonant
corrections to the transition frequency, has been widely studied in
applications to various atomic systems for one-photon scattering Jent-Mohr ;
HH-2010 ; HH-2011 ; Sansonetti ; Brown-2013 ; MHH-2015 ; Amaro-2015 ; Amaro-
mH-2015  and recently for two-photon absorption experiments AZSL-2022 ; Anikin
; Anikin2021 .

The main feature of QIE is dependence on the angles between the directions (or
polarizations) of the absorbed and emitted photons involved in the process.
This opens up the possibility of avoiding such effects by choosing the
geometry of the experimental setup or by using the asymmetry parameter in the
appropriate fitting of the line profile fit H-exp . While remaining
significant, nonresonant corrections cannot be always reduced to zero by the
choice of geometry: for example, corrections arising through the total cross
section LSPS ; PRA-LSPS , for specific states of atoms in the differential
cross section Anikin  or for a special experimental scheme Schwob ; deB-0 ;
deB-1 ; deB-2 ; Anikin2021 . On this basis, it can be argued that the
procedure of fitting the observed line shape by the profile, having regard to
the asymmetry parameter, remains the immutable way to determine the transition
frequency precisely. In this case, it becomes necessary to calculate the
corresponding asymmetry parameter (nonresonant correction) for each particular
experiment.

In this paper, we will focus on another type of line profile asymmetry. In
contrast to QIE and other nonresonant corrections, where resonant and nearest
neighboring states can be resolved experimentally, this asymmetry is the
result of deriving the line profile beyond the resonant approximation and
naturally arises for specific transitions, i.e., one-photon electric, magnetic
dipole or higher multipole transition rates (with the same concepts for
multiphoton processes).

##  II Natural line profile asymmetry

The natural asymmetry of the line profile was originally discussed in PRA-LSPS
, where it was shown that the level width does not depend on frequency only
within the resonant approximation. As an example, the profile of the Lymanα
line in the hydrogen atom was considered in PRA-LSPS  (see also Andr ), where
the ”velocity” form of the level width was found as
$\Gamma(\omega)=(2^{11}/3^{9})\alpha^{3}\omega$ (in atomic units, a.u.,
$\alpha$ is the fine structure constant) instead of the resonant one
$\Gamma(\omega)=(2/3)^{8}\alpha^{3}$ a.u. Alongside this, the former initially
arises in the QED theory, while the latter is the result of the approximation,
see AZSL-2022  for details and further extensions. This immediately leads to
(by inserting $\Gamma(\omega)$) a modification of the Lorentz profile,
$L(\omega)$, which becomes asymmetric. An investigation of the frequency
dependence of the numerator of the Lorentzian line profile can also be found
in PhysRevA.25.3079 , along with consideration of cascade emission/absorption
processes.

The asymmetry parameter can be considered as an additional frequency shift in
the approximation of its smallness with respect to the level width. To
determine the correction to the transition frequency, it suffices to fulfill
the extremum condition $dL(\omega)/d\omega=0$, from which the frequency can be
found as $\omega=\omega_{0}+\delta_{a}$. Here $\omega_{0}$ represents the
resonant quantity, and $\delta_{a}$ is responsible for the asymmetry (due to
the different frequency dependence of the line profile). The additional
asymmetry shift $\delta_{a}$, which follows from the ”velocity” form of
$\Gamma(\omega)$, reads

$\displaystyle\delta_{a}=\frac{1}{8}\Gamma(\omega_{0})\left[\frac{d\Gamma(\omega)}{d\omega}\right]_{\omega=\omega_{0}}.$
(1)

For the Lymanα line in the hydrogen atom it gives $1$ Hz with the parametric
estimation of $(\alpha Z)^{6}$ a.u.

The value of $\delta_{a}$ lies beyond the current accuracy of the Lymanα
frequency, which is at the level of few MHz Eikema . The correction (1) can
easily be extended to the case of two-photon $1s-2s$ absorption Parthey ; Mat
, where the relative experimental error is of the order $10^{-15}$. For this,
the estimates $\Gamma_{2s}^{(2\gamma)}(\omega_{0})\sim m\alpha^{2}(\alpha
Z)^{6}$ and $\omega_{0}\sim m(\alpha Z)^{2}$ in relativistic units (r.u.) lead
to a negligible value of the order of $\left[m\alpha^{2}(\alpha
Z)^{6}\right]^{2}/m(\alpha Z)^{2}=m\alpha^{4}(\alpha Z)^{10}$ r.u. or
$\alpha^{2}(\alpha Z)^{10}$ a.u. However, considering the explicit frequency
dependence of the level width for the two-photon transition rate LabKlim ; Zon
, see also Andr , we find the following expression

$\displaystyle\Gamma^{(2\gamma)}_{2s}(\omega,\omega^{\prime})\sim\omega^{3}\omega^{\prime
3}\times$ (2) $\displaystyle\times\int
d\bm{n}_{\bm{k}}d\bm{n}_{\bm{k}^{\prime}}\Bigg{|}\sum_{\bm{e}\bm{e}^{\prime}}\sum_{n}\Bigg{(}\frac{\langle
2s|A_{\mathrm{abs}}|n\rangle\langle
n|A^{\prime}_{\mathrm{abs}}|1s\rangle}{E_{n}-E_{1s}+\omega}+$
$\displaystyle+\frac{\langle 2s|A^{\prime}_{\mathrm{abs}}|n\rangle\langle
n|A_{\mathrm{abs}}|1s\rangle}{E_{n}-E_{1s}+\omega^{\prime}}\Bigg{)}\Bigg{|}^{2},$

in which $A_{\rm abs}=(\bm{e}\bm{\alpha})e^{-\mathrm{i}\bm{k}\bm{r}}$ and
$A^{\prime}_{\rm
abs}=(\bm{e}^{\prime}\bm{\alpha})e^{-\mathrm{i}\bm{k}^{\prime}\bm{r}}$ denotes
the wave functions of the absorbed photons, where $\bm{e},\bm{e}^{\prime}$ –
polarization vectors, $\bm{k},\bm{k}^{\prime}$ – wave vectors
($|\bm{k}|\equiv\omega$ and $|\bm{k}^{\prime}|\equiv\omega^{\prime}$ – photon
frequencies) and $\bm{\alpha}$ represents the corresponding Dirac matrix.
Summation in (2) runs over photons’ polarization vectors and over entire
spectrum of Dirac equation. In this expression one can neglect the
$\omega$-dependence in the energy denominators by putting
$\omega=\frac{\omega_{0}}{2}$, so that $\Gamma^{(2\gamma)}_{2s}(\omega)$ can
be written in the form

$\displaystyle\Gamma^{(2\gamma)}_{2s}(\omega)=\left(\frac{\omega}{\omega_{0}/2}\right)^{6}\Gamma^{(2\gamma)}_{2s}\left(\frac{\omega_{0}}{2}\right).$
(3)

where we took into account that the excitation process occurs for two
equivalent photons ($\omega=\omega^{\prime}$) at the half-resonant frequency
Parthey ; Mat .

Then, the asymmetry shift arises from the extremum condition for the profile

$\displaystyle\frac{dL(\delta_{a})}{d\delta_{a}}$ $\displaystyle=$
$\displaystyle\frac{d}{d\delta_{a}}\frac{\left(\frac{\delta_{a}+\omega_{0}}{\omega_{0}}\right)^{6}\Gamma^{(2\gamma)}_{2s}\left(\frac{\omega_{0}}{2}\right)}{\delta_{a}^{2}+\frac{1}{4}\left(\left(\frac{\delta_{a}+\omega_{0}}{\omega_{0}}\right)^{6}\Gamma^{(2\gamma)}_{2s}\left(\frac{\omega_{0}}{2}\right)\right)^{2}}=0,$
(4) $\displaystyle\delta_{a}$ $\displaystyle\approx$
$\displaystyle-\frac{3\left(\Gamma^{(2\gamma)}_{2s}\left(\frac{\omega_{0}}{2}\right)\right)^{2}}{4\omega_{0}}.$
(5)

Here $\delta_{a}\equiv 2\omega-\omega_{0}$ can be evaluated using the value of
$\Gamma^{(2\gamma)}_{2s}\approx 8s^{-1}\approx 1$ Hz (see, for example,
Jentschura_2007 ) and $\omega_{0}\approx 10^{16}$ Hz Parthey ; Mat  which
gives a negligible value of about $10^{-14}$ Hz. Nevertheless, this result is
more of a model character, since the general process of measuring the $1s-2s$
transition frequency requires a more detailed theoretical description. We will
address it in the next section. It should be noted that similar approximation
for the asymmetry shift of the $2s-ns/nd$ (here $n=3,\,4,\,6,\,8,\,12$ is the
principal quantum number) transition frequencies are valid Schwob ; deB-0 ;
deB-1 ; deB-2 ; GM-1s-3s ; Mat-2s-8s .

The smallness of the natural asymmetry shift arises from the width of the $2s$
metastable state level in hydrogen. However, for the $ns/nd$ states the level
width is formed mainly by the one-photon decay rates to the lower states,
$\Gamma_{ns/nd}=\sum\limits_{k<n}W^{(1\gamma)}_{ns/nd\rightarrow kp}$. The
frequency dependence in ”velocity” form can be introduced through the energy
conservation law for the scattering process: $E_{f}+\omega_{\rm em}-2\omega-
E_{i}=0$, where $E_{i}$, $E_{f}$ are the energies of the initial and final
states (in our case $E_{i}=1s$ or $2s$ and $E_{f}=kp$), respectively, and
$\omega_{\rm em}$ is the frequency of the emitted photon. Then substitution of
the resonance energy $\pm E_{ns/nd}$ yields $\omega_{\rm
em}=E_{ns/nd}-E_{kp}+\delta_{a}$, resulting in the line profile

$\displaystyle
L(\delta_{a})=\frac{\left(\frac{\delta_{a}+\omega_{0}}{\omega_{0}}\right)^{6}\Gamma^{(2\gamma)}_{ns/nd}\left(\frac{\omega_{0}}{2}\right)}{\delta_{a}^{2}+\frac{1}{4}\left[\sum_{k<n}\left(\frac{\Delta
E_{ns/nd,kp}+\delta_{a}}{\Delta
E_{ns/nd,kp}}\right)W^{(1\gamma)}_{ns/nd\rightarrow kp}\right]^{2}}.$ (6)

Employing the extremum condition, the natural asymmetry shift can be found as

$\displaystyle\delta_{a}\approx\frac{3}{4}\frac{\Gamma_{ns/nd}^{2}}{\omega_{0}}-\frac{\Gamma_{ns/nd}}{4}\sum\limits_{k<n}\frac{W^{(1\gamma)}_{ns/nd\rightarrow
kp}}{\Delta E_{ns/nd,kp}}.$ (7)

Numerical values of the correction (7) are collected in Table 1.

As follows from Table 1, the natural line asymmetry is much less than the
actual measurement accuracy, which for all considered excited states is of the
order of several kHz. The maximum value of about $10^{-2}$ Hz can be found for
the $3d$ and $4d$ states. The main conclusion, however, is that the correction
depends on the level width, which, in turn, depends on the experimental
conditions. In the next section, utilizing the example of $1s-2s$ two-photon
absorption, we will demonstrate the importance of a detailed description of
the experimental setup for analyzing the asymmetry of the natural line
profile.

##  III $1s-2s$ absorption profile asymmetry

According to the experiment on measuring the $1s-2s$ transition frequency
Parthey ; Mat , the de-excitation process of the metastable $2s$ state occurs
due to the admixture of the $2p$ state in an external electric field. Thus,
the level width of the $2s$ atomic state is determined by the decay rate in
the presence of mixed state, which can be found in Ans  (see also
Solovyev_2010 ; Sol-2015 ). It is known that the electric field strength of
$475$ V/cm completely mixes the $2s$ and $2p$ states, which leads to the decay
of the $2s$ state at a rate equal to the $2p-1s$ transition probability Ans .
Then, an estimation for the natural asymmetry shift can be obtained from Eq.
(5) by substituting the magnitude of $\Gamma^{(1\gamma)}_{2p}$ instead of
$\Gamma^{(2\gamma)}_{2s}$, result is $-3$ Hz.

This estimation is of the order of the experimental error of $10$ Hz. However,
the width of the $2s$ mixed state should depend quadratically on the electric
field strength Ans ; Solovyev_2010 ; Sol-2015 . For example, decreasing the
field strength to $10$ V/cm used in the experiment Mat  immediately gives a
value of $-6\times 10^{-7}$ Hz. Varying strongly with the field strength these
estimations demonstrate the necessity of an accurate analysis. To do this, we
use the $S$-matrix formalism for the Feynman diagram shown in Fig. 1.

Figure 1: The Feynman graph corresponding to the schematic description of the
photon scattering process $1s+2\gamma\rightarrow\overline{2s}\rightarrow
1s+1\gamma$ in an external electric field ($\overline{2s}$ denotes the mixed
$2s$, $2p$ state). Wavy lines with arrows represent emitted ($\omega_{\rm
em}$) or absorbed ($\omega$) photons. The dashed line represents the
interaction with an external field. The double solid line denotes the bound
electron (Furry picture), the final and initial states denoted as $f$ and $i$,
respectively, correspond to the $1s$ state in hydrogen, the intermediate
resonance states are given by the $2s$ and $2p$ atomic levels.

The time which takes atoms to propagate between the excitation and the
deexcitation regions, presented in experiments like Parthey ; Mat , deserves
special discussion. Since the $2s$-state for which two-photon absorption
occurs is metastable, it makes it possible to separate the excitation region
from the interaction with the external electric field. But due to the fact
that the action of an electric field leads to immediate decay of the excited
$2s-$state and that propagation time, which lies in the region
$\tau\in[10,1610]$ $\mu$s Mat , is much smaller then the $2s-$state lifetime,
during the propagation between these two regions one should estimate the level
width as the inverse to the propagation time. However, the estimation of the
level width in an electric field should rather be attributed to the
calculation of the transition probabilities Ans ; Solovyev_2010 ; Sol-2015 ,
and the delay corresponds to the control of the $2s$ population in time. The
latter can be considered according to the exponential decay law with the
natural level width of the state $2s$ (assuming that during this time nothing
happens to the atom). Thus, the additional factor
$\exp(-\Gamma_{2s}^{(2\gamma)}\tau)$ ranges from 1 to $0.987$ and is therefore
irrelevant for our estimates.

Performing a cumbersome but standard calculation for the $S$-matrix element
corresponding to Fig. 1, we find

$\displaystyle S_{fi}=-2\pi\mathrm{i}\delta(E_{f}+\omega_{\rm em}-2\omega-
E_{i})\frac{e^{3}(2\pi)^{3/2}}{\omega\sqrt{\omega_{\rm em}}}\qquad$ (8)
$\displaystyle\sum\limits_{n_{1}n_{2}n_{3}}\frac{\langle f|A^{*}_{\rm
em}|n_{1}\rangle\langle n_{1}|(e\bm{D}\bm{r})|n_{2}\rangle}{[E_{f}+\omega_{\rm
em}-E_{n_{1}}+\mathrm{i}0][E_{f}+\omega_{\rm em}-E_{n_{2}}+\mathrm{i}0]}$
$\displaystyle\times\frac{\langle n_{2}|A_{\rm abs}|n_{3}\rangle\langle
n_{3}|A_{\rm abs}|i\rangle}{E_{i}+\omega-E_{n_{3}}+\mathrm{i}0}.\qquad$

Here $A_{\rm em}=(\bm{e}^{*}\bm{\alpha})e^{-\mathrm{i}\bm{k}_{\rm em}\bm{r}}$,
$A_{\rm abs}=(\bm{e}\bm{\alpha})e^{-\mathrm{i}\bm{k}\bm{r}}$ denotes the wave
functions of the emitted and absorbed photons, respectively, where $\bm{e}$ –
polarization vector, $\bm{k}$ – wave vector ($|\bm{k}|\equiv\omega$ – photon
frequency) and $\bm{\alpha}$ represents the corresponding Dirac matrix,
$\bm{D}$ denotes the electric field strength, and the scalar product
$(e\bm{D}\bm{r})$ represents the dipole interaction of the bound electron with
an external field. We also used that the excitation process occurs by two
equivalent photons with the same frequencies $\omega$. Summation over
$n_{1},n_{2},n_{3}$ runs over entire spectrum of Dirac equation including the
continuum.

For an accurate description, it is necessary to consider all variants of the
interaction of an atom with photons, i.e. to take into account permutations of
photon lines in Fig. 1. Proceeding in this way, it can be found that 1) the
two-photon absorption amplitude can be arranged with the last energy
denominator in Eq. (8) in conjunction with the summation over $n_{3}$, and 2)
all other terms representing the nonresonant contribution, see Andr ; ZSLP-
report ; AZSL-2022 , can be omitted for our purposes. To form the line
profile, we fix the states $n_{2}=2s$ and $n_{1}=2p$ (resonant approximation),
which yields

$\displaystyle U_{fi}=\frac{\frac{e^{3}(2\pi)^{3/2}}{\omega\sqrt{\omega_{\rm
em}}}\langle f|A^{*}_{\rm em}|2p\rangle\langle 2p|(e\bm{D}\bm{r})|2s\rangle
A^{(2\gamma)}_{2s,1s}}{[E_{f}+\omega_{\rm
em}-E_{2p}+\mathrm{i}0][E_{f}+\omega_{\rm em}-E_{2s}+\mathrm{i}0]}.$ (9)

Here the amplitude $U_{fi}$ is defined via
$S_{fi}=-2\pi\mathrm{i}\delta(E_{f}+\omega_{\rm em}-2\omega-E_{i})U_{fi}$ and
$A^{(2\gamma)}_{2s,1s}$ denotes the two-photon absorption amplitude LabKlim ,
summation over projections of intermediate states is assumed.

Using the energy conservation law, we can find that the energy denominators in
Eq. (9) diverge. The regularization of these divergences can be performed in
the framework of the QED theory by inserting one-loop self-energy corrections
in Fig. 1 Low ; Andr . In contrast to the case with no external fields the
self-energy operator for the denominator with $E_{2s}$ should be averaged over
the mixed state $\overline{2s}$ Ans , while for the denominator with $E_{2p}$,
it should be averaged over the pure $2p$ state. Then, the imaginary part of
the averaged one-loop self-energy operator leads to the corresponding level
widths. Finally, the amplitude $U_{fi}$ takes the form

$\displaystyle\frac{\frac{e^{3}(2\pi)^{3/2}}{\omega\sqrt{\omega_{\rm
em}}}\langle f|A^{*}_{\rm em}|2p\rangle\langle 2p|(e\bm{D}\bm{r})|2s\rangle
A^{(2\gamma)}_{2s,1s}}{\left[E_{f}+\omega_{\rm
em}-E_{2p}-\frac{\mathrm{i}\Gamma_{2p}}{2}\right]\left[E_{f}+\omega_{\rm
em}-E_{2s}-\frac{\mathrm{i}\Gamma_{\overline{2s}}}{2}\right]}.$ (10)

By squaring the modulus of the expression (10), one can find the line shape of
considered process. It should be noted especially that substituting
$\omega_{\rm em}=E_{2s}-E_{f}$ into the first energy denominator of Eq. (10)
leads directly to the result used in Ans , i.e. to the representation of the
mixed state in the form $|\overline{2s}\rangle=|2s\rangle+\eta\langle
2p|(e\bm{D}\bm{r})|2s\rangle|2p\rangle$, with $\eta=(\Delta
E_{L}-\mathrm{i}\Gamma_{2p}/2)^{-1}$, see also Solovyev_2010 ; Sol-2015 . In
turn, the energy conservation law given by the $\delta$-function in Eq. (8),
$E_{f}+\omega_{\rm em}=2\omega+E_{i}$, in conjunction with $\omega_{\rm
em}=E_{2s}-E_{f}$ results in the Lorentz contour when the resonant
approximation is employed for numerator. In other words,

$\displaystyle L(\omega)\sim\frac{W_{2p,1s}^{(1\gamma)}\left|\langle
2p|(e\bm{D}\bm{r})|2s\rangle\right|^{2}W_{2s,1s}^{(2\gamma)}}{[\Delta
E_{L}^{2}+\frac{1}{4}\Gamma_{2p}^{2}][(2\omega-
E_{2s}+E_{1s})^{2}+\frac{1}{4}\Gamma_{\overline{2s}}^{2}]},$ (11)

where $\Delta E_{L}=E_{2s}-E_{2p}$ is the Lamb shift, $W_{2p,1s}^{(1\gamma)}$
is the one-photon $2p\rightarrow 1s$ transition probability and
$W_{2s,1s}^{(2\gamma)}$ corresponds to the two-photon absorption rate
$1s\rightarrow 2s$.

It is easy to see that the line profile (11) is symmetrical and gives maximum
at $\omega=(E_{2s}-E_{1s})/2$. The same result can be obtained for
$\omega_{\rm em}$ considering that for the case $E_{f}=E_{i}=1s$ $\omega_{\rm
em}=2\omega$. However, according to (10), in order to take into account the
natural line profile asymmetry, the frequency dependence should be preserved
in both energy denominators and in the numerator. By discarding all
unimportant constants, we can write

$\displaystyle
L(\delta_{a})\sim\frac{(\delta_{a}+\omega_{0})^{7}}{\left[\omega_{0}^{2}\,\delta_{a}^{2}+\frac{1}{4}\Gamma_{\overline{2s}}^{2}\omega_{\rm
em}^{2}\right]\left[\omega_{0}^{2}(\delta_{a}+\Delta
E_{L})^{2}+\frac{1}{4}\Gamma_{2p}^{2}\omega_{\rm em}^{2}\right]},$ (12)

where $\omega_{0}=E_{2s}-E_{1s}$ and $\delta_{a}\equiv\omega_{\rm
em}-\omega_{0}$ or $\omega_{\rm em}=\delta_{a}+\omega_{0}$.

Using the extremum condition with respect to $\delta_{a}$ for the profile
(12), one finds the natural asymmetry shift to be

$\displaystyle\delta_{a}\approx\frac{5}{8}\frac{\Gamma_{\overline{2s}}^{2}}{\omega_{0}}-\frac{1}{4}\frac{\Gamma_{\overline{2s}}^{2}}{\Delta
E_{L}}+\frac{1}{16}\frac{\Gamma_{\overline{2s}}^{2}\Gamma_{2p}^{2}}{\Delta
E_{L}^{3}}-\frac{1}{16}\frac{\Gamma_{\overline{2s}}^{2}\Gamma_{2p}^{2}}{\Delta
E_{L}^{2}\omega_{0}}.$ (13)

The difference in the coefficients for the first term in Eq. (13) and Eq. (5)
is explained by the frequency dependence of the numerators and denominators in
the corresponding expressions. Note that in Eq. (12), the additional power
$\omega_{\rm em}$ is due to the one-photon emission probability
$W_{2p,1s}^{(1\gamma)}$, as well as different frequency dependence of the
level widths in the denominators of Eq. (13).

The expression (13) can be evaluated using the results of Ans ; Solovyev_2010
; Sol-2015 . For a field-quadratic contribution (as a dominant one) to
$\Gamma_{\overline{2s}}$, one can write

$\displaystyle\Gamma_{\overline{2s}}=W_{2p,1s}^{(1\gamma)}\left[\frac{D}{475{\rm
V/cm}}\right]^{2}.$ (14)

Then, using the values $W_{2p,1s}^{(1\gamma)}=6.26826\times 10^{8}$ s-1,
$\omega_{0}=E_{2s}-E_{1s}=2\,466\,061\,413\,187.035$ kHz, $\Delta
E_{L}=E_{2s}-E_{2p}=1\,057\,845.0$ kHz Mohr-2016-RMP  and the field strength
$D=10$ V/cm Parthey , we find $\delta_{a}=-2.9$ s-1 or $-0.46$ Hz, which is
consistent with the estimates above.

##  IV Conclusions

In this paper, we analyzed the natural asymmetry of the line profile that
arises in the process of photon scattering. It is shown that within the
framework of the rigorous QED theory, the line shape differs from the
”ordinary” Lorentz contour, which is a consequence of the resonance
approximation. A more precise description leads to a frequency dependence of
the partial transition rates and, as a consequence, of the level widths. Such
a deviation, under the assumption that the correction is small, can be
regarded as an additional shift in the resonant transition frequency.

In contrast to the known non-resonant corrections and, in particular, the
effect of quantum interference, the asymmetry shift occurs even for an
”isolated” atomic level. As usual in such studies, the correction due to the
natural line profile asymmetry depends on the conditions of experiment. We
found that for highly excited states in the hydrogen atom, it does not exceed
$10^{-2}$ Hz, i.e. is negligible with respect to the experimental error.
However, a rough estimate of the most accurately measured frequency in the
hydrogen atom for the $1s-2s$ two-photon transition is close to the
experimental error of $10$ Hz Mat . Due to the continuous improvement of
experiments, it can be expected that such a correction may be of significance
in the near future.

## Acknowledgements

The work of A.A. was supported by foundation for the advancement of
theoretical physics ”BASIS”. T. Z. also acknowledges the grant number
MK-4796.2022.1.2.

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Table 1: Natural asymmetry shift, Eq. (7), to the transition frequencies $2s/1s-ns/nd$ ($n=3,\,4,\,6,\,8$) with excited states listed in the first column. In the second and in the third columns the level widths, $\Gamma_{ns/nd}$, and $W^{(1\gamma)}_{ns/nd\rightarrow kp}$ are listed, respectively. In the fourth and in the fifth columns resonant transition frequencies and the energy intervals $\Delta E_{ns/nd,kp}$ are listed, respectively. Finally, in the last two columns the first and the second terms of $\delta_{a}$ (see Eq. (7)) are given. The total values of $\delta_{a}$ are listed in the substring with empty cells. All values are obtained in a purely nonrelativistic theory and are given in Hz. state | $\Gamma_{ns/nd}$ | $W^{(1\gamma)}_{ns/nd\rightarrow kp}$ | $\omega_{0}$ | $\Delta E_{ns/nd,kp}$ | $\frac{3}{4}\frac{\Gamma_{ns/nd}^{2}}{\omega_{0}}$ | $-\frac{\Gamma_{ns/nd}}{4}\frac{W^{(1\gamma)}_{ns/nd\rightarrow kp}}{\Delta E_{ns/nd,kp}}$
---|---|---|---|---|---|---
$3s$ | $1.005\times 10^{6}$ |  $1.005\times 10^{6}$ (2p) | $2.924\times 10^{15}$ | $4.569\times 10^{14}$ | $2.59\times 10^{-4}$ | $-5.53\times 10^{-4}$
|  |  |  |  |  | $-2.94\times 10^{-4}$
$3d$ | $1.029\times 10^{7}$ |  $1.029\times 10^{7}$ (2p) | $2.924\times 10^{15}$ | $4.569\times 10^{14}$ | $2.72\times 10^{-2}$ | $-5.79\times 10^{-2}$
|  |  |  |  |  | $-3.08\times 10^{-2}$
$4s$ | $7.028\times 10^{5}$ |  $2.923\times 10^{5}$ (3p) | $6.168\times 10^{14}$ | $1.599\times 10^{14}$ | $6.01\times 10^{-4}$ | $-3.21\times 10^{-4}$
| $7.028\times 10^{5}$ |  $4.105\times 10^{5}$ (2p) | $6.168\times 10^{14}$ | $6.168\times 10^{14}$ | $6.01\times 10^{-4}$ | $-1.17\times 10^{-4}$
|  |  |  |  |  | $1.62\times 10^{-4}$
$4d$ | $4.405\times 10^{6}$ |  $1.121\times 10^{6}$ (3p) | $6.168\times 10^{14}$ | $1.599\times 10^{14}$ | $2.36\times 10^{-2}$ | $-7.72\times 10^{-3}$
| $4.405\times 10^{6}$ |  $3.284\times 10^{6}$ (2p) | $6.168\times 10^{14}$ | $6.168\times 10^{14}$ | $2.36\times 10^{-2}$ | $-5.86\times 10^{-3}$
|  |  |  |  |  | $1.00\times 10^{-2}$
$6s$ | $2.975\times 10^{5}$ |  $4.270\times 10^{4}$ (5p) | $7.311\times 10^{14}$ | $4.021\times 10^{13}$ | $9.08\times 10^{-5}$ | $-7.90\times 10^{-5}$
| $2.975\times 10^{5}$ |  $5.705\times 10^{4}$ (4p) | $7.311\times 10^{14}$ | $1.142\times 10^{14}$ | $9.08\times 10^{-5}$ | $-3.72\times 10^{-5}$
| $2.975\times 10^{5}$ |  $8.076\times 10^{4}$ (3p) | $7.311\times 10^{14}$ | $2.741\times 10^{14}$ | $9.08\times 10^{-5}$ | $-2.19\times 10^{-5}$
| $2.975\times 10^{5}$ |  $1.170\times 10^{5}$ (2p) | $7.311\times 10^{14}$ | $7.311\times 10^{14}$ | $9.08\times 10^{-5}$ | $-1.19\times 10^{-5}$
|  |  |  |  |  | $-5.92\times 10^{-5}$
$6d$ | $1.337\times 10^{6}$ |  $7.158\times 10^{4}$ (5p) | $7.311\times 10^{14}$ | $4.021\times 10^{13}$ | $1.83\times 10^{-3}$ | $-5.95\times 10^{-4}$
| $1.337\times 10^{6}$ |  $1.373\times 10^{5}$ (4p) | $7.311\times 10^{14}$ | $1.142\times 10^{14}$ | $1.83\times 10^{-3}$ | $-4.02\times 10^{-4}$
| $1.337\times 10^{6}$ |  $2.990\times 10^{5}$ (3p) | $7.311\times 10^{14}$ | $2.741\times 10^{14}$ | $1.83\times 10^{-3}$ | $-3.65\times 10^{-4}$
| $1.337\times 10^{6}$ |  $8.193\times 10^{5}$ (2p) | $7.311\times 10^{14}$ | $7.311\times 10^{14}$ | $1.83\times 10^{-3}$ | $-3.75\times 10^{-4}$
|  |  |  |  |  | $9.77\times 10^{-5}$
$8s$ | $1.441\times 10^{5}$ |  $1.046\times 10^{4}$ (7p) | $7.710\times 10^{14}$ | $1.573\times 10^{13}$ | $2.02\times 10^{-5}$ | $-2.40\times 10^{-5}$
| $1.441\times 10^{5}$ |  $1.290\times 10^{4}$ (6p) | $7.710\times 10^{14}$ | $3.998\times 10^{13}$ | $2.02\times 10^{-5}$ | $-1.16\times 10^{-5}$
| $1.441\times 10^{5}$ |  $1.656\times 10^{4}$ (5p) | $7.710\times 10^{14}$ | $8.019\times 10^{13}$ | $2.02\times 10^{-5}$ | $-7.44\times 10^{-6}$
| $1.441\times 10^{5}$ |  $2.260\times 10^{4}$ (4p) | $7.710\times 10^{14}$ | $1.542\times 10^{14}$ | $2.02\times 10^{-5}$ | $-5.28\times 10^{-6}$
| $1.441\times 10^{5}$ |  $3.291\times 10^{4}$ (3p) | $7.710\times 10^{14}$ | $3.141\times 10^{14}$ | $2.02\times 10^{-5}$ | $-3.77\times 10^{-6}$
| $1.441\times 10^{5}$ |  $4.863\times 10^{4}$ (2p) | $7.710\times 10^{14}$ | $7.710\times 10^{14}$ | $2.02\times 10^{-5}$ | $-2.27\times 10^{-6}$
|  |  |  |  |  | $-3.41\times 10^{-5}$
$8d$ | $5.724\times 10^{5}$ |  $1.201\times 10^{4}$ (7p) | $7.710\times 10^{14}$ | $1.573\times 10^{13}$ | $3.19\times 10^{-5}$ | $-1.09\times 10^{-4}$
| $5.724\times 10^{5}$ |  $1.878\times 10^{4}$ (6p) | $7.710\times 10^{14}$ | $3.998\times 10^{13}$ | $3.19\times 10^{-5}$ | $-6.72\times 10^{-5}$
| $5.724\times 10^{5}$ |  $3.079\times 10^{4}$ (5p) | $7.710\times 10^{14}$ | $8.019\times 10^{13}$ | $3.19\times 10^{-5}$ | $-5.49\times 10^{-5}$
| $5.724\times 10^{5}$ |  $5.612\times 10^{4}$ (4p) | $7.710\times 10^{14}$ | $1.542\times 10^{14}$ | $3.19\times 10^{-5}$ | $-5.21\times 10^{-5}$
| $5.724\times 10^{5}$ |  $1.204\times 10^{5}$ (3p) | $7.710\times 10^{14}$ | $3.141\times 10^{14}$ | $3.19\times 10^{-5}$ | $-5.49\times 10^{-5}$
| $5.724\times 10^{5}$ |  $3.268\times 10^{5}$ (2p) | $7.710\times 10^{14}$ | $7.710\times 10^{14}$ | $3.19\times 10^{-5}$ | $-6.07\times 10^{-5}$
|  |  |  |  |  | $-8.03\times 10^{-5}$