# Effective two-level approximation of a multi-level system driven by coherent
and incoherent fields

R. Veyron  V. Mancois  J.B. Gerent  G. Baclet  P. Bouyer  S. Bernon
<EMAIL_ADDRESS>LP2N, Laboratoire Photonique, Numérique et
Nanosciences, Université Bordeaux-IOGS-CNRS:UMR 5298, rue F.Mitterrand,
F-33400 Talence, France

(September 3, 2024)

###### Abstract

The numerical simulation of multiple scattering in dense ensembles is the
mostly adopted solution to predict their complex optical response. While the
scalar and vectorial light mediated interactions are accurately taken into
account, the computational complexity still limits current simulations to the
low saturation regime and ignores the internal structure of atoms. Here, we
propose to go beyond these restrictions, at constant computational cost, by
describing a multi-level system (MLS) by an effective two-level system (TLS)
that best reproduces the coherent and total scattering properties in any
saturation regime. The correspondence of our model is evaluated for different
experimentally realistic conditions such as the modification of the driving
field polarization, the presence of stray magnetic fields or an incoherent
resonant electromagnetic field background. The trust interval of the model is
quantified for the D$\phantom{}{}_{2}$-line of 87Rb atoms but it could be
generalized to any closed transition of a multi-level quantum system.

multi-level atom, scattering rates, saturation, coherent field, incoherent
background

###### pacs:

32.80.-t, 32.80.Cy, 32.80.Bx, 32.80.Wr

##  I  Introduction

The response of dense ensembles to coherent optical illumination is a
paradigmatic situation to study multiple scattering dynamics in which
collective effects can be prominent. They lead, for instance, to modifications
of the scattering properties such as line shifts and broadening [1] in 1D [2]
and 2D systems [3], sub- [4] and super-radiance [5], the optical phase profile
engineering [6] to control the reflection properties [7] of a single atomic
layer or the localization of light in different regimes [8, 9]. Simulations of
the coupled dipole equations in the linear-optics regime include interference
effects such as coherent backscattering [10, 11]which was also predicted using
random walk simulations including the atomic internal structure complexity
[12, 13].

In contrast with the preceding cases, when increasing the saturation
parameter, the system deviates from its linear response [14], requiring a full
quantum treatment that scales dramatically with the atom number. An ensemble
of $N$ Multi-Level Systems (MLS) with $k$-levels each yields the
diagonalization of a $k^{N}\times k^{N}$ matrix and is computationally out-of-
range when considering more than a few particles.

In a mean-field approach where entanglement between atoms is neglected, the
full density matrix can be factorized as the product state of single atom
density matrices reducing the matrix dimensions to $kN\times kN$ thus
improving the simulation capabilities up to few thousand particles. For a
given computational power, the traditional trade-off for numerical simulations
is either to consider the internal atomic structure [15, 16] while reducing
the number of particles or to model real atoms by two-level systems (TLS)
[17]. The latter allows a quantitative comparison with experiments only in
specific situations where the experimental conditions allow one to suppress
spurious transitions [1]. It represents a loss of generality and a limitation
for the general comparison of theory and experiments in the limit of dense and
saturated ensembles.

In this paper, we show that an effective TLS can properly approximate the
scattering properties of a MLS, with exact correspondence in certain
conditions. To this end, we numerically solve the optical Bloch equations
(OBE) for a single MLS driven by a coherent field that originates either from
a probe laser or from neighboring atoms via coherent scattering. We then fit
the effective TLS model parameters to the coherent and total scattering rates
obtained from the density matrix calculations. In this study, we detail the
influence of the driving field polarization, stray magnetic fields and
incoherent resonant electromagnetic field background on the effective TLS
parameters.

Sec. II introduces the relevant quantities discussed throughout this
manuscript and derives the optical response of a TLS driven by a coherent
field and an incoherent background. Sec. III details the calculation for the
exact solution of a MLS and its comparison to an effective TLS. As an example,
Sec. IV quantitatively compares the effective TLS that best corresponds to the
closed transition of the D2-line of 87Rb which is formed by the hyperfine
states $F_{g}=2$ and $F_{e}=3$. The numerical simulations performed in Sec. IV
could be carried for any multi-level quantum system with a closed transition.

##  II  Two-level system dynamics

Our study begins with the scattering dynamics of two-level systems. We remind
in Sec. II.1 the standard expressions [18, 19] of the density matrix elements
from which the coherent, incoherent and total scattering rates are derived. In
Sec. II.2, we derive the expressions of the same quantities when the two-level
systems TLS is driven by a coherent field and an incoherent field background.

###  II.1  Scattering rate under coherent drive

The total scattering rate is an essential quantity describing the influence of
the light field on the atom. It gives the number of photons emitted per unit
of time [18]: $R^{(tot)}_{sca}=\Gamma\rho_{ee}$ where $\Gamma$ is the natural
linewidth and $\rho_{ee}$ the excited state population. This total scattering
rate can be decomposed in terms of a coherent scattering rate
$R^{(coh)}_{sca}=\Gamma|{\rho}_{eg}|^{2}$ that represents scattering events
that are temporally coherent with respect to the driving field and an
incoherent scattering rate $R^{(inc)}_{sca}=R^{(tot)}_{sca}-R^{(coh)}_{sca}$
that, by energy conservation, is the difference between the two previous ones.
Computing these rates requires deriving expressions for the density matrix
population $\rho_{ee}$ and slowly varying coherence ${\rho}_{eg}$, both
obtained by solving the steady-state regime of the OBE driven by a coherent
field:

$\displaystyle{\rho}_{eg}=$
$\displaystyle\frac{-i}{\sqrt{2}}\frac{\sqrt{s_{c}^{\prime}}}{1+s_{c}^{\prime}}\frac{1+2i\delta}{\sqrt{1+4\delta^{2}}},$
(1) $\displaystyle\rho_{ee}=$
$\displaystyle\frac{1}{2}\frac{s_{c}^{\prime}}{1+s_{c}^{\prime}},$

where $s_{c}=2\Omega_{c}^{2}/\Gamma^{2}$ is the on-resonance saturation
parameter, $s_{c}^{\prime}=s_{c}/(1+4\delta^{2})$ is the effective saturation
parameter, $\Omega_{c}$ is the coherent Rabi frequency and
$\delta=(\omega-\omega_{0})/\Gamma$ the normalized detuning between the laser
frequency $\omega$ and the atomic transition $\omega_{0}$.

From Eqs. (1) the coherent and total scattering rates are given by:

$\displaystyle R^{(coh)}_{sca}=$
$\displaystyle\frac{\Gamma}{{2}}\frac{{s_{c}^{\prime}}}{\left(1+s_{c}^{\prime}\right)^{2}},$
(2) $\displaystyle R^{(tot)}_{sca}=$
$\displaystyle\frac{\Gamma}{2}\frac{s_{c}^{\prime}}{1+s_{c}^{\prime}}.$

In the regime of weak saturation ($s_{c}\ll 1$), the atom response is linear
in $s_{c}$ and temporally coherent. This is the regime of the linear dipole
approximation that is convenient for coupled dipole simulations. In the
opposite strong saturation regime ($s_{c}\gg 1$), the TLS can be saturated,
and the temporally incoherent scattering dominates. This is the regime of the
Mollow triplet where the incoherent scattering rate scales as ${s_{c}^{\prime
2}}/{\left(1+s_{c}^{\prime}\right)^{2}}$.

To move from an ideal TLS to the effective TLS, we follow the work of [20]
where the author computes analytically the scattering rates for the case of a
$\pi$-polarization and shows that the saturation intensity is reduced by a
factor $\alpha$. We then introduce an effective TLS ansatz under a saturating
driving field and in perturbed conditions:

$\displaystyle R^{(coh)}_{sca}=$
$\displaystyle\frac{\Gamma}{{2}}\frac{\beta}{\alpha}\frac{s_{c}^{\prime}/\alpha}{\left(1+s_{c}^{\prime}/\alpha\right)^{2}},$
(3) $\displaystyle R^{(tot)}_{sca}=$
$\displaystyle\frac{\Gamma}{2}\frac{s_{c}^{\prime}/\alpha}{1+s_{c}^{\prime}/\alpha}.$

Eqs. (3) correspond to an effective TLS with a corrected saturation parameter
$s_{c}^{\prime}/\alpha$. The factor $\beta/\alpha$ accounts for multi-level
corrections (Sec. IV) where $\alpha$ comes from a geometric factor due to the
coupling strength of the transitions and $\beta$ is an amplitude factor of the
coherent scattering field.

###  II.2  Scattering rate in coherent and incoherent drives

For a TLS, coherences between atomic states are driven by the field complex
amplitude. Therefore, a temporally incoherent field (frequency broadband
and/or temporally isotropic polarization) gives an average coherence of zero
for averaging time longer than the field spectral width. Thus, only the
intensity of the incoherent field affects the OBE by incoherently pumping the
populations at a rate of $\Gamma s_{i}$/2 where $s_{i}$ is an effective
saturation parameter for the incoherent intensity [18]. For a coherent field
with Rabi frequency $\Omega_{c}$ and detuning $\delta$, and an incoherent
intensity with saturation parameter $s_{i}$, one obtains the following OBE:

$\displaystyle\frac{\partial{{\rho}}_{ge}}{{\partial}t}=$
$\displaystyle-(i{\delta\Gamma}+\frac{\Gamma}{2}){\rho}_{ge}+\frac{i\Omega_{c}}{2}(\rho_{gg}-\rho_{ee}),$
(4) $\displaystyle\frac{\partial{\rho}_{gg}}{{\partial}t}=$
$\displaystyle-\frac{s_{i}}{2}\Gamma\left(\rho_{gg}-\rho_{ee}\right)+\frac{i\Omega_{c}}{2}({\rho}_{ge}-{\rho}_{eg})+\Gamma\rho_{ee}.$

The steady-state solution of Eqs.(4) is:

$\displaystyle{\rho}_{eg}=$
$\displaystyle{\frac{-i}{\sqrt{2}}\frac{\sqrt{s_{c}^{\prime}}}{\left(1+s_{c}^{\prime}+s_{i}\right)}\frac{1+2i\delta}{\sqrt{1+4\delta^{2}}}},$
(5) $\displaystyle\rho_{ee}=$
$\displaystyle\frac{1}{2}\frac{s_{c}^{\prime}+s_{i}}{1+s_{c}^{\prime}+s_{i}}.$

Eqs. (5) shows that populations can be transferred by both the coherent and
incoherent light while coherences are driven only by the coherent field but
damped in the saturation regime by both fields i.e. coherences are reduced by
the total saturation parameter $s_{c}^{\prime}+s_{i}$.

The above expressions of the density matrix elements of a TLS in an incoherent
background (Eqs. 5) allow generalizing the effective TLS ansatz (Eqs. 3) in:

$\displaystyle R^{(coh)}_{sca}=$
$\displaystyle{\frac{\Gamma}{2}\frac{\beta}{\alpha_{\rm
eff}}\frac{{\frac{s_{c}^{\prime}}{\alpha_{\rm
eff}}}}{\left(1+\frac{s_{c}^{\prime}}{\alpha_{\rm eff}}\right)^{2}}},$ (6)
$\displaystyle R^{(tot)}_{sca}=$
$\displaystyle\frac{\Gamma}{2}\left(\frac{\frac{s_{c}^{\prime}}{\alpha_{\rm
eff}}}{1+\frac{s_{c}^{\prime}}{\alpha_{\rm
eff}}}+\frac{\frac{s_{i}}{\alpha_{c}}}{1+\frac{s_{i}}{\alpha_{c}}}\right),$

where the corrections to the scattering rates are:

$\displaystyle\alpha_{\rm eff}=$ $\displaystyle\alpha\left(1+s_{i}\right),$
(7) $\displaystyle\alpha_{c}=$ $\displaystyle 1+s_{c}^{\prime}.$

##  III  Multi-level system dynamics

In this section, we give the master equation that describes the density matrix
evolution of a multi-level atomic system and the coherent and total scattering
rates [18].

The master equation in the rotating frame at $\omega$ includes the hyperfine
splitting Hamiltonian ${{H}}_{HF}$ coming from the atomic energy structure,
the first order Zeeman magnetic shift Hamiltonian ${{H}}_{B}$ and the
electric-dipole interaction Hamiltonian between the atom and the driving
field:

$\displaystyle\frac{d{\rho}}{dt}=$
$\displaystyle-\frac{i}{\hbar}\left[{{H}}_{HF}+{{H}}_{B}+{{H}}_{AF},{\rho}\right]$
(8)
$\displaystyle+\Gamma\left(\frac{2J_{e}+1}{2J_{g}+1}\right)\sum_{q}D[\Sigma_{q}]{\rho},$

where $\Sigma_{q}$ is the lowering operator for the polarization
$q=m_{g}-m_{e}$, $D$ the Lindblad superoperator defined as
$D[\Sigma_{q}]{\rho}=\Sigma_{q}{\rho}\Sigma_{q}^{\dagger}-1/2\left(\Sigma_{q}^{\dagger}\Sigma_{q}{\rho}+{\rho}\Sigma_{q}^{\dagger}\Sigma_{q}\right)$.
The electric-dipole Hamiltonian is ${{H}}_{AF}=-\hat{\mathbf{d}}.{\mathbf{E}}$
where $\hat{\mathbf{d}}=-e.\hat{\mathbf{r}}$ is the dipole moment and
$\mathbf{E}=E_{0}\boldsymbol{\epsilon}e^{i\omega t}+c.c.$. In this work, the
master equation has been traced over the environment to obtain the OBE for the
multi-level atom. Rewriting the density matrix components as a column vector
$\boldsymbol{\rho}$, we obtain a set of linearly coupled equations:

$\frac{d\boldsymbol{\rho}}{dt}=\mathbb{M}\boldsymbol{\rho},$ (9)

where the matrix $\mathbb{M}$ contains the matrix elements for all
interactions and are detailed in Eq. (19) in the Appendix.

The last interaction to be added to the master equation is an incoherent field
background. The interaction between a broadband incoherent field such as a
black-body radiation and a MLS has been previously studied [21, 22, 23].
Master equations enable one a detailed description of the system composed of
an atom and a bath. We obtain the Lindblad equations via two simplifications.
First the Born-Markov approximation assumes a weak coupling
($\Gamma\ll\omega_{0}$) and a vanishing bath memory time [21, 22, 23].
Secondly the secular approximation, where the coherences and the populations
evolve independently, neglects interference effects (Zeeman coherences). Under
these approximations, the master equation of an incoherently driven system
simplifies to rate equations on the populations only. As detailed in the
Appendix Appendix, under the influence of an on resonance temporally
incoherent field, the master equation of a coherently driven system is
modified by including terms in the matrix $M$ to account for incoherent
population transfer.

For a multi-level atom, the coherent and total scattering rates in the steady-
state are defined as:

$\displaystyle R^{(coh)}_{sca}=$
$\displaystyle\Gamma\sum_{q}|\langle\Sigma_{q}\rangle|^{2},$ (10)
$\displaystyle R^{(tot)}_{sca}=$
$\displaystyle\Gamma\sum_{q}\langle\Sigma_{q}^{\dagger}\Sigma_{q}\rangle.$

In our method, these rates are obtained by solving the MLS master equation in
the steady-state by setting ${d{\boldsymbol{\rho}}}/{dt}=0$ either with
symbolic computations for pure polarizations to determine the exact analytical
formulas or numerically otherwise. We checked that the results of both methods
are totally consistent. From the density matrix solution, we compute the exact
coherent and total scattering rates using Eq. (10). The parameters
$\alpha_{\rm eff}$ and $\beta_{\rm eff}$ are then obtained by fitting the
exact rates with the effective TLS model from Eq. (6). For simplicity, in the
following, these parameters will be noted as $\alpha$ and $\beta$.

##  IV  Effective TLS of the D2-line of rubidium 87

To study the role of experimental imperfections such as polarization
orientation, DC magnetic fields and incoherent background on the scattering
rates, we restrict our model to multiple degenerate closed states. As an
example, we choose to simulate our model on all Zeeman states of the
transition $\ket{F_{g}=2}\rightarrow\ket{F_{e}=3}$ of the 87Rb
D$\phantom{}{}_{2}$-line and restrict ourselves to situations where the power
broadening is much smaller than the hyperfine energy splitting. Also, the
ratio of scattering rates between the $\ket{F_{e}=3}$ and $\ket{F_{e}=2}$
states (44 $\Gamma$ detuned from $\ket{F_{e}=3}$) is about 1000 for $s_{c}=60$
on the closed transition. As a result, the transition from $\ket{F_{g}=2}$ to
$\ket{F_{e}=3}$ is considered to be closed in the range $s_{c}\in[0.1,30]$. In
the following, we therefore neglect the residual coupling to other hyperfine
excited and ground states. In the steady-state regime, these coupling would
lead to depumping out of the considered transition. Our study is therefore
valid only before depumping occurs and is robust for closed transitions.

The role of the driving field polarization (Sec. IV.1), DC magnetic field
(Sec. IV.2) and isotropic incoherent field (Sec. IV.3) are studied
independently. The quantization axis is taken as $\boldsymbol{\epsilon}_{z}$.

###  IV.1 Role of polarization: $\sigma_{\pm}$, $\pi$ and elliptical

(a) $\Theta=0$$y$$x$$\zeta=0$$\zeta=1$$0<\zeta<1$$z$ |  (b) $\Theta=\frac{\pi}{2}$$z$$x$$\zeta=0$$\zeta=1$$0<\zeta<1$$y$
---|---
Figure 1: Two standard polarization cases and their dependence on ellipticity
$\zeta$ and quantization axis (${z}$): a) For $\Theta=0$, the polarization
lies in the (x,y) plane and is circular for $\zeta=1$ with a perpendicular
quantization axis, b) For $\Theta=\pi/2$, the polarization lies in the (x,z)
plane and is linear for $\zeta=0$ with a parallel quantization axis.

In this section, only the coherent drive field polarization is being changed
at zero magnetic field and zero incoherent field. We use the spherical basis
with respect to the Cartesian as follows:

$\displaystyle\boldsymbol{\epsilon}_{\pm}=$
$\displaystyle\mp\frac{\left(\boldsymbol{\epsilon}_{x}\pm
i\boldsymbol{\epsilon}_{y}\right)}{\sqrt{2}},$ (11)
$\displaystyle\boldsymbol{\epsilon}_{0}=$
$\displaystyle\boldsymbol{\epsilon}_{z}.$ (12)

This polarization
${\boldsymbol{\epsilon}}=({\epsilon}_{-},{\epsilon}_{+},{\epsilon}_{0})$ is
parametrized in the spherical basis
$(\boldsymbol{\epsilon}_{-},\boldsymbol{\epsilon}_{+},\boldsymbol{\epsilon}_{0})$
by an ellipticity $\zeta=E_{0x}/E_{0y}$ and $\pi$ polarization projection
angle $\Theta$ (Fig. 1) as:

$\displaystyle{\epsilon}_{\pm}=$
$\displaystyle\mp\frac{1}{\sqrt{2}\norm{{\boldsymbol{\epsilon}}}}(\zeta\mp\cos\Theta),$
(13) $\displaystyle{\epsilon}_{0}=$
$\displaystyle\frac{i}{\norm{{\boldsymbol{\epsilon}}}}\sin\Theta,$

where $\norm{{\boldsymbol{\epsilon}}}=\sqrt{1+\zeta^{2}}$. Using the TLS
ansatz of Eq. (3), we evaluate the parameters
$\alpha\equiv\alpha_{\boldsymbol{\epsilon}}$ and
$\beta\equiv\beta_{\boldsymbol{\epsilon}}$ that best match the exact
scattering rates. For a given $\zeta$, both scattering rates
$R^{(coh)}_{sca},R^{(tot)}_{sca}$ are computed numerically as a function of
the coherent saturation parameter $s_{c}$. $\alpha_{\boldsymbol{\epsilon}}$
and $\beta_{\boldsymbol{\epsilon}}$ are the best fitting parameter for
$s_{c}\in[0.1,30]$.
In the limit case of a $\sigma_{-}$ circular polarization
$\boldsymbol{\epsilon}=(1,0,0)$ parametrized by $(\zeta,\Theta)=(1,0)$, the
atom is pumped in a perfect two-level cycling transition and is expected to
reach the maximal scattering cross section $\sigma_{0}={3\lambda^{2}}/{2\pi}$
with $\alpha_{\sigma}=\beta_{\sigma}=1$. In the opposite limit of a $\pi$
linear polarization $\boldsymbol{\epsilon}=(0,0,1)$, the populations and
coherences have been computed analytically [20]. The system formed by five
$\pi$-transitions for $\ket{F_{g}=2}$ to $\ket{F_{e}=3}$ is equivalent to a
TLS with a reduced cross section for which $\alpha_{\pi}=461/252=1.829$ and
$\beta_{\pi}=1$.

Figure 2: a) $\alpha$ and b) $\beta$ as a function of the ellipticity $\zeta$
for a polarization in the (x,y) plane ($\Theta=0$), c) input and output
intensities per polarization, d) scalar product of the input and output
fields.

In Fig. 2, intermediate ellipticities are obtained by varying $\zeta$ from 1
(circular) to $0$ (linear) at $\Theta=0$. The values of
$\alpha_{\boldsymbol{\epsilon}}$ and $\beta_{\boldsymbol{\epsilon}}$ are exact
(error bars go to 0) which means that the scattering rates are also exactly
described by Eq. (3). It does not necessarily mean that this situation is
exactly equivalent to a TLS, that is a scalar scattering problem, since the
MLS is vectorial. The circular and linear polarizations yield the expected
values $\alpha_{\sigma}=1$ and $\alpha_{\pi}=1.829$. Note that the linear
polarization is at a $None$ angle of the $x$ and $y$ axis. In absence of
magnetic field bias, the electric field sets the quantization axis. The same
curves would be obtained for $\Theta=\pi/2$. An imperfect polarization as
could occur in experiments induces little changes for the circular
polarization ($\zeta=0$) -below $10\%$ variation on the parameters- even up to
$\zeta=0.5$, while it has more effect for a linear polarization. This is due
to optical pumping in the closed transition that protects the atomic state.
For the same reason, the polarization of the radiated field differs from the
input polarization at maximum by 8$\%$ for an ellipticity about $\zeta=0.2$.

###  IV.2 Role of a DC magnetic field

Under a constant magnetic field, magneto-optical effects occur. We refer to
the review [24] for more detail. The Faraday effect, for example, results in
the optical rotation and ellipticity change of the output scattering. The
scattering process is therefore a truly vectorial problem that cannot be
exactly mapped onto the TLS solution due to the output polarization.

Figure 3: a) $\alpha$ and b) $\beta$ as a function of the ellipticity $\zeta$
in the case of $\Theta=0$. $\zeta=1$ is circular and $\zeta=0$ is linear along
$\boldsymbol{\epsilon}_{x}$ and does not correspond to a $\pi$ polarization.
For any ellipticity, $\mathbf{B}$ is perpendicular to the electric field. The
maximum and the standard deviation of the relative error between the
approximated TLS solution and the exact calculation of the c) excited state
population and d) density matrix coherence.

In this section, we focus the analysis on the TLS parameters $\alpha$ and
$\beta$ that best mimic the scattering rate amplitude when the Zeeman
degeneracy is lifted by a magnetic field bias such that
$\mathbf{B}=B\boldsymbol{\epsilon}_{z}$. The driving field frequency is kept
constant and is equal to the unshifted transition. The magnetic Zeeman shift
between two states $\ket{F_{e},m_{e}}$ and $\ket{F_{g},m_{g}}$ is
$\Delta(B)=\mu_{b}B/\hbar\left(g_{F_{e}}m_{g}-g_{F_{g}}m_{g}\right)$ where
$g_{F_{e/g}}$ are the Landé factors and $\mu_{b}$ is the Bohr magneton (see
Appendix Appendix). The strength of magnetic field considered in this study
are up to $2$G which is well below the restriction to first order Zeeman
perturbation and results in a frequency shift smaller than $\Gamma$. The state
$\ket{F_{g}=2}$ (resp. $\ket{F_{e}=2}$) has a frequency sensibility to
magnetic shift of 0.12$\Gamma$/G (resp. 0.15$\Gamma$/G).

In a $\sigma_{-}$ polarization case,
$\alpha=\beta=\alpha_{\sigma}(1+4\left(\delta_{\omega}+\delta_{B}\right)^{2})$
where $\delta_{B}=\mu_{b}B/\hbar\Gamma$. It simply corresponds to a TLS probed
off-resonantly due to the Zeeman shift and the driving field detuning. Due to
optical pumping, only the two states $\ket{F_{g}=2,m_{g}=-2}$ and
$\ket{F_{e}=3,m_{e}=-3}$ are occupied in the steady-state. The atomic response
is the one of a TLS and the results of Eq. (2) are exactly recovered. In this
situation, the Zeeman shift can be experimentally compensated by the driving
field detuning. Interestingly, for a linear polarization aligned with a
magnetic bias ($\pi$ polarization), the scattering rates are also exactly
given by the effective TLS Eqs. (2) with
$\alpha=\alpha_{\pi}\left(1+4\delta_{B}^{2}\frac{41}{1008\alpha_{\pi}}+4\delta_{\omega}^{2}\right)$
and $\beta=1$. These expressions can be derived by solving the linear system
of Eqs. (9) containing all $\pi-$transitions. In this situation, the
sensitivity of the scattering rates to the detuning is reduced by a factor
$\frac{41}{1008\alpha_{\pi}}\approx 0.02$ with respect to the $\sigma$
polarization case. As a result, for pure $\sigma$ or $\pi$ polarizations,
$\alpha$ and $\beta$ have well known lower and upper limits.

Figure 4: a) $\alpha$ and b) $\beta$ as a function of $\epsilon$ in the case
of $\Theta=\pi/2$. $\zeta=1$ is circular and $\zeta=0$ is linear along
$\boldsymbol{\epsilon}_{z}$ and does correspond to a $\pi$ polarization. The
maximum and the standard deviation of the relative error between the
approximated TLS solution and the exact calculation of the c) excited state
population and d) density matrix coherence.

We consider now deviations from these ideal cases by studying the influence of
the ellipticity $\zeta$ in the $(x,y)$ plane corresponding to $\Theta=0$, and
in the $(x,z)$ plane corresponding to $\Theta=\pi/2$. $\alpha$ and $\beta$
from Eq. (6) are together fitted from the scattering rates obtained by
numerically solving Eq. (9) and plotted with error bars within $95\%$
confidence interval of the fits parameters.

As expected, Fig. 3 ($\Theta=0$) shows that the scattering rates are sensitive
to the polarization. The growing error bars for an increased ellipticity
indicate a growing deviation from the effective TLS behavior. The offset
between the curves is merely due to the Zeeman shift. The relative error
between the simulated scattering rate and the model of Eq. (3) is plotted in
terms of the maximum and the standard deviation of the relative error over the
range of the saturation parameter $s_{c}$ used for the fitting. It shows that
the model has a maximum relative error which does not exceed $6\%$ for the
total scattering rate, which happens only for an ellipticity of 1. The
coherent scattering rate is more sensitive to the ellipticity with up to
$50\%$ relative error. Nevertheless, for quasi-circular polarizations
($\zeta\approx 1$) as typically used in $\sigma$ absorption imaging, the MLS
reduces to a TLS.

Fig. 4 presents the influence of the ellipticity in the ($x,z$) plane. For
$\zeta=0$, corresponding to a $\pi$ polarization, the TLS is exact with little
dependence on magnetic field. As the ellipticity is increased, this TLS
behavior becomes less accurate with increasing error bars on the fitting
parameters. Consequently, $\pi$ polarization imaging will be less accurate to
define the absorption cross section and the atom numbers.

###  IV.3 Role of an incoherent drive

The scattering cross section is also modified in the presence of an incoherent
field. It could be for example generated by a thermal lamp, or by the
temporally incoherent response of the surrounding atomic gas to the coherent
excitation. Here, we consider the influence of a partially polarized
incoherent field on the coherent and incoherent scattering response of a MLS
probed by a coherent $\sigma_{-}$ polarized light under zero magnetic field
offset. Electromagnetic fields are here considered as temporally incoherent if
their coherence time is smaller than the Rabi period, meaning that the density
matrix coherences are zero on average while the excited state populations are
non-zero. An incoherent field yields optical pumping without coherence.

We consider a partially $\sigma_{-}$ polarized incoherent field that could for
example be generated by the incoherent scattering of a $\sigma_{-}$ polarized
coherent field. This incoherent field is parametrized by a polarization degree
$r\in[0,1]$ in the form $s_{i}^{\sigma}=rs_{i}$ and $s_{i}^{\rm
iso}=(1-r)s_{i}$ where the $s_{i}^{\sigma}$/$s_{i}^{\rm iso}$/$s_{i}$ are
respectively the $\sigma_{-}$ polarized / isotropic / total saturation
intensities expressed relatively to the saturation intensity of $\sigma_{-}$
polarized light. With such definition, $r=1$ describes a $\sigma_{-}$
polarized incoherent field while $r=0$ describes an isotropic incoherent
field.

In the limit $r=1$, both the coherent and incoherent fields are $\sigma_{-}$
polarized. The atomic population are pumped in the closed transition and the
situation is exactly described as in Sec. II.2 with scattering rates given by
Eq. (6) where $\delta=0$ and $\alpha=1$. In the opposite limit $r=0$, a purely
isotropic incoherent field drive will redistribute the ground state
population. In the absence of the coherent field drive ($s_{c}=0$), the
populations are equally redistributed among the Zeeman states. Interestingly,
we notice that for large saturation parameters $s_{i}$, the total population
in the excited state can be higher than $1/2$. It is indeed bounded by the
number of excited Zeeman states over the total number of states which is
$7/12$ for the considered transition of 87Rb. The total excited state
population $\rho_{ee}$ in the absence of coherent field $s_{c}=0$ can be
analytically derived from the master equation (Appendix Appendix) and is given
by:

$\displaystyle\rho_{ee}=$
$\displaystyle\frac{7}{12}\frac{s_{i}/\frac{30}{12}}{1+s_{i}/\frac{30}{12}}.$
(14)

In the low saturation regime i.e. $s_{i}\ll 1$, Eq. (14) reduces to
$\rho_{ee}=\frac{1}{2}\frac{s_{i}}{\alpha_{\rm iso}}$ where $\alpha_{\rm
iso}=\frac{15}{7}$ is the reduction of the cross section for an isotropic and
incoherent field.

We consider now the general case of a MLS driven simultaneously by a partially
polarized incoherent field ($r\in[0,1]$) and by a coherent drive $s_{c}$. In
the Appendix Appendix, the master equation given by Eq. (19) is expressed in
the form of Eq. (24) and the steady-state solutions of all elements of the
density matrix are obtained by an algebraic solver. The algebraic solution has
the form of a ratio of polynomials containing few 100 terms. This exact
algebraic solution of the scattering rates is compared to a phenomenological
model of a TLS:

$\displaystyle R_{sca}^{(coh)}/\Gamma$ $\displaystyle=$
$\displaystyle\frac{\beta}{2\alpha}\frac{s_{c}/\alpha}{\left(1+s_{i}^{\rm
iso}/\frac{30}{12}+s_{i}^{\sigma}/\alpha+s_{c}/\alpha\right)^{2}},$
$\displaystyle R_{sca}^{(tot)}/\Gamma$ $\displaystyle=$
$\displaystyle\eta(r,s_{i},s_{c})+\frac{1}{2}\frac{s_{c}/\alpha}{1+s_{i}^{\rm
iso}/\frac{30}{12}+s_{i}^{\sigma}/\alpha+s_{c}/\alpha},$
$\displaystyle\eta(r,s_{i},s_{c})$ $\displaystyle=$
$\displaystyle\frac{\eta_{0}-\eta_{\infty}}{1+\frac{s_{c}}{\alpha\left(1+s_{i}^{\rm
iso}/\frac{30}{12}+s_{i}^{\sigma}/\alpha\right)}}+\eta_{\infty},$ (15)

where $\alpha$, $\beta$, $\eta_{0}$ and $\eta_{\infty}$ are fitting
parameters. To minimize the number of fitted parameters, the saturation
intensity correction of the isotropic incoherent field was fixed to $30/12$.
In this model, $\alpha$ quantifies the reduction of the coherent absorption
cross section, $\beta<1$ quantifies the reduction of coherent field emission
with respect to a perfect TLS. $\eta(r,s_{i},s_{c})$ is the part of the total
excited state population induced by the incoherent field drive. This
contribution of the excited state population includes states which are coupled
to all three fields (e.g. $\ket{F_{e}=3,m_{F}=-3,-2,-1,0,1}$) and states
coupled only to the isotropic polarization (e.g. $\ket{F_{e}=3,m_{F}=2,3}$).
This excited state population $\eta(r,s_{i},s_{c})$ has a complex dynamic that
depends on the polarization ratio of the incoherent field $r$, the value of
the incoherent $s_{i}$ and coherent $s_{c}$ saturation intensities. For a
given couple of parameters $r$ and $s_{i}$, we observe that this contribution
varies monotonically from $\eta_{0}$ in absence of coherent drive and
saturates at $\eta_{\infty}$ for large coherent drive $s_{c}\gg s_{i}$. The
model given in Eq. (15) reproduces the saturation behavior with a crossover at
$s_{c}/\alpha=1+s_{i}^{\rm iso}/\frac{30}{12}+s_{i}^{\sigma}/\alpha$.
As an example, the total scattering rate and coherent scattering rate are
given as a function of $s_{c}$ for $r=0.5$ and $s_{i}=5$ in Fig. 5. We observe
that the equivalent TLS given by Eqs. (15) describes well the dynamics with a
standard deviation of the relative error below $1\%$ on this example. The
excited state population offset at $s_{c}=0$ is due to the incoherent drive.

Figure 5: a) Variations of the total excited state population and b)
$\sigma_{-}$-polarized coherences with $r=0.5$ and $s_{i}=5$ in blue and the
fits in red using Eq. (15).

We compare the TLS model and the exact solution by fitting the parameters
$\alpha(s_{i},r)$, $\beta(s_{i},r)$ $\eta_{0}(s_{i},r)$ and
$\eta_{\infty}(s_{i},r)$. The fits are realized at a fixed incoherent
intensity $s_{i}\in[0,10]$ and polarization degree $r\in[0,1]$ and for a
coherent intensity $s_{c}$ varying between 0 and $10s_{i}$. In this parameter
range ($s_{c}\leq 100$) the closed transition approximation holds at least for
500 scattering events. The results are shown in Fig. 6. As a comparison, the
expected parameters for an exact TLS are $\alpha=1,\beta=1$ and
$\eta_{0}=\eta_{\infty}$ given by the second term in Eq. (6) which are indeed
the exact solutions found for a polarized incoherent field ($r=1$). For $r=1$,
the MLS is optically pumped in an exact TLS. As the incoherent drive
polarization is randomized $r\rightarrow 0$, the fitted parameters slightly
deviate from these initial values with a complex behavior that is mostly
driven by optical pumping mechanisms. In absence of coherent drive, the
excited state population $\eta_{0}$ increases as a function of the incoherent
saturation intensity. For an isotropic incoherent drive, this excited state
population can even exceed 1/2 (cf. Eq. (14)) as all 12 states become equally
populated. In our model, the coherent scattering rate only depends on $\alpha$
and $\beta$. For large and unpolarized incoherent drive, the coherent
scattering rate can be reduced by up to a factor 1/2 with respect to the pure
TLS. This reduction is essentially due to Clebsch-Gordon coefficients entering
in the calculation of $\beta$. As checked numerically, the polarization of the
coherently scattered field is exactly aligned with the input field
($\sigma$-polarized). This was expected given that the incoherent field cannot
drive coherence between the Zeeman sub-levels and therefore does not alter the
scattered polarization (Eq. (23) in the Appendix).

Figure 6: Model parameters a) $\alpha$, b) $\beta$, c) $\eta_{0}$ and d)
$\eta_{\infty}$ as a function of the incoherent intensity $s_{i}$ for
polarization degrees $r\in[0,1]$ by step of 0.1. The shaded area represents
the $95\%$ confidence interval of the fits and is very small on the plots. In
c), the limit case $r=0$ and $r=1$ exactly coincide with the analytical
solutions given in Eq. (14) and the second term of Eq. (6) respectively.

##  V Discussion

The combined coherent and incoherent response of atoms in the saturated regime
has a strong impact on the interpretation of fluorescence and absorption
imaging of ensembles. Experimental imperfections are the common explanations
[25] for the reduction of the atomic cross section. With this argument, the
absolute determination of atom numbers is subject to a precise calibration of
the experimental conditions which is often questioned. In Fig. 2, we have
shown that the effective TLS description of a multi-level atom is very robust
to polarization imperfections especially for the case of $\sigma$ polarized
light. We attribute this robustness to optical pumping mechanisms that
protects the stretched-state hyperfine transition ($|F_{g}=2,m_{F}=-2\rangle$
to $|F_{e}=3,m_{F}=-3\rangle$ in $\sigma_{-}$). This pumping was additionally
observed in the steady-state solutions of the OBE via a strong imbalance of
the repartition of the population that favored the $|F_{g}=2,m_{F}=-2\rangle$
state even for $\zeta\in[0.5,1]$. This optical pumping protection of $\sigma$
transitions is not specific to the imaging transition of the
D$\phantom{}{}_{2}$ line of Rubidium and will be applicable to any Zeeman
degenerated closed transition. In the absence of Zeeman degeneracy, it was
additionally shown that MLS behaves exactly as an effective TLS for any drive
polarization. As expected for 87Rb, the cross section reduction factor
$\alpha$ varies from 1 ($\sigma$) to 1.829 ($\pi$) depending on the driving
field polarization.

On the other hand, stray magnetic field strongly impact the calibration of the
scattering cross section of a $\sigma$ polarized probe (Fig. 3(a)).
Nevertheless, this correction which is solely given by detuning of the
coherent drive from the extreme TL transition can be exactly compensated for
by tuning the drive frequency on resonance with the extreme TL transition in
experiments. It is therefore not a concern for cross section calibrations. In
addition, due to symmetry of the first order Zeeman splitting, the cross
section correction of a $\pi$ polarized probe is independent of the stray
magnetic field amplitude if it is aligned along the linear polarization axis.

As mentioned earlier, in the presence of a stray magnetic field, light
scattering is a vectorial process, and the coherent scattered field
polarization is not aligned with the drive. Nevertheless, in the specific case
of $\sigma$ and $\pi$ polarized light with a magnetic bias well-defined with
respect to light polarization, both atom-light and magnetic interaction are
diagonalized in the same basis leading to an aligned output field. The
scattering process is therefore scalar in these two situations with additional
robustness to imperfections for the $\sigma$ polarized drive (Fig. 2(d)).
Polarization and stray magnetic field imperfections have therefore little
influence on the reduction of the cross section.

On the other hand, we have observed that in the presence of an incoherent
background that would mimic the temporally and spatially incoherent scattering
from other atoms in the ensemble, the cross section is notably reduced which
results in systematics errors on absorption imaging measurements of atom
number in the saturation regime and/or at large optical thickness [26, 3]. For
large optical depth, most of the coherent drive is converted to an incoherent
field via multiple scattering mechanism. A total conversion of field leads to
$s_{i}=s_{c}$ which gives an upper bound for the modification of
$\alpha=1+s_{i}$.

For large saturations, the atom is driven in the Mollow triplet regime, and we
expect this large value of $\alpha$ to be mitigated by a reduction of the
reabsorption cross section [27]. In other words, the converted light will be
partly off-resonant with the atomic transition.

##  VI Conclusion

In this paper, we have thoroughly studied the total and coherent scattering
rates of a MLS atoms illuminated by different configurations of the
electromagnetic field that correspond to situations often encountered in
experimental realization. We proposed to map these scattering properties to
the one of an effective TLS model which is particularly relevant to reduce the
complexity of multiple scattering simulations. We have shown that, at zero
magnetic field, our effective TLS model describes exactly the scattering rates
of a MLS for any saturation parameter. In the presence of stray magnetic
fields that lift the Zeeman degeneracy, the amplitude of the scattered fields
is well described by the TLS model, with an exact mapping for the specific
case of $\sigma$ and $\pi$ polarizations. For other polarizations, vectorial
scattering (magneto-optical effects) can occur. Our scalar model cannot
exactly render such rotations but proved to be robust for $\sigma$ polarized
scattering. In the limit of strong saturation, incoherent scattering
dominates. In dense ensembles, the dynamic of a single atom will be affected
by this incoherent electromagnetic background. We have shown that, while the
MLS dynamic becomes complex, an equivalent TLS model can be adapted to
enlighten the general behavior and main scattering response. In particular, we
noticed that even though the intrinsic reduction $\alpha$ of the saturation
intensity stays close to 1, the effective reduction $\alpha_{\rm eff}$ is
proportional to the incoherent background intensity thus reducing the coherent
scattered field. This work gives an upper bound to situations in which a TLS
model applies and can be extended to any atom having a cycling transition.

## Acknowledgements

The authors thank W. Guerin, R. Bachelard and K. Vinck for helpful
discussions. R.V acknowledge PhD support from the university of Bordeaux,
J-B.G and V.M. acknowledge support from the French State, managed by the
FrenchNational Research Agency (ANR) in the frame of the Investments for the
future Programme IdEx Bordeaux-LAPHIA (ANR-10-IDEX-03-02). This work was also
supported by the ANR contract ANR-18-CE47-0001-01.

## Appendix

In the following, we briefly give the formalism used to calculate the steady-
states solutions of the density matrix that were used to evaluate the atomic
scattering cross section. A detailed derivation of the formalism can be found
in [18].

The lowering operator between two hyperfine states $F_{g}$ and $F_{e}$ is
given by:

$\Sigma_{q}=\sum_{\begin{subarray}{c}F_{g},m_{g},\\\
F_{e},m_{e}\end{subarray}}\lambda(F_{g},m_{g},F_{e},m_{e},q)\ket{F_{g},m_{g}}\bra{F_{e},m_{e}},$
(16)

where

$\displaystyle\lambda(F_{g},m_{g},F_{e},m_{e},q)=$
$\displaystyle(-1)^{F_{e}+J_{g}+1+I}\sqrt{(2F_{e}+1)(2J_{g}+1)}$
$\displaystyle\innerproduct{F_{g},m_{g}}{F_{e},m_{e};1,q}\begin{Bmatrix}J_{e}&J_{g}&1\\\
F_{g}&F_{e}&I\end{Bmatrix},$ (17)

with the curly brackets denoting the Wigner-6j symbol.
The expectation values involved in the coherent and total scattering rates of
Eqs. (10) using Eq. (16) are:

$\displaystyle\langle\Sigma_{q}\rangle=$
$\displaystyle\sum_{\begin{subarray}{c}F_{g},m_{g},\\\
F_{e},m_{e}\end{subarray}}\lambda(F_{g},m_{g},F_{e},m_{e},q){\rho}_{F_{e},m_{e},F_{g},m_{g}}$
$\displaystyle\sum_{q}\langle\Sigma_{q}^{\dagger}\Sigma_{q}\rangle=$
$\displaystyle(2J_{g}+1)/(2J_{e}+1)\sum_{F_{e},m_{e}}{\rho}_{F_{e},m_{e},F_{e},m_{e}}.$
(18)

So the coherent scattering is given by the optical coherences and the total
scattering by the population of the excited state.
Projecting Eq. (8) onto the general states $\bra{\alpha,m_{\alpha}}$ and
$\ket{\beta,m_{\beta}}$, the full master equation for an arbitrary number of
hyperfine and Zeeman states reads:

$\begin{split}\dot{{\rho}}_{\alpha,m_{\alpha},\beta,m_{\beta}}=&-\frac{i}{2}[\sum_{F_{e},m_{e}}\Omega^{*}(\alpha,m_{\alpha},F_{e},m_{e})\delta_{\alpha
g}{\rho}_{F_{e},m_{e},\beta,m_{\beta}}+\sum_{F_{g},m_{g}}\Omega(F_{g},m_{g},\alpha,m_{\alpha})\delta_{\alpha
e}{\rho}_{F_{g},m_{g},\beta,m_{\beta}}\\\
&-\sum_{F_{g},m_{g}}\Omega^{*}(F_{g},m_{g},\beta,m_{\beta})\delta_{e\beta}{\rho}_{\alpha,m_{\alpha},F_{g},m_{g}}-\sum_{F_{e},m_{e}}\Omega(\beta,m_{\beta},F_{e},m_{e})\delta_{g\beta}{\rho}_{\alpha,m_{\alpha},F_{e},m_{e}}]\\\
&-i\left(\delta_{\alpha g}\delta_{e\beta}-\delta_{g\beta}\delta_{\alpha
e}\right)\Delta_{F_{g}F_{e}}{\rho}_{\alpha,m_{\alpha},\beta,m_{\beta}}-i\Delta_{\alpha,m_{\alpha},\beta,m_{\beta}}(B_{z}){\rho}_{\alpha,m_{\alpha},\beta,m_{\beta}}\\\
&-\frac{\Gamma}{2}\delta_{\alpha e}{\rho}_{\alpha,m_{\alpha},\beta
m_{\beta}}-\frac{\Gamma}{2}\delta_{\beta e}{\rho}_{\alpha,m_{\alpha},\beta
m_{\beta}}\\\ &+\ \delta{\alpha
g}\delta{g\beta}\sum_{q,F_{e},F_{e}^{\prime}}\Gamma(-1)^{-\alpha-\beta}(2J_{e}+1)\begin{Bmatrix}J_{g}&J_{e}&1\\\
F_{e}&\alpha&I\end{Bmatrix}\begin{Bmatrix}J_{g}&J_{e}&1\\\
F_{e}^{\prime}&\beta&I\end{Bmatrix}\sqrt{2\alpha+1}\sqrt{2\beta+1}\\\
&\innerproduct{F_{e},m_{\alpha}-q}{\alpha,m_{\alpha};1,-q}\innerproduct{F_{e}^{\prime},m_{\beta}-q}{\beta,m_{\beta};1,-q}{\rho}_{F_{e},m_{\alpha}-q,F_{e}^{\prime},m_{\beta}-q}.\end{split}$
(19)

The first 4 terms in Eq. (19) are the field terms, followed by the laser
detuning, the Zeeman splitting, and finally 3 decay terms proportional to
$\Gamma$. ${\rho}_{i,m_{i},j,m_{j}}$ are the density matrix elements,
$\delta_{i,j}$ is the Kronecker symbol,
$\Delta_{F_{g},F_{e}}=\omega-\omega_{Fg,Fe}$ is the detuning of the laser
compared to the atomic hyperfine transition from $F_{g}$ to $F_{e}$. The
magnetic field is along $\boldsymbol{\epsilon}_{z}$, the Landé factors are
$g_{i}$. The Zeeman shift is given by
$\Delta_{\alpha,m_{\alpha},\beta,m_{\beta}}(B_{z})=\frac{\mu_{b}B_{z}}{\hbar}\left(g_{\alpha}m_{\alpha}-g_{\beta}m_{\beta}\right)$.
The Rabi frequency depends on Clebsch-Gordan coefficients and the field
complex amplitude:

$\displaystyle\Omega(F_{e},m_{e},F_{g},m_{g})=$
$\displaystyle(-1)^{F_{e}+J_{g}+1+I}\sqrt{{(2F_{e}+1)(2J_{g}+1)}}$
$\displaystyle\bra{F_{g},m_{g}}\ket{F_{e},m_{e};1,m_{g}-m_{e}}$
$\displaystyle\begin{Bmatrix}J_{g}&J_{e}&1\\\
F_{e}&F_{g}&I\end{Bmatrix}\Omega_{m_{g}-m_{e}}^{(J_{g},J_{e})},$ (20)

where
$\Omega_{q}^{(J_{g},J_{e})}=-{2\bra{J_{g}}\mid{e\mathbf{r}}\mid\ket{J_{e}}E_{0q}^{(+)}}/{\hbar}$
with $\bra{J_{g}}|e\mathbf{r}|\ket{J_{e}}$ being the reduced dipole matrix
element between the states $J_{g}$ and $J_{e}$ and $E_{0q}^{+}$ the positive
rotating amplitude of the electric field. The coupling between two Zeeman
states is expressed with the Wigner-6j symbol and Clebsch-Gordan coefficients
expressed with the Wigner-3j symbols:

$\displaystyle\bra{F_{g},m_{g}}\ket{F_{e},1;m_{e},q}=$
$\displaystyle(-1)^{F_{e}-1+m_{g}}\sqrt{2F_{g}+1}$
$\displaystyle\begin{pmatrix}F_{e}&1&F_{g}\\\ m_{e}&q&-m_{g}\end{pmatrix},$
(21) $\displaystyle\bra{F_{e},m_{e}}\ket{F_{g},1;m_{g},-q}=$
$\displaystyle(-1)^{F_{g}-m_{e}-1}\sqrt{2F_{e}+1}$
$\displaystyle\begin{pmatrix}F_{e}&1&F_{g}\\\ m_{e}&q&-m_{g}\end{pmatrix}.$
(22)

The optical coherence, under the adiabatic approximation where the coherences
are always in equilibrium with respect to the population evolution
($\dot{{\rho}}_{eg}\approx 0$), for a transition between $m_{e}$ and $m_{g}$
on resonance is given by (23). It depends on the population difference and on
Zeeman coherences. In the case of a $pure$ polarization (aligned with only one
element of the spherical basis), if the Zeeman coherences start at zero then
they remain zero at steady-state, so the optical coherences are directly given
by the population difference. Otherwise, Zeeman coherences can be driven and
contribute to the scattering rates.

$\begin{split}{\rho}_{e,m_{e},g,m_{g}}=&-\frac{i}{\Gamma}\sum_{m_{g}^{\prime}}\Omega(m_{g}^{\prime},m_{e}){\rho}_{g,m_{g}^{\prime},g,m_{g}}\\\
&-\frac{i}{\Gamma}\sum_{m_{e}^{\prime}}\Omega(m_{g}^{\prime},m_{e}^{\prime}){\rho}_{e,m_{e},e,m_{e}^{\prime}}.\end{split}$
(23)

To include an incoherent field background in the master equation, Zeeman
coherences are neglected in Eq. (23). Writing a master equation only for
incoherent field terms leads to the following rate equations with saturation
$s_{i}=2\Omega_{i}^{2}/\Gamma^{2}$:

$\begin{split}\Gamma\dot{{\rho}}_{\alpha,m_{\alpha},\alpha,m_{\alpha}}=&\delta_{\alpha
e}\sum_{i}\Omega_{i}^{2}(m_{\alpha},i){\rho}_{g,i,g,i}\\\ &-\delta_{\alpha
e}\left(\sum_{i}\Omega_{i}^{2}(m_{\alpha},i)\right){\rho}_{\alpha,m_{\alpha},\alpha,m_{\alpha}}\\\
&\delta_{\alpha g}\sum_{i}\Omega_{i}^{2}(i,m_{\alpha}){\rho}_{e,i,e,i}\\\
&-\delta_{\alpha
g}\left(\sum_{i}\Omega_{i}^{2}(i,m_{\alpha})\right){\rho}_{\alpha,m_{\alpha},\alpha,m_{\alpha}}.\\\
\end{split}$ (24)

Finally, the full master equation including all effects is obtained by adding
the terms of Eq. (24) to Eq. (19). Also, solving the full master equation at
zero magnetic field, without coherent driving field ($s_{c}=0$) and keeping
the decay terms, one obtains the total excited state population given in the
main text in Eq. (14).

## References

  * Jennewein  _et al._ [2018] S. Jennewein, L. Brossard, Y. R. P. Sortais, A. Browaeys, P. Cheinet, J. Robert, and P. Pillet, Coherent scattering of near-resonant light by a dense, microscopic cloud of cold two-level atoms: Experiment versus theory, Phys. Rev. A 97, 053816 (2018).
  * Glicenstein  _et al._ [2020] A. Glicenstein, G. Ferioli, N. Šibalić, 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).
  * Corman  _et al._ [2017] L. Corman, J. L. Ville, R. Saint-Jalm, M. Aidelsburger, T. Bienaimé, S. Nascimbène, J. Dalibard, and J. Beugnon, Transmission of near-resonant light through a dense slab of cold atoms, Phys. Rev. A 96, 053629 (2017).
  * Guerin  _et al._ [2016] W. Guerin, M. O. Araújo, and R. Kaiser, Subradiance in a large cloud of cold atoms, Phys. Rev. Lett. 116, 083601 (2016).
  * Cottier  _et al._ [2018] F. Cottier, R. Kaiser, and R. Bachelard, Role of disorder in super- and subradiance of cold atomic clouds, Phys. Rev. A 98, 013622 (2018).
  * Jenkins and Ruostekoski [2012] S. D. Jenkins and J. Ruostekoski, Controlled manipulation of light by cooperative response of atoms in an optical lattice, Phys. Rev. A 86, 031602 (2012).
  * Rui  _et al._ [2020] J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M. Stamper-Kurn, C. Gross, and I. Bloch, A subradiant optical mirror formed by a single structured atomic layer, Nature 583, 369 (2020).
  * Cottier  _et al._ [2019] F. Cottier, A. Cipris, R. Bachelard, and R. Kaiser, Microscopic and macroscopic signatures of 3d anderson localization of light, Phys. Rev. Lett. 123, 083401 (2019).
  * Wiersma  _et al._ [1997] D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Localization of light in a disordered medium, Nature 390, 671 (1997).
  * Chabé  _et al._ [2014] J. Chabé, M.-T. Rouabah, L. Bellando, T. Bienaimé, N. Piovella, R. Bachelard, and R. Kaiser, Coherent and incoherent multiple scattering, Phys. Rev. A 89, 043833 (2014).
  * Sokolov and Guerin [2019] I. M. Sokolov and W. Guerin, Comparison of three approaches to light scattering by dilute cold atomic ensembles, J. Opt. Soc. Am. B 36, 2030 (2019).
  * Labeyrie  _et al._ [2003] G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura, and R. Kaiser, Coherent backscattering of light by an inhomogeneous cloud of cold atoms, Phys. Rev. A 67, 033814 (2003).
  * Labeyrie  _et al._ [2000] G. Labeyrie, C. A. Müller, D. S. Wiersma, C. Miniatura, and R. Kaiser, Observation of coherent backscattering of light by cold atoms, J. Opt. B: Quantum Semiclass. Opt 2, 672 (2000).
  * do Espirito Santo  _et al._ [2020] T. S. do Espirito Santo, P. Weiss, A. Cipris, R. Kaiser, W. Guerin, R. Bachelard, and J. Schachenmayer, Collective excitation dynamics of a cold atom cloud, Phys. Rev. A 101, 013617 (2020).
  * ller and Miniatura [2002] C. A. M. ller and C. Miniatura, Multiple scattering of light by atoms with internal degeneracy, J. Phys. A: Math. Gen. 35, 10163 (2002).
  * Lee  _et al._ [2016] M. D. Lee, S. D. Jenkins, and J. Ruostekoski, Stochastic methods for light propagation and recurrent scattering in saturated and nonsaturated atomic ensembles, Phys. Rev. A 93, 063803 (2016).
  * Chomaz  _et al._ [2012] L. Chomaz, L. Corman, T. Yefsah, R. Desbuquois, and J. Dalibard, Absorption imaging of a quasi-two-dimensional gas: a multiple scattering analysis, New J. Phys. 14, 055001 (2012).
  * Steck [2019] D. A. Steck, Quantum and atom optics (2019), revision 0.12.6.
  * Steck [2001] D. Steck, Rubidium 87 D line data (2001).
  * Gao [1993] B. Gao, Effects of zeeman degeneracy on the steady-state properties of an atom interacting with a near-resonant laser field: Analytic results, Phys. Rev. A 48, 2443 (1993).
  * Dodin  _et al._ [2018] A. Dodin, T. Tscherbul, R. Alicki, A. Vutha, and P. Brumer, Secular versus nonsecular redfield dynamics and fano coherences in incoherent excitation: An experimental proposal, Phys. Rev. A 97, 013421 (2018).
  * Dodin  _et al._ [2016] A. Dodin, T. V. Tscherbul, and P. Brumer, Quantum dynamics of incoherently driven v-type systems: Analytic solutions beyond the secular approximation, J. Chem. Phys. 144, 244108 (2016).
  * Tscherbul and Brumer [2015] T. V. Tscherbul and P. Brumer, Partial secular bloch-redfield master equation for incoherent excitation of multilevel quantum systems, J. Chem. Phys. 142, 104107 (2015).
  * Budker  _et al._ [2002] D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, Resonant nonlinear magneto-optical effects in atoms, Rev. Mod. Phys. 74, 1153 (2002).
  * Reinaudi  _et al._ [2007] G. Reinaudi, T. Lahaye, Z. Wang, and D. Guéry-Odelin, Strong saturation absorption imaging of dense clouds of ultracold atoms, Opt. Lett. 32, 3143 (2007).
  * [26] R. Veyron, J.-B. Gerent, V. Mancois, P. Bouyer, and S. Bernon, Quantitative absorption imaging of dense clouds at high saturation intensity and short time scales, In preparation .
  * Mollow [1972] B. R. Mollow, Stimulated emission and absorption near resonance for driven systems, Phys. Rev. A 5, 2217 (1972).


* [44] E. Szemerédi.  Regular partitions of graphs.  In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 399–401. CNRS, Paris, 1978.
  * [45] T. Tao.  Szemerédi’s regularity lemma revisited.  Contrib. Discrete Math., 1(1):8–28, 2006.
  * [46] T. Tao.  An epsilon of room, II.  American Mathematical Society, Providence, RI, 2010.  Pages from year three of a mathematical blog.
  * [47] C. Thomassen.  The graph genus problem is NP-complete.  J. Algorithms, 10(4):568–576, 1989.
  * [48] C. Thomassen.  Triangulating a surface with a prescribed graph.  J. Combin. Theory Ser. B, 57(2):196–206, 1993.
  * [49] C. Thomassen.  The genus problem for cubic graphs.  J. Combin. Theory Ser. B, 69(1):52–58, 1997.
  * [50] V. V. Vazirani.  Approximation algorithms.  Springer-Verlag, Berlin, 2001.
  * [51] S. G. Williamson.  Depth-first search and Kuratowski subgraphs.  J. Assoc. Comput. Mach., 31(4):681–693, 1984.
  * [52] J. W. T. Youngs.  Minimal imbeddings and the genus of a graph.  J. Math. Mech., 12:303–315, 1963.
  * [53] R. Yuster.  Integer and fractional packing of families of graphs.  Random Structures Algorithms, 26(1-2):110–118, 2005.