# Atomic-motion-induced spectroscopic effects nonlinear in atomic density in a
gas

V. I. Yudin<EMAIL_ADDRESS>Novosibirsk State University, ul. Pirogova 1,
Novosibirsk, 630090, Russia  Institute of Laser Physics SB RAS, Pr. Lavrentyev
15-B, Novosibirsk, 630090, Russia  Novosibirsk State Technical University, Pr.
Karl Marks 20, Novosibirsk, 630073, Russia  A. V. Taichenachev  Novosibirsk
State University, ul. Pirogova 1, Novosibirsk, 630090, Russia  Institute of
Laser Physics SB RAS, Pr. Lavrentyev 15-B, Novosibirsk, 630090, Russia  M. Yu.
Basalaev  Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090,
Russia  Institute of Laser Physics SB RAS, Pr. Lavrentyev 15-B, Novosibirsk,
630090, Russia  Novosibirsk State Technical University, Pr. Karl Marks 20,
Novosibirsk, 630073, Russia  O. N. Prudnikov  Novosibirsk State University,
ul. Pirogova 1, Novosibirsk, 630090, Russia  Institute of Laser Physics SB
RAS, Pr. Lavrentyev 15-B, Novosibirsk, 630090, Russia  S. N. Bagayev
Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090, Russia
Institute of Laser Physics SB RAS, Pr. Lavrentyev 15-B, Novosibirsk, 630090,
Russia

###### Abstract

The interatomic dipole-dipole interaction is commonly thought to be the main
physical reason for spectroscopic effects nonlinear in atomic density.
However, we have found that the free motion of atoms can lead to other effects
nonlinear in atomic density $n$, using a previously unknown self-consistent
solution of the Maxwell-Bloch equations in the mean-field approximation for a
gas of two-level atoms with an optical transition at unperturbed frequency
$\omega_{0}$. These effects distort the Doppler lineshape (shift, asymmetry,
broadening), but are not associated with an atom-atom interaction. In
particular, in the case of $nk^{-3}_{0}<1$ (where $k_{0}=\omega_{0}/c$) and
significant Doppler broadening (with respect to collisional broadening),
atomic-motion-induced nonlinear effects significantly exceed the well-known
influence of the dipole-dipole interatomic interaction (e.g., Lorentz-Lorenz
shift) by more than one order of magnitude. Moreover, under some conditions a
frequency interval appears in which a non-trivial self-consistent solution of
the Maxwell-Bloch equations is absent due to atomic motion effects. Thus, the
existing physical picture of spectroscopic effects nonlinear in atomic density
in a gas medium should be substantially revised.

##  I Introduction

Collective effects are one of the most interesting and important issues in
fundamental physics. With regard to laser spectroscopy of resonant gas media
and high-precision atomic clocks, collective effects, such as the interatomic
dipole-dipole interaction, distort the resonance lineshape (shift, broadening,
asymmetry) Friedberg_1973 ; Lewis_1980 ; Lorentz_2011 ; Kazantsev_1967 ;
Vdovin_1967 ; Vdovin_1969 ; Srivastava_1976 ; Fleischhauer_1999 ; Pfau_2002 ;
Pfau_2005 ; Pfau_2009 ; Fofanov_2011 ; Friedberg_2011 ; Weller_2011 ;
Sokolov_2011 ; Adams_2012 ; Javanainen_2014 ; Jenkins_2014 ; Sokolov_2014 ;
Bettles_2015 ; Scully_2015 ; Scully_2016 ; Adams_2016 ; Jenkins_2016 ;
Jenkins_2016_PRA ; Sokolov_2016 ; Pfau_2017 ; Adams_2018 ; Sokolov_2020 ;
Chang_2004 ; Ostermann_2012 ; Ostermann_2013 ; Kramer_2016 ; Henriet_2019 ;
Qu_2019 ; Liu_2020 ; Cidrim_2021 ; Bettles_2016 ; Glicenstein_2020 ;
Masson_2020 ; Andreoli_2021 ; Fayard_2021 ; Sierra_2021 . The need to use a
quantum mechanical many particle formalism (for example, a many-atomic density
matrix), significantly complicates the theoretical description of such
effects. In this case, most theoretical papers investigate the limit where the
collisional broadening of the line exceeds the inhomogeneous Doppler
broadening caused by the free motion of atoms as this somewhat simplifies the
mathematical calculations. In real experiments, this condition is satisfied
either for strongly heated atomic cells or for an ensemble of laser-cooled
atoms. However, the opposite case of significant Doppler broadening is studied
quite rarely as atomic density effects are assumed to be small.

As is well-known Lorentz_2011 , in an ensemble of two-level atoms with an
unperturbed frequency $\omega_{0}$ for a closed optical transition
$|g\rangle\to|e\rangle$ (see Fig. 1), the scale of the dipole-dipole
interaction is determined by the value of Lorentz-Lorenz shift $\Delta_{\rm
LL}=-\pi nk^{-3}_{0}\gamma_{0}$, where $n$ is the atomic density (number of
atoms per unit volume), $k_{0}=\omega_{0}/c$ is the wave number ($c$ is the
speed of light in vacuum), $\gamma_{0}$ is the spontaneous decay rate of the
upper level (see Fig. 1). In particular, for an ensemble of atoms confined
within a layer of thickness ${\cal L}$, the total redshift induced by dipole-
dipole interaction is (see Ref. Friedberg_1973 )

$\Delta_{\rm dd}=\Delta_{\rm LL}-\frac{3}{4}\Delta_{\rm LL}\left(1-\frac{\sin
2k_{0}{\cal L}}{2k_{0}{\cal L}}\right)<0\,,$ (1)

where the second term is the collective Lamb shift. For a thick layer
($k_{0}{\cal L}\gg 1$), the redshift (1) is equal to

$\Delta_{\rm dd}=\frac{1}{4}\Delta_{\rm
LL}\approx-0.79nk^{-3}_{0}\gamma_{0}\,.$ (2)

However, when accounting for atomic motion in a gas, the results may differ
substantially from Eq. (1). For example, numerical calculations for large
inhomogeneous broadening in Ref. Javanainen_2014  show that the shift of
Doppler lineshape significantly differs from the dependence (1) (see Fig. 2
(right panel) in Ref. Javanainen_2014 ). In the context of quantum microscopic
theory, the interatomic dipole-dipole interaction is the result of the
exchange of resonant photons (e.g., see Ref. Sokolov_2011 ). Therefore, atomic
motion modifies the dipole-dipole interaction, since a photon emitted by one
atom becomes non-resonant for an atom moving at a different velocity.

In addition to fundamental interest, effects nonlinear in atomic density are
of great applied importance for high-precision laser spectroscopy and atomic
clocks. In particular, the dependence of a reference resonance shift on atomic
density $n$ determines the inaccuracy and long-term instability of atomic
clocks due to the temperature variations, since the value of $n$ depends on
the vapor cell temperature $T$.

Figure 1: Two-level atom.

In this paper, we describe spectroscopic effects nonlinear in the atomic
density due to atomic motion, which follow from a self-consistent solution of
the Maxwell-Bloch equations for atomic gases. These effects can significantly
exceed the contribution of the interatomic dipole-dipole interaction to the
distortion of the spectroscopic lineshape. In particular, the frequency shift
of the Doppler lineshape for a monochromatic plane wave has an opposite sign
and is more than one order of magnitude larger than the estimate (2). We
emphasize that our results are obtained in the standard mean-field
approximation without taking into account atom-atom interactions. The
presented atomic-motion-induced effects have not previously been discussed in
the scientific literature, as far as we know. Moreover, these effects cannot
be described with a stochastic model of motionless atoms, in which
inhomogeneous Doppler broadening is introduced by assigning to each atom a
shift of the resonance frequency drawn at random from a Gaussian distribution.

##  II Basic formalism

Let us consider propagation along the $z$ axis of a plane monochromatic wave
with a real electric field $E(t,z)$ in a homogeneous gas medium of free-moving
two-level atoms (see Fig. 1). The atom-field interaction is described by the
electro-dipole interaction operator $-\hat{d}E$. Our analysis is carried out
within the framework of a self-consistent solution of the well-known Maxwell-
Bloch equations, which include the wave equation for the field (in the CGS
system):

$\left(\frac{\partial^{2}}{\partial
z^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\right)E(t,z)=\frac{4\pi}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}P(t,z)\,,$ (3)

where $P(t,z)$ is the mean-field polarization of the medium. The atomic gas is
described by the one-atomic density matrix $\hat{\rho}(v)$ ($v$ is the
velocity of the atom) with the components $\rho_{jl}(v)=\langle
j|\hat{\rho}(v)|l\rangle$ (where $j,l=\\{e,g\\}$). In a linear approximation
of the electric field $E$ (i.e., in the small saturation limit corresponding
to the condition $|d_{eg}E/\gamma_{eg}|^{2}\ll 1$), we have the following
Bloch equations:

$\displaystyle\left(\frac{\partial}{\partial t}+v\frac{\partial}{\partial
z}+\gamma_{eg}+i\omega_{0}\right)\rho_{eg}(v)=id_{eg}\rho_{gg}(v)E(t,z)/\hbar\,,$
(4)
$\displaystyle\;\;\rho_{ge}(v)=\rho^{\ast}_{eg}(v)\,,\quad\rho_{gg}(v)=f(v)\,,\quad\rho_{ee}(v)=0\,,$
$\displaystyle\;\int^{\infty}_{-\infty}f(v)dv=1\,,$

where $\gamma_{eg}$ is the optical coherence relaxation rate, $d_{eg}=\langle
e|\hat{d}|g\rangle=d^{\ast}_{ge}$ is the matrix element of the dipole moment
operator, and the function $f(v)$ describes the velocity distribution of
atoms. The operator $v({\partial}/{\partial z})$ in the left-hand side of Eq.
(4) is a one-dimensional version (for the plane wave case) of the scalar
operator $({\bf v}\cdot\nabla)$ for free-moving atoms in a gas, which is a
fundamental difference from Bloch equations for an ensemble of motionless
atoms where this operator is absent (e.g., for impurity resonant atoms in a
solid). The polarization of a gas medium is defined as:

$\displaystyle P(t,z)=n\langle D\rangle_{v}\,,$ (5) $\displaystyle\langle
D\rangle_{v}=\int^{\infty}_{-\infty}D(v)dv,\quad
D(v)=d_{ge}\rho_{eg}(v)+c.c.\,,$

where $\langle D\rangle_{v}$ is the velocity-averaged dipole moment of the
atom. Eqs. (3)-(5) constitute the Maxwell-Bloch system of equations in our
case. Note also that we set the condition for the thickness of the gas medium
${\cal L}\gg\lambda$ to exclude the significant influence of various boundary
effects Dicke_1953 ; Batygin_1953 .

In the case of a monochromatic plane wave traveling in a gas medium, the
density matrix equations (4) typically use $E(t,z)=[E_{0}\exp\\{-i(\omega
t-kz)\\}+c.c.]$ (where $\omega\approx\omega_{0}$ the light frequency,
$k=\omega/c=2\pi/\lambda$ is the wave number in vacuum, $\lambda$ is the
wavelength in vacuum), which leads to the standard expression for the Doppler
lineshape. However, it is more correct to define the function $E(t,z)$ as a
self-consistent solution of the equations system (3)-(5), which we will seek
in the form:

$E(t,z)=E_{0}e^{-i\omega t+Kz}+c.c.\,,$ (6)

where $K$ is an unknown complex number. Substituting the expression (6) into
right-hand side of Eq. (4), we find in the rotating wave approximation,

$\rho_{eg}(v)=\frac{id_{eg}f(v)/\hbar}{\gamma_{eg}-i\delta+Kv}\,E_{0}e^{-i\omega
t+Kz},$ (7)

where $\delta=\omega-\omega_{0}$ is the frequency detuning of the laser field
(6) in the laboratory reference frame. Then, using Eqs. (7) and (5), we obtain
from Eq. (3) the following equation with respect to the unknown $K$:

$K^{2}+k^{2}=-\frac{i4\pi
k^{2}n|d_{eg}|^{2}}{\hbar}\left\langle\frac{f(v)}{\gamma_{eg}-i\delta+Kv}\right\rangle_{v},$
(8)

where $\langle...\rangle_{v}$ denotes an integral over the velocity,
$\int_{-\infty}^{+\infty}...dv$. Note that Eq. (8) is obtained within the
framework of a canonical approach. However, this equation and its solution
have not been presented in scientific literature.

For the convenience in further analysis, we will make two transformations:

First, we represent the value of $K$ as follows

$K=(i+\alpha)k\,,$ (9)

where the dimensionless $\alpha$ describes the contribution of the gas medium.
In this case, the correct (physical) solution must satisfy the condition ${\rm
Re}[\alpha]<0$, which corresponds to the attenuation of the wave during
propagation along the positive direction of the $z$ axis.

Second, using the well-known expression
$\gamma_{0}=4k_{0}^{3}|d_{eg}|^{2}/(3\hbar)$ for the spontaneous decay rate of
the upper level $|e\rangle$ (see Fig. 1), we can represent Eq. (8) as an
equation with respect to the unknown $\alpha$:

$\alpha^{2}+2i\alpha=-i3\pi
nk^{-3}_{0}\gamma_{0}\left\langle\frac{f(v)}{\gamma_{eg}-i\delta+(i+\alpha)k\bar{v}v/\bar{v}}\right\rangle_{v},$
(10)

where we also introduced the parameter $\bar{v}$ characterizing the width of
the velocity distribution $f(v)$. For example, in the case of the Maxwellian
distribution, we have

$f(v)=f_{M}(v)=e^{-(v/\bar{v})^{2}}/(\bar{v}\sqrt{\pi})\;,\quad\bar{v}=\sqrt{2k_{B}T/m}\;,$
(11)

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature of the gas,
$m$ is the mass of the atom.

Since we are interested in the behavior of the lineshape under conditions
close to the optical resonance ($\omega\approx\omega_{0}$) in the range of
several gigahertz, the magnitude of the relative change in the wave vector
$\Delta k/k_{0}$ is less than $10^{-4}$. Thus, we can safely use

$k\bar{v}\approx k_{0}\bar{v}=\Omega_{D}\,,$ (12)

in the right-hand side of Eq. (10) to obtain the final equation

$\displaystyle\alpha^{2}+2i\alpha=$ (13) $\displaystyle-i3\pi
nk^{-3}_{0}\gamma_{0}\left\langle\frac{f_{M}(v)}{\gamma_{eg}-i\delta+(i+\alpha)\Omega_{D}v/\bar{v}}\right\rangle_{v},\;{\rm
Re}[\alpha]<0,$

which we will use to determine the frequency dependence $\alpha(\delta)$ for
each set of the fixed parameters
$\\{nk^{-3}_{0},\gamma_{0},\gamma_{eg},\Omega_{D}\\}$. It should be emphasized
that although Eq. (13) was obtained in the framework of a one-atomic density
matrix, there is the parameter $nk^{-3}_{0}\gamma_{0}$, which plays a key role
in the theory of dipole-dipole interatomic interaction (i.e., in the case of
many-atomic formalism).

##  III Atomic-motion-induced effects

Note that if we remove the $\alpha^{2}$ contribution in the left-hand side of
Eq. (13), the remaining equation

$\alpha=-(3/2)\pi
nk^{-3}_{0}\gamma_{0}\left\langle\frac{f_{M}(v)}{\gamma_{eg}-i\delta+(i+\alpha)\Omega_{D}v/\bar{v}}\right\rangle_{v},$
(14)

corresponds to the case when the reduced Maxwell equation is used instead of
the full wave equation (3). If we put $\alpha=0$ in the right-hand side
denominator of the reduced equation (14), we get the well-known expression for
the Doppler-broadened lineshape,

$\alpha(\delta)=-(3/2)\pi
nk^{-3}_{0}\gamma_{0}\left\langle\frac{f_{M}(v)}{\gamma_{eg}-i\delta+i\Omega_{D}v/\bar{v}}\right\rangle_{v},$
(15)

described by the standard Voigt profile, which is linearly proportional to the
parameter $nk^{-3}_{0}\gamma_{0}$. However, as follows from the basic equation
(13), $\alpha(\delta)$ should have a more complex nonlinear dependence on
$nk^{-3}_{0}\gamma_{0}$. Moreover, the presence of a nonlinear dependence on
$nk^{-3}_{0}\gamma_{0}$ follows even from the reduced equation (14).

Figure 2: Spectroscopic dependence $\alpha(\delta)$ as a solution of Eq. (13):
(a) the absorption coefficient Re[$\alpha(\delta)$] (red solid line); the
bottom panel clearly shows the frequency shift $\bar{\delta}_{\rm Re}$ in
comparison to the standard Voigt profile (15) (see dashed line);
(b) the correction to the dispersion law Im[$\alpha(\delta)$] (green solid
line); the bottom panel shows the frequency shift $\bar{\delta}_{\rm Im}$ of
the dispersion curve in comparison to the standard Voigt profile (15) (see
dashed line).
The calculations are done with the following parameters: $nk^{-3}_{0}=0.1$;
$\gamma_{eg}/\gamma_{0}=1/2$; $\Omega_{D}/\gamma_{0}=50$.

Let us numerically investigate the solution of Eq. (13). Fig. 2(a) shows the
dependence of the absorption coefficient $4\pi{\rm Re}[\alpha(\delta)]$, where
we use the factor $4\pi$ because the spatial dependence of the intensity
$I(z,\delta)$ ($I\propto|E|^{2}$) during propagation in a medium has the form,

$I(z,\delta)=I_{0}e^{4\pi{\rm Re}[\alpha(\delta)]z/\lambda}\,.$ (16)

As can be seen from Fig. 2(a) (bottom panel), there is a positive shift
$\bar{\delta}_{\rm Re}$ for the top of the Doppler absorption lineshape. Our
calculations show that under the conditions $\gamma_{eg}/\Omega_{D}\ll 1$ and
$nk^{-3}_{0}<0.5$ this shift is well described by the formula

$\bar{\delta}_{\rm Re}\approx 18.7\,nk^{-3}_{0}\gamma_{0}\,.$ (17)

In the Fig. 2(b), we see the correction to the dispersion law ${\rm
Im}[\alpha(\delta)]$, which is shifted by

$\bar{\delta}_{\rm Im}\approx 11\,nk^{-3}_{0}\gamma_{0}\,.$ (18)

Comparing (17) and (18) with the estimate (2), we can assert that in the case
of $\gamma_{eg}/\Omega_{D}\ll 1$ and $nk^{-3}_{0}<0.5$ atomic-motion-induced
effects are more than one order of magnitude greater than effects due to the
dipole-dipole interaction. Moreover, shifts (17)-(18) have a positive sign
(blueshifts), unlike the redshift (2).

Note that in Eq. (13), there are two sources of nonlinearity on the density
parameter $nk^{-3}_{0}$ for $\alpha(\delta)$:
1) the quadratic contribution $\alpha^{2}$ in the left-hand side of Eq. (13),
which is not related to atomic motion;
2) the term $\alpha\Omega_{D}v/\bar{v}$ in the right-hand side denominator
caused by the free motion of atoms in a gas.
In order to understand which of these factors is more significant, we use the
reduced equation (14), where the quadratic contribution $\alpha^{2}$ in the
left-hand side is absent. In this case, we get the following results:

$\bar{\delta}_{\rm Re}\approx
14\,nk^{-3}_{0}\gamma_{0}\,,\quad\bar{\delta}_{\rm Im}\approx
7.5\,nk^{-3}_{0}\gamma_{0}\,.$ (19)

In addition, to clarify the role of atomic motion, consider the other equation

$\alpha^{2}+2i\alpha=-i3\pi
nk^{-3}_{0}\gamma_{0}\left\langle\frac{f_{M}(v)}{\gamma_{eg}-i\delta+i\Omega_{D}v/\bar{v}}\right\rangle_{v},$
(20)

obtained by ejecting $\alpha$ from the denominator in the right-hand side of
Eq. (13). The physical solution of the quadratic equation (20) has the form

$\alpha(\delta)=-i\left[1-\left(1+i\left\langle\frac{3\pi
nk^{-3}_{0}\gamma_{0}f_{M}(v)}{\gamma_{eg}-i\delta+i\Omega_{D}v/\bar{v}}\right\rangle_{v}\right)^{1/2}\right],$
(21)

which gives us the following result:

$\bar{\delta}_{\rm Re}\approx
5\,nk^{-3}_{0}\gamma_{0}\,,\quad\bar{\delta}_{\rm Im}\approx
5\,nk^{-3}_{0}\gamma_{0}\,.$ (22)

Comparing this with (17)-(18), we can assert that the atomic motion
contribution [see the term $\alpha\Omega_{D}v/\bar{v}$ in the right hand side
denominator in Eq. (13)] to nonlinear effects is dominate in the case of
$nk^{-3}_{0}<0.5$ and $\gamma_{eg}/\Omega_{D}\ll 1$. However, it is
interesting to note that even the shift (22), caused only by the quadratic
contribution $\alpha^{2}$ in the left-hand side of Eq. (13), is six times
greater than the value (2).

Figure 3: Frequency dependence of the absorption coefficient
Re[$\alpha(\delta)$] for the 87Rb D1 line in the region of hyperfine
transitions $F_{g}=1\to F_{e}=1$ and $F_{g}=1\to F_{e}=2$ ($\delta$ is the
detuning from the transition $F_{g}=1\to F_{e}=2$):
(a) lineshape for $nk^{-3}_{0}=0.5$ and $\gamma_{eg}/\gamma_{0}=1/2$;
(b) lineshape for $nk^{-3}_{0}=3$ and $\gamma_{eg}/\gamma_{0}=1/2$, where
there are two frequency intervals $\Delta_{\rm break}$, within which there is
no solution of Eq. (24);
(c) lineshape for $nk^{-3}_{0}=3$ and broadening coefficient $A=\pi$ [see Eq.
(28)], where the AMCMBE-catastrophe disappears.
All calculations are done for $\Omega_{D}/\gamma_{0}=60$.

Note that a simple two-level system in Fig. (1) corresponds to the transition
$J_{g}=0\to J_{e}=1$, where $J_{g}$ and $J_{e}$ are angular momenta in the
ground and excited states, respectively. In the general case of an arbitrary
closed transition $J_{g}\to J_{e}$, the following equation should be used:

$\displaystyle\alpha^{2}+2i\alpha=$ (23)
$\displaystyle-i\frac{2J_{e}+1}{2J_{g}+1}\pi
nk^{-3}_{0}\gamma_{0}\left\langle\frac{f_{M}(v)}{\gamma_{eg}-i\delta+(i+\alpha)\Omega_{D}v/\bar{v}}\right\rangle_{v},$

instead of Eq. (13).

In the case of atoms with hyperfine structure, there are several closely lying
transitions with resonance frequencies $\omega^{(q)}_{0}$ ($q=1,2,...$), which
result in a more general equation:

$\displaystyle\alpha^{2}+2i\alpha=$ (24) $\displaystyle-i\pi
nk^{-3}_{0}\gamma_{0}\sum_{q}\left\langle\frac{C_{q}\,f_{M}(v)}{\gamma^{(q)}_{eg}-i\delta_{q}+(i+\alpha)\Omega_{D}v/\bar{v}}\right\rangle_{v},$

where $\delta_{q}=\omega-\omega^{(q)}_{0}$. The weight coefficients $C_{q}$
are determined by the quantum numbers of the corresponding hyperfine levels
(total angular momentum $F$, electronic angular momentum $J$, nuclear spin
$S_{n}$) using the quantum theory of angular momentum. As an example of
solving the equation (24), Fig. 3(a) shows the spectroscopic dependencies of
Re[$\alpha(\delta)$] for the D1 line in a gas of 87Rb atoms ($\lambda=795$ nm;
$\gamma_{0}/2\pi=5.75$ MHz) when scanning the laser frequency $\omega$ in the
region of hyperfine transitions $F_{g}=1\to F_{e}=1$ and $F_{g}=1\to F_{e}=2$
(where $F_{g}$ and $F_{e}$ are the total angular momenta of hyperfine levels
in the ground and excited states, respectively). In this case, we have the
weight factors $C_{1}=1/16$ for the transition $F_{g}=1\to F_{e}=1$ and
$C_{2}=5/16$ for the transition $F_{g}=1\to F_{e}=2$.

##  IV The atomic-motion-conditioned catastrophe for the Maxwell-Bloch
equations

Figure 4: Spectroscopic dependence $\alpha(\delta)$ in the case where there is
a frequency interval $\Delta_{\rm break}$, within which there is no solution
of Eq. (13): (a) Doppler absorption lineshape Re[$\alpha(\delta)$]; (b)
correction to the dispersion law Im[$\alpha(\delta)$]; (c) frequency
dependence $\xi(\delta)$ [see Eq. (27)].
The calculations correspond to the following parameters: $nk^{-3}_{0}=0.25$;
$\gamma_{eg}/\gamma_{0}=1/2$; $\Omega_{D}/\gamma_{0}=50$.

In addition to the above, we have found an extremely unexpected result, where
there exists no solution to Eq. (13) within a frequency interval $\Delta_{\rm
break}$ starting from some value $nk^{-3}_{0}$ [i.e., only the trivial
solution $E=0$ takes place for the Maxwell-Bloch equations (3)-(5)]. This is
clearly seen in Figs. 4(a) and (b), as a discontinuity in the functional
dependence $\alpha(\delta)$. Numerical analysis shows that when
$\gamma_{eg}/\Omega_{D}\ll 1$ such a frequency interval exists (i.e.,
$\Delta_{\rm break}\neq 0$) under the condition

$nk^{-3}_{0}>0.27\,\gamma_{eg}/\gamma_{0}\,.$ (25)

In the case of purely spontaneous relaxation ($\gamma_{eg}=\gamma_{0}/2$),
this corresponds to $nk^{-3}_{0}>0.135$. Note also that the absence of a
solution of Eq. (13) is due only to the atomic motion, because the solution
(21) for the simplified equation (20) exists for any set of values
{$nk^{-3}_{0}$, $\gamma_{0}$, $\gamma_{eg}$, $\delta$, $\Omega_{D}$}.

The possibility of discontinuities in $\alpha(\delta)$ is mathematically
substantiated in Appendix A. In particular, under the condition

$\frac{3\pi nk^{-3}_{0}\gamma_{0}}{\Omega_{D}}\ll 1\,,$ (26)

the existence of the solution for Eq. (13) additionally requires the
inequality

$\xi={\rm
Im}\left\\{\frac{\gamma_{eg}-i\delta}{(i+\alpha)\Omega_{D}}\right\\}<0\,.$
(27)

Moreover, at the points of discontinuity, $\xi=0$. Since
Re$[\alpha(\delta)]<0$, the inequality (27) always holds for $\delta<0$.
Therefore, in the case of the condition (33), the location of the interval
$\Delta_{\rm break}$ should be expected in the opposite area, i.e. for
$\delta>0$.

Figure 4 shows numerical calculations that confirm the above reasoning. As
seen in Fig. 4(c), in the region where the solution of Eq. (13) exists, the
condition $\xi<0$ is satisfied [see Eq. (27)], while at the boundaries of
$\Delta_{\rm break}$ we have $\xi=0$. Moreover, we have verified that these
results hold for other velocity distributions $f(v)$ (e.g., for the stepwise
and Lorentzian). Note also that the absence of a solution of Eq. (13) cannot
be explaned by the linear approximation in the electric field $E$ for the
Bloch equations (4), while the nonlinear contributions (in $E$) will solve
this problem. Indeed, we can always choose the initial amplitude $E_{0}$ small
enough that the linear approximation in the field $E$ for the Bloch equations
(4) is quite correct.

Figure 5: Spectroscopic dependence $\alpha(\delta)$ in the presence of
broadening (28) with proportionality coefficient $A=5.5$ for different atomic
densities $n$:
(a) Doppler absorption Re[$\alpha(\delta)$] for $nk^{-3}_{0}=0.3$ at $A=5.5$
(solid red line) and at $A=0$ (dashed black line, where a break in the
dependence is seen in the region $\delta>0$);
(b) frequency dependencies Re[$\alpha(\delta)$], Im[$\alpha(\delta)$],
$\xi(\delta)$ in the case of $nk^{-3}_{0}=3$ and $A=5.5$, where there is a
frequency interval $\Delta_{\rm break}$, within which there is no solution of
Eq. (13).
All calculations are done for $\Omega_{D}/\gamma_{0}=50$.

As follows from Fig. 3(b), in the general case of hyper-fine structure, there
may be several frequency intervals $\Delta_{\rm break}$ for which there is no
solution of Eq. (24). Note that due to the presence of several resonant terms
on the right-hand side of Eq. (24), the condition for the existence of a
solution will differ from Eq. (27).

Logically, the absence of a solution to Eq. (13) can be interpreted in two
ways. One, if we assume that this fact has a physical meaning, then we can
expect to experimentally observe some features in the frequency interval
$\delta\in\Delta_{\rm break}$, when only the trivial self-consistent solution
to the Maxwell-Bloch equations exists, $E=0$. In this case, the situation
looks like that the atomic medium “does not let in” light, which may be
manifested as a total reflection of light from the boundary of the atomic
cell.

Two, if the absence of the solution of Eq. (13) has no physical meaning, then
this is a serious mathematical problem, where for a certain range of
parameters the Maxwell-Bloch equations system (3)-(5) has no solution at all.
In this case, we call this problem the atomic-motion-conditioned catastrophe
for Maxwell-Bloch equations (AMCMBE-catastrophe). Alternatively, the problem
may be related to the mean-field approximation, which we call the atomic-
motion-conditioned catastrophe of the mean-field approximation (AMCMF-
catastrophe). In any case, this mathematical problem concerns the basic
principles of the theoretical description of light propagation in a gas and
requires further research.

In particular, if the mean free path of an atom is much larger than the
wavelength $\lambda$, we can combine (at least phenomenologically) our
approach with collective effects due to the interatomic dipole-dipole
interaction using the following replacement in Eqs. (13), (23) and (24):

$\gamma_{eg}=\gamma_{0}/2+Ank^{-3}_{0}\gamma_{0}\,,\quad\delta\to\delta-
Bnk^{-3}_{0}\gamma_{0}\,,$ (28)

where $Ank^{-3}_{0}\gamma_{0}$ and $Bnk^{-3}_{0}\gamma_{0}$ are the broadening
and shift (both induced by the dipole-dipole interaction), respectively. The
proportionality coefficients $A$ and $B$ are determined theoretically or
experimentally. In this case, the total shift of the Doppler lineshape is the
sum of the shift due to atomic-motion-induced effects [i.e., as a result of
the solution of Eqs. (13), (23), (24)] and the value $Bnk^{-3}_{0}\gamma_{0}$.
Note that for a more detailed theory, we can also assume a velocity dependence
for the coefficients $A(v)$ and $B(v)$ in Eq. (28) (e.g., $A(v),B(v)\propto
f_{M}(v)$ in the case of $\gamma_{eg}/\Omega_{D}\ll 1$).

Using above approach, we have found that the homogeneous broadening
$Ank^{-3}_{0}\gamma_{0}$ in Eq. (28) can noticeably affect the Doppler
lineshape and the regime where there is no solution to Eqs. (13),(23), (24)
[e.g., see Fig. 5(a)]. For example, if we use the known values $A=\pi$ and
$A=\pi\sqrt{2}$ for the D1 and D2 lines of alkali atoms (see in Ref.
Lewis_1980 ), respectively, then from numerical calculations we see that the
AMCMBE-catastrophe problem disappears [e.g., see Fig. 3(c)] (i.e., the
solution of Eq. (24) exists for any value of $nk^{-3}_{0}$). Some difficulty
remains only for a simple two-level atom, because the value $A=5.4$-5.7 (see
Refs. Kazantsev_1967 ; Vdovin_1967 ; Lewis_1980 ) is not enough to completely
exclude the AMCMBE-catastrophe regime [see Fig. 5(b)]. In the case of
transition $J_{g}=0\to J_{e}=1$, our calculations show total disappearance of
AMCMBE-catastrophe only for $A>6.9$. A possible solution of this “conundrum”
is to take into account the reabsorption of spontaneously emitted photons
Fleischhauer_1999 , which can make an additional contribution to the
broadening coefficient $A$ such that $A>6.9$. Thus, the use of Eq. (28) is one
of possible methods to solve the AMCMBE-catastrophe problem. However, as of
now we still cannot completely exclude the presence of the physical meaning
for the AMCMBE-catastrophe regime comment .

In addition, our calculations show that taking into account the homogeneous
broadening $Ank^{-3}_{0}\gamma_{0}$ in Eq. (28) has small effect on the
magnitudes of the shifts $\bar{\delta}_{\rm Re}$ and $\bar{\delta}_{\rm Im}$
in Eqs. (17)-(18) (at the percentages level for $A=5.7$) in the case of
$nk^{-3}_{0}<1$ and significant Doppler broadening
($\Omega_{D}\gg\gamma_{eg}$).

##  V Conclusion

We have developed a theory of the Doppler-broadened lineshape in an atomic
gas, based on the previously unknown self-consistent solution to the Maxwell-
Bloch equations in the mean-field approximation and one-atomic density matrix.
The effects nonlinear in the atomic density caused by the free motion of atoms
were found, which affect the lineshape (shift, asymmetry, broadening). It was
shown that in the regime of significant Doppler broadening, these effects can
exceed by more than one order of magnitude the contribution of interatomic
dipole-dipole interactions (e.g., Lorentz-Lorenz shift). Moreover, in certain
area of parameters, we have found that there exists a frequency interval
$\Delta_{\rm break}$ with only the trivial self-consistent solution to the
Maxwell-Bloch equations (i.e., only $E=0$). This problem (called by us AMCMBE-
or AMCMF-catastrophe) concerns the basic theoretical description of light
propagation in a gas and, therefore, requires further research. In particular,
we have shown that this problem disappears when we take into account the
homogeneous broadening due to atom-atom interaction.

Note that in numerical simulations, inhomogeneous Doppler broadening is often
described using a stochastic model of motionless atoms, where the resonance
frequency of each atom in an ensemble is shifted by a Gaussian-distributed
random variable with zero mean and the rms value $\Omega_{D}$. However, it
should be emphasized that in this model the above-described atomic-motion-
induced effects cannot be taken into account, because these effects are
rigorously based on the presence of the differential operator $({\bf
v}\cdot\nabla)$ in the Bloch equations for the density matrix of moving atoms.

Thus, in a resonant gas medium, the physical picture of spectroscopic effects
nonlinear in atomic density is a complicated mixture of both effects due to
the interatomic interaction and atomic motion. The obtained results are
important for laser spectroscopy, atomic clocks and fundamental physics.

The possibilities of experimental verification of the presented results are
discussed in Appendix B, where we estimate the conditions for observing the
central part of the lineshape in a wide range of atomic density parameter
$nk^{-3}_{0}$ for a transmission signal $I(\delta,{\cal L})$ through an atomic
cell of length ${\cal L}$.

We thank I. M. Sokolov, V. L. Velichansky, and J. W. Pollock for useful
discussions and comments.

## Appendix A

We will consider the basic equation (13). From a mathematical viewpoint, it is
possible to assume the existence of discontinuity points in the analytical
dependence $\alpha(\delta)$ from the following considerations. Let us
represent the integral in the right-hand side of Eq. (13) as follows:

$\displaystyle\left\langle\frac{f_{M}(v)}{\gamma_{eg}-i\delta+(i+\alpha)\Omega_{D}v/\bar{v}}\right\rangle_{v}=$
$\displaystyle\frac{1}{(i+\alpha)\Omega_{D}}\left\langle\frac{f_{M}(v)}{\frac{\gamma_{eg}-i\delta}{(i+\alpha)\Omega_{D}}+v/\bar{v}}\right\rangle_{v}.$
(29)

In this case, the fraction in the integrand can be rewritten in the form

$\frac{1}{\frac{\gamma_{eg}-i\delta}{(i+\alpha)\Omega_{D}}+v/\bar{v}}=\frac{1}{i\xi+\theta}=-i\frac{\xi}{\xi^{2}+\theta^{2}}+\frac{\theta}{\xi^{2}+\theta^{2}}\,,$
(30)

where

$\xi=\text{Im}\left\\{\frac{\gamma_{eg}-i\delta}{(i+\alpha)\Omega_{D}}\right\\}\,,\quad\theta=v/\bar{v}+\text{Re}\left\\{\frac{\gamma_{eg}-i\delta}{(i+\alpha)\Omega_{D}}\right\\}.$
(31)

Then, in the case of $\xi\rightarrow 0$ for Eq. (30), we have

$\frac{\xi}{\xi^{2}+\theta^{2}}\rightarrow\pi\,\text{sign}[\xi]\,\delta_{\rm
Dirac}(\theta)\,,$ (32)

where $\delta_{\rm Dirac}(...)$ denotes the Dirac $\delta$-function.
Therefore, in principle, one can expect a violation of the analyticity (for
example, a discontinuity) of the function $\alpha(\delta)$ at the points where
$\xi=0$ (due to the non-analyticity of the function $\text{sign}[\xi]$ at the
point $\xi=0$).

Figure 6:  The dependence of the absolute value $|F(x+iy)|$ on the complex
variable $\alpha=x+iy$ in the plane ($x,y$):
(a) the case when a point for which $|F(x+iy)|=0$ exists,
$\\{nk^{-3}_{0}=0.1;\,\Omega_{D}/\gamma_{0}=50;\,\gamma_{eg}/\gamma_{0}=0.5;\,\delta/\gamma_{0}=20\\}$;
(b) the case when a point for which $|F(x+iy)|=0$ does not exist,
$\\{nk^{-3}_{0}=0.2;\,\Omega_{D}/\gamma_{0}=50;\,\gamma_{eg}/\gamma_{0}=0.5;\,\delta/\gamma_{0}=20\\}$.

Let us show that if the following condition

$\frac{3\pi nk^{-3}_{0}\gamma_{0}}{\Omega_{D}}\ll 1$ (33)

is satisfied [see Eq. (26) in the main text], the presence of a root of the
equation (13) must be accompanied by the inequality

$\xi=\text{Im}\left\\{\frac{\gamma_{eg}-i\delta}{(i+\alpha)\Omega_{D}}\right\\}<0\,.$
(34)

Indeed, using the expressions (A) and (30), we can rewrite Eq. (13) in the
following form:

$(\alpha^{2}+2i\alpha)(i+\alpha)=-\frac{i3\pi
nk^{-3}_{0}\gamma_{0}}{\Omega_{D}}\left\langle\frac{f_{M}(v)}{i\xi+\theta}\right\rangle_{v}.$
(35)

Obviously, if the condition Eq. (33) holds, $|\alpha|\ll 1$ holds also. Then,
discarding the small terms $\alpha^{2}$ and $\alpha^{3}$ in the right-hand
side of Eq. (35), we obtain an approximate equation

$-2\alpha\approx-\frac{i3\pi
nk^{-3}_{0}\gamma_{0}}{\Omega_{D}}\left\langle\frac{f_{M}(v)}{i\xi+\theta}\right\rangle_{v},$
(36)

whence follows

$2{\rm Re}[\alpha]\approx\frac{3\pi
nk^{-3}_{0}\gamma_{0}}{\Omega_{D}}\left\langle\frac{\xi
f_{M}(v)}{\xi^{2}+\theta^{2}}\right\rangle_{v}.$ (37)

Because in our case the wave propagates from left to right, the physical
solution must satisfy the condition ${\rm Re}[\alpha]<0$, i.e., $\xi<0$. Thus,
Eq. (37) leads to the inequality (34), which corresponds to Eq. (27) in the
main text.

Note that it is possible to easily check the presence/absence of a solution
for Eq. (13) using a graphical method. To do this, we define the complex
function $F(\alpha)$ as follows:

$F(\alpha)=\alpha^{2}+2i\alpha+i\left\langle\frac{3\pi
nk^{-3}_{0}\gamma_{0}f_{M}(v)}{\gamma_{eg}-i\delta+(i+\alpha)\Omega_{D}v/\bar{v}}\right\rangle_{v}.$
(38)

Now you can build a three-dimensional graph of the dependence of the absolute
value $|F(x+iy)|$ on the complex variable $\alpha=x+iy$ in the plane ($x,y$)
for fixed values
$\\{nk^{-3}_{0}\gamma_{0};\,\Omega_{D};\,\gamma_{eg};\,\delta\\}$. In this
case, the existence of a solution to the equation (13) corresponds to the
presence of a point for which $|F(x+iy)|=0$. Indeed, Fig. 6(a) shows the
dependence $|F(x+iy)|$ in the case when such a point exists. However, if we
take another set of parameters
$\\{nk^{-3}_{0}\gamma_{0};\,\Omega_{D};\,\gamma_{eg};\,\delta\\}$, for which
the solution of the equation (13) is absent, then the strict inequality
$|F(x+iy)|>0$ always holds, as can be seen in Fig. 6(b).

## Appendix B

Let us consider experimental verification of the results obtained above when
scanning the frequency inside the Doppler absorption line,
$|\delta|<2\Omega_{D}$, in the transmission signal. First of all, the initial
amplitude of the field $E_{0}$ at the entrance to the medium should be small
enough to satisfy the linear approximation for the Bloch equations [see Eq.
(4) in the main text], i.e., small saturation limit. In the case of the
transition $J_{g}=0\rightarrow J_{e}=1$, which exactly corresponds to the two-
level atom, this requires the following condition:

$\frac{|d_{eg}E_{0}|^{2}}{\gamma^{2}_{eg}}\ll 1\,.$ (39)

For an optical dipole transition ($\gamma_{0}/2\pi\sim 1$-10 MHz), this
approximately corresponds to the initial intensity $I_{0}<0.1$-1 mW/cm2, and
the diameter of the light beam can be arbitrary. Considering the high level of
the absorption coefficient Re[$\alpha(\delta)$] (see Figs. 2-5 and Eq. (16) in
the main text), we can estimate the thickness of the atomic cell ${\cal L}<$
10 $\mu$m (depending on the wavelength $\lambda$), which allows study of the
Doppler lineshape in the transmission signal right up to $nk^{-3}_{0}\sim 1$.

Figure 7:  Frequency dependence of the normalised transmitted intensity
$I(\delta,{\cal L})/I_{0}$ through atomic cells of length ${\cal
L}$$=$$5\lambda$ and ${\cal L}$$=$$10\lambda$ for various atomic densities
(from top to bottom):
(a) $nk^{-3}_{0}=0.01;\,0.03;\,0.05;\,0.1;\,0.15;\,0.2;\,0.3;\,0.5$ for two-
level atom with coefficient of the homogeneous broadening $A=5.7$ (see Eq.
(28) in the main text);
(b) $nk^{-3}_{0}=0.1;\,0.3;\,0.5;\,1.0;\,3.0;\,10.0$ for the 87Rb D1 line in
the region of hyperfine transitions $F_{g}=1\to F_{e}=1$ and $F_{g}=1\to
F_{e}=2$ with coefficient of the homogeneous broadening $A=\pi$ (see Eq. (28)
in the main text).
The calculations correspond to the value $\Omega_{D}/\gamma_{0}=60$.

An additional difficulty lies in choosing of suitable atom with an isolated
transition $J_{g}=0\rightarrow J_{e}=1$. The even isotopes (with zero nuclear
spin) of alkaline earth atoms (e.g., Mg, Ca, Sr, Yb), which have a closed
1S0$\to^{1}$P1 transition in the optical range, formally seem most convenient.
However, the melting point for these elements is very high ($\sim 1000$ K),
which makes it difficult to use an atomic vapor-cell. Nevertheless, for even
isotopes of calcium atoms (40Ca) vapor-cell spectroscopy is possible [A. S.
Zibrov, et al., Appl. Phys. B, 59, 327 (1994)]. Also it is possible to use
even isotopes 196Hg-204Hg of the mercury atom (melting point 234 K) with a
convenient for our purposes intercombination transition 1S0$\to^{3}$P1
($\lambda=253.7$ nm, $\gamma_{0}/2\pi=1.3$ MHz).

As for alkali metal atoms (e.g., Rb, Cs), which are most often used in laser
spectroscopy of atomic media, there always are several hyperfine degenerated
states (i.e., with Zeeman sublevels). This leads to a complicated lineshape
due to several closely lying Doppler absorption lines, and a redistribution of
populations over the lower hyperfine states and Zeeman sublevels due to
spontaneous transitions. As a result, the general condition for the linear
regime of atom-light interaction has a form different from (39):

$\displaystyle\gamma_{0}\bar{\tau}\frac{|d_{eg}E_{0}|^{2}}{\gamma^{2}_{eg}}\ll
1\quad{\rm if}\quad(\gamma_{0}\bar{\tau})>1\,,$ (40)
$\displaystyle\frac{|d_{eg}E_{0}|^{2}}{\gamma^{2}_{eg}}\ll 1\quad{\rm
if}\quad(\gamma_{0}\bar{\tau})\leq 1\,,$

where $\bar{\tau}$ is the flight average time of atoms through the light beam.
If the beam diameter $d_{b}$ is much less than the cell thickness ${\cal L}$,
then $\bar{\tau}\sim d_{b}/\bar{v}$. And vice versa, if $d_{b}>{\cal L}$, then
$\bar{\tau}\sim{\cal L}/\bar{v}$. Thus, for D1 and D2 lines of alkali metal
atoms, the condition (40) is satisfied for the initial intensity $I_{0}<0.1$-1
mW/cm2, if the beam diameter $d_{b}\sim$ 1-10 mm and cell thickness ${\cal
L}<$ 10 $\mu$m [i.e., ${\cal L}\ll d_{b}$ and $(\gamma_{0}\bar{\tau})\leq 1$].
Similar estimates are also valid for molecular gases with hyperfine and
vibrational-rotational structures of energy levels.

To illustrate our estimates, Fig. 7 shows the dependences of the transmitted
intensity described by the Eq. (16):

$I({\delta,\cal L})=I_{0}e^{4\pi{\rm Re}[\alpha(\delta)]{\cal L}/\lambda}\,,$
(41)

for the transition $J_{g}=0\to J_{e}=1$ [see Fig. 7(a)] and the D1 line of
87Rb [see Fig. 7(b)] for various values of atomic density at atomic cell
thicknesses of ${\cal L}=5\lambda$ and ${\cal L}=10\lambda$.

In the case of large detunings $|\delta|\gg\Omega_{D}$, the linear regime of
atom-light interaction is essentially simplified in relation to (39)-(40),
that allows the use of thick atomic cells (${\cal L}\sim$ 1-10 mm) and
significantly higher light intensities. However, in our opinion, the
information on possible quasi-collective effects will be noticeably leveled
out in this case. In particular, the frequency interval $\Delta_{\rm break}$,
within which the solution of Eq. (13) [or Eqs. (23),(24)] may be absent, will
be outside of the investigated frequency range.

## References

  * (1) R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems ot two-level atoms,” Phys. Rep. 7, 101 (1973).
  * (2) E. Lewis, “Collisional relaxation of atomic excited states, line broadening and interatomic interactions,” Phys. Rep. 58, 1 (1980).
  * (3) H. A. Lorentz, The Theory of Electrons: and Its Applications to the Phenomena of Light and Radiant Heat (Dover, New York, 2011).
  * (4) A. P. Kazantsev, “Kinetic equation for a gas of exited atoms,” Sov. Phys. JETP 24, 1183 (1967).
  * (5) Yu. A. Vdovin and V. M. Galitskii, “Dielectric constant of a gas of resonant atoms,” Sov. Phys. JETP 25, 894 (1967).
  * (6) Yu. A. Vdovin and N. A. Dobrodeev, “Dielectric constant of alkali-metal vapor or hydrogen at resonance frequencies,” Sov. Phys. JETP 28, 544 (1969).
  * (7) R. P. Srivastava and H. R. Zaidi, “Lorentz-Lorenz shift in gases,” Journal of Quantitative Spectroscopy and Radiative Transfer 16, 301 (1976).
  * (8) M. Fleischhauer and S. F. Yelin, “Radiative atom-atom interactions in optically dense media: Quantum corrections to the Lorentz-Lorenz formula,” Phys. Rev. A 59, 2427 (1999).
  * (9) S. Giovanazzi, A. G$\ddot{\rm o}$rlitz, and T. Pfau, “Tuning the Dipolar Interaction in Quantum Gases,” Phys. Rev. Lett. 89, 130401 (2002).
  * (10) R. L$\ddot{\rm o}$w, R. Gati 2, J. Stuhler, and T. Pfau, “Probing the light-induced dipole-dipole interaction in momentum space,” Europhysics Letters 71, 214 (2005).
  * (11) T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Reports on Progress in Physics 72, 126401 (2009).
  * (12) Ya. A. Fofanov, A. S. Kuraptsev, I. M. Sokolov, and M. D. Havey, “Dispersion of the dielectric permittivity of dense and cold atomic gases,” Phys. Rev. A 84, 053811 (2011).
  * (13) R. Friedberg and J. T. Manassah, “Initial cooperative decay rate and cooperative Lamb shift of resonant atoms in an infinite cylindrical geometry,” Phys. Rev. A 84, 023839 (2011).
  * (14) L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D1 line including resonant dipole-dipole interactions,” J. Phys. B 44, 195006 (2011).
  * (15) I. M. Sokolov, D. V. Kupriyanov, and M. D. Havey, “Microscopic Theory of Scattering of Weak Electromagnetic Radiation by a Dense Ensemble of Ultracold Atoms,” Journal of Experimental and Theoretical Physics 112, 246 (2011).
  * (16) J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, and C. S. Adams, “Cooperative Lamb Shift in an Atomic Vapor Layer of Nanometer Thickness,” Phys. Rev. Lett. 108, 173601 (2012).
  * (17) J. Javanainen, J. Ruostekoski, Y. Li, and S.-M. Yoo, “Shifts of a Resonance Line in a Dense Atomic Sample,” Phys. Rev. Lett. 112, 113603 (2014).
  * (18) J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sortais, A. Browaeys, S. D. Jenkins, and J. Ruostekoski, “Observation of Suppression of Light Scattering Induced by Dipole-Dipole Interactions in a Cold-Atom Ensemble,” Phys. Rev. Lett. 113, 133602 (2014).
  * (19) S. E. Skipetrov and I. M. Sokolov, “Absence of Anderson Localization of Light in a Random Ensemble of Point Scatterers,” Phys. Rev. Lett. 112, 023905 (2014).
  * (20) R. J. Bettles, S. A. Gardiner, and C. S. Adams, “Cooperative ordering in lattices of interacting two-level dipoles,” Phys. Rev. A 92, 063822 (2015).
  * (21) M. O. Scully, “Single Photon Subradiance: Quantum Control of Spontaneous Emission and Ultrafast Readout,” Phys. Rev. Lett. 115, 243602 (2015).
  * (22) A. A. Svidzinsky, F. Li, H. Li, X. Zhang, C. H. Raymond Ooi, and M. O. Scully, “Single-photon superradiance and radiation trapping by atomic shells,” Phys. Rev. A 93, 043830 (2016).
  * (23) R. J. Bettles, S. A. Gardiner, and C. S. Adams, “Enhanced Optical Cross Section via Collective Coupling of Atomic Dipoles in a 2D Array,” Phys. Rev. Lett. 116, 103602 (2016).
  * (24) S. D. Jenkins, J. Ruostekoski, J. Javanainen, R. Bourgain, S. Jennewein, Y. R. P. Sortais, and A. Browaeys, “Optical Resonance Shifts in the Fluorescence of Thermal and Cold Atomic Gases,” Phys. Rev. Lett. 116, 183601 (2016).
  * (25) S. D. Jenkins, J. Ruostekoski, J. Javanainen, S. Jennewein, R. Bourgain, J. Pellegrino, Y. R. P. Sortais, and A. Browaeys, “Collective resonance fluorescence in small and dense atom clouds: Comparison between theory and experiment,” Phys. Rev. A 94, 023842 (2016).
  * (26) A. S. Kuraptsev and I. M. Sokolov, “Light trapping in an ensemble of pointlike impurity centers in a Fabry-Perot cavity,” Phys. Rev. A 94, 022511 (2016).
  * (27) M. Wenzel, F. B$\ddot{\rm o}$ttcher, T. Langen, I. Ferrier-Barbut, and T. Pfau, “Striped states in a many-body system of tilted dipoles,” Phys. Rev. A 96, 053630 (2017).
  * (28) T. Peyrot, Y. R. P. Sortais, A. Browaeys, A. Sargsyan, D. Sarkisyan, J. Keaveney, I. G. Hughes, and C. S. Adams, “Collective Lamb Shift of a Nanoscale Atomic Vapor Layer within a Sapphire Cavity,” Phys. Rev. Lett. 120, 243401 (2018).
  * (29) A. S. Kuraptsev and I. M. Sokolov, “Influence of atomic motion on the collective effects in dense and cold atomic ensembles,” Phys. Rev. A 101, 033602 (2020).
  * (30) D. E. Chang, J. Ye, and M. D. Lukin, “Controlling dipole-dipole frequency shifts in a lattice-based optical atomic clock,” Phys. Rev. A 69, 023810 (2004).
  * (31) L. Ostermann, H. Zoubi, and H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express 20, 29634 (2012).
  * (32) L. Ostermann, H. Ritsch, and C. Genes, “Protected State Enhanced Quantum Metrology with Interacting Two-Level Ensembles,” Phys. Rev. Lett. 111, 123601 (2013).
  * (33) S. Krämer, L. Ostermann, and H. Ritsch, “Optimized geometries for future generation optical lattice clocks,” Europhys. Lett. 114, 14003 (2016).
  * (34) L. Henriet, J. S. Douglas, D. E. Chang, and A. Albrecht, “Critical open-system dynamics in a one-dimensional optical-lattice clock,” Phys. Rev. A 99, 023802 (2019).
  * (35) C. Qu and A. M. Rey, “Spin squeezing and many-body dipolar dynamics in optical lattice clocks,” Phys. Rev. A 100, 041602(R) (2019).
  * (36) G. Liu, Y. Huang, Z. Cheng, Z. Chen, and Z. Yu, “Rabi spectroscopy of three-dimensional optical lattice clocks,” Phys. Rev. A 101, 012504 (2020).
  * (37) A. Cidrim, A. Piñeiro Orioli, C. Sanner, R. B. Hutson, J. Ye, R. Bachelard, and A. M. Rey, “Dipole-Dipole Frequency Shifts in Multilevel Atoms,” Phys. Rev. Lett. 127, 013401 (2021).
  * (38) R. J. Bettles, S. A. Gardiner, and C. S. Adams, “Enhanced Optical Cross Section via Collective Coupling of Atomic Dipoles in a 2D Array,” Phys. Rev. Lett. 116, 103602 (2016).
  * (39) A. Glicenstein, G. Ferioli, N. $\check{\rm S}$ibali$\acute{\rm c}$, L. Brossard, I. Ferrier-Barbut, and A. Browaeys, “Collective Shift in Resonant Light Scattering by a One-Dimensional Atomic Chain,” Phys. Rev. Lett. 124, 253602 (2020).
  * (40) S. J. Masson, I. Ferrier-Barbut, L. A. Orozco, A. Browaeys, and A. Asenjo-Garcia, “Many-Body Signatures of Collective Decay in Atomic Chains,” Phys. Rev. Lett. 125, 263601 (2020).
  * (41) F. Andreoli, M. J. Gullans, A. A. High, A. Browaeys, and D. E. Chang, “Maximum Refractive Index of an Atomic Medium,” Phys. Rev. X 11, 011026 ( 2021).
  * (42) N. Fayard, L. Henriet, A. Asenjo-Garcia, and D. E. Chang, “Many-body localization in waveguide quantum electrodynamics,” Phys. Rev. Research 3, 033233 (2021).
  * (43) E. Sierra, S. J. Masson, and A. Asenjo-Garcia, “Dicke Superradiance in Ordered Lattices: Dimensionality Matters,” Phys. Rev. Research 4, 023207 (2022).
  * (44) R. H. Dicke, “The Effect of Collisions upon the Doppler Width of Spectral Lines,” Physical Review 89, 472 (1953).
  * (45) V. V. Batygin and I. M. Sokolov, “Some features of relaxation and spatial distribution of atomic polarization momenta in stationary optical pumping,” Physical Letters A 108, 29 (1985).
  * (46) The situation seems as follows: In order to solve the formal mathematical problem of the AMCMBE-catastrophe for the one-atom approximation and Doppler broadening ($\gamma_{0}\ll\Omega_{D}$), we should unexpectedly use a multi-atomic consideration that takes into account the interatomic dipole-dipole interaction. From a logical viewpoint, this looks rather strange. Indeed, in the opposite case of cold atoms, no mathematical catastrophe arises at all, because in the case of $\gamma_{0}\gg\Omega_{D}$ the solution of Eq. (13) formally exists for any value of $nk^{-3}_{0}$ and, therefore, there is no formal (i.e., purely mathematical) reason to take into account the atom-atom interaction. Therefore, we believe that this problem requires further careful analysis and as of now we still cannot completely exclude the existence of a physical meaning for the AMCMBE-catastrophe regime.