Effective two-level approximation of a multi-level system driven by coherent andincoherent fields

R. Veyron, V. Mancois, J.B. Gerent, G. Baclet, P. Bouyer, and S. Bernon∗LP2N, Laboratoire Photonique, Numérique et Nanosciences,

Université Bordeaux-IOGS-CNRS:UMR 5298, rue F.Mitterrand, F-33400 Talence, France(Dated: October 19, 2021)

The numerical simulation of multiple scattering in dense ensembles is the mostly adopted solutionto predict their complex optical response. While the scalar and vectorial light mediated interactionsare accurately taken into account, the computational complexity still limits current simulations tothe low saturation regime and ignores the internal structure of atoms. Here, we propose to go beyondthese restrictions, at constant computational cost, by describing a multi-level system (MLS) by aneffective two-level system (TLS) that best reproduces the coherent and total scattering propertiesin any saturation regime. The correspondence of our model is evaluated for different experimentallyrealistic conditions such as the modification of the driving field polarization, the presence of straymagnetic fields or an incoherent resonant electromagnetic field background. The trust interval ofthe model is quantified for the D2-line of 87Rb atoms but it could be generalized to any closedtransition of a multi-level quantum system.

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

I. INTRODUCTION

The response of dense ensembles to coherent opticalillumination is a paradigmatic situation to study mul-tiple scattering dynamics in which collective effects canbe prominent. They lead, for instance, to modifica-tions of the scattering properties such as line shifts andbroadening [1] in 1D [2] and 2D systems [3], sub- [4]and super-radiance [5], the optical phase profile engi-neering [6] to control the reflection properties [7] of asingle atomic layer or the localization of light in differentregimes [8, 9]. Simulations of the coupled dipole equa-tions in the linear-optics regime include interference ef-fects such as coherent backscattering [10, 11]which wasalso predicted using random walk simulations includingthe atomic internal structure complexity [12, 13].

In contrast with the preceding cases, when increasingthe saturation parameter, the system deviates from itslinear response [14], requiring a full quantum treatmentthat scales dramatically with the atom number. Anensemble of N Multi-Level Systems (MLS) with k-levelseach yields the diagonalization of a kN×kN matrix andis computationally out-of-range when considering morethan a few particles.

In a mean-field approach where entanglement be-tween atoms is neglected, the full density matrix can befactorized as the product state of single atom densitymatrices reducing the matrix dimensions to kN × kNthus improving the simulation capabilities up to fewthousand particles. For a given computational power,

∗ Correspondence email address: si-mon.bernon@institutoptique.fr

the traditional trade-off for numerical simulations is ei-ther to consider the internal atomic structure [15, 16]while reducing the number of particles or to model realatoms by two-level systems (TLS) [17]. The latter al-lows a quantitative comparison with experiments onlyin specific situations where the experimental conditionsallow one to suppress spurious transitions [1]. It repre-sents a loss of generality and a limitation for the generalcomparison of theory and experiments in the limit ofdense and saturated ensembles.

In this paper, we show that an effective TLS can prop-erly approximate the scattering properties of a MLS,with exact correspondence in certain conditions. To thisend, we numerically solve the optical Bloch equations(OBE) for a single MLS driven by a coherent field thatoriginates either from a probe laser or from neighboringatoms via coherent scattering. We then fit the effec-tive TLS model parameters to the coherent and totalscattering rates obtained from the density matrix cal-culations. In this study, we detail the influence of thedriving field polarization, stray magnetic fields and in-coherent resonant electromagnetic field background onthe effective TLS parameters.

Sec. II introduces the relevant quantities discussedthroughout this manuscript and derives the optical re-sponse of a TLS driven by a coherent field and an in-coherent background. Sec. III details the calculationfor the exact solution of a MLS and its comparison toan effective TLS. As an example, Sec. IV quantitativelycompares the effective TLS that best corresponds to theclosed transition of the D2-line of 87Rb which is formedby the hyperfine states Fg = 2 and Fe = 3. The numer-ical simulations performed in Sec. IV could be carriedfor any multi-level quantum system with a closed tran-sition.

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II. TWO-LEVEL SYSTEM DYNAMICS

Our study begins with the scattering dynamics oftwo-level systems. We remind in Sec. II A the stan-dard expressions [18, 19] of the density matrix elementsfrom which the coherent, incoherent and total scatter-ing rates are derived. In Sec. II B, we derive the expres-sions of the same quantities when the two-level systemsTLS is driven by a coherent field and an incoherent fieldbackground.

A. Scattering rate under coherent drive

The total scattering rate is an essential quantity de-scribing the influence of the light field on the atom. Itgives the number of photons emitted per unit of time[18]: R

(tot)sca = Γρee where Γ is the natural linewidth

and ρee the excited state population. This total scat-tering rate can be decomposed in terms of a coherentscattering rate R(coh)

sca = Γ|ρeg|2 that represents scat-tering events that are temporally coherent with respectto the driving field and an incoherent scattering rateR

(inc)sca = R

(tot)sca − R(coh)

sca that, by energy conservation,is the difference between the two previous ones. Com-puting these rates requires deriving expressions for thedensity matrix population ρee and slowly varying co-herence ρeg, both obtained by solving the steady-stateregime of the OBE driven by a coherent field:

ρeg =−i√

2

√s′c

1 + s′c

1 + 2iδ√1 + 4δ2

,

ρee =1

2

s′c1 + s′c

,

(1)

where sc = 2Ω2c/Γ

2 is the on-resonance saturation pa-rameter, s′c = sc/(1 + 4δ2) is the effective satura-tion parameter, Ωc is the coherent Rabi frequency andδ = (ω − ω0)/Γ the normalized detuning between thelaser frequency ω and the atomic transition ω0.

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

R(coh)sca =

Γ

2

s′c

(1 + s′c)2 ,

R(tot)sca =

Γ

2

s′c1 + s′c

.

(2)

In the regime of weak saturation (sc 1), the atomresponse is linear in sc and temporally coherent. Thisis the regime of the linear dipole approximation thatis convenient for coupled dipole simulations. In theopposite strong saturation regime (sc 1), the TLScan be saturated, and the temporally incoherent scat-tering dominates. This is the regime of the Mollow

triplet where the incoherent scattering rate scales ass′2c /(1 + s′c)

2.To move from an ideal TLS to the effective TLS,

we follow the work of [20] where the author computesanalytically the scattering rates for the case of a π-polarization and shows that the saturation intensity isreduced by a factor α. We then introduce an effectiveTLS ansatz under a saturating driving field and in per-turbed conditions:

R(coh)sca =

Γ

2

β

α

s′c/α

(1 + s′c/α)2 ,

R(tot)sca =

Γ

2

s′c/α

1 + s′c/α.

(3)

Eqs. (3) correspond to an effective TLS with a correctedsaturation parameter s′c/α. The factor β/α accounts formulti-level corrections (Sec. IV) where α comes from ageometric factor due to the coupling strength of thetransitions and β is an amplitude factor of the coherentscattering field.

B. Scattering rate in coherent and incoherentdrives

For a TLS, coherences between atomic states aredriven by the field complex amplitude. Therefore,a temporally incoherent field (frequency broadbandand/or temporally isotropic polarization) gives an aver-age coherence of zero for averaging time longer than thefield spectral width. Thus, only the intensity of the in-coherent field affects the OBE by incoherently pumpingthe populations at a rate of Γsi/2 where si is an effec-tive saturation parameter for the incoherent intensity[18]. For a coherent field with Rabi frequency Ωc anddetuning δ, and an incoherent intensity with saturationparameter si, one obtains the following OBE:

∂ρge∂t

=− (iδΓ +Γ

2)ρge +

iΩc2

(ρgg − ρee), (4)

∂ρgg∂t

=− si2

Γ (ρgg − ρee) +iΩc2

(ρge − ρeg) + Γρee.

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

ρeg =−i√

2

√s′c

(1 + s′c + si)

1 + 2iδ√1 + 4δ2

,

ρee =1

2

s′c + si1 + s′c + si

.

(5)

Eqs. (5) shows that populations can be transferred byboth the coherent and incoherent light while coherencesare driven only by the coherent field but damped in

3

the saturation regime by both fields i.e. coherences arereduced by the total saturation parameter s′c + si.

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

R(coh)sca =

Γ

2

β

αeff

s′cαeff(

1 +s′cαeff

)2 ,

R(tot)sca =

Γ

2

s′cαeff

1 +s′cαeff

+siαc

1 + siαc

,

(6)

where the corrections to the scattering rates are:

αeff =α (1 + si) ,

αc =1 + s′c.(7)

III. MULTI-LEVEL SYSTEM DYNAMICS

In this section, we give the master equation that de-scribes the density matrix evolution of a multi-levelatomic system and the coherent and total scatteringrates [18].

The master equation in the rotating frame at ω in-cludes the hyperfine splitting Hamiltonian HHF comingfrom the atomic energy structure, the first order Zee-man magnetic shift Hamiltonian HB and the electric-dipole interaction Hamiltonian between the atom andthe driving field:

dρ

dt=− i

~[HHF +HB +HAF , ρ]

+ Γ

(2Je + 1

2Jg + 1

)∑q

D[Σq]ρ,(8)

where Σq is the lowering operator for the polarizationq = mg −me, D the Lindblad superoperator defined asD[Σq]ρ = ΣqρΣ†q−1/2

(Σ†qΣqρ+ ρΣ†qΣq

). The electric-

dipole Hamiltonian is HAF = −d.E where d = −e.ris the dipole moment and E = E0εe

iωt + c.c.. In thiswork, the master equation has been traced over the en-vironment to obtain the OBE for the multi-level atom.Rewriting the density matrix components as a columnvector ρ, we obtain a set of linearly coupled equations:

dρ

dt= Mρ, (9)

where the matrix M contains the matrix elements forall interactions and are detailed in Eq. (19) in the Ap-pendix.

The last interaction to be added to the master equa-tion is an incoherent field background. The interactionbetween a broadband incoherent field such as a black-body radiation and a MLS has been previously studied[21–23]. Master equations enable one a detailed descrip-tion of the system composed of an atom and a bath. Weobtain the Lindblad equations via two simplifications.First the Born-Markov approximation assumes a weakcoupling (Γ ω0) and a vanishing bath memory time[21–23]. Secondly the secular approximation, where thecoherences and the populations evolve independently,neglects interference effects (Zeeman coherences). Un-der these approximations, the master equation of an in-coherently driven system simplifies to rate equations onthe populations only. As detailed in the Appendix VI,under the influence of an on resonance temporally inco-herent field, the master equation of a coherently drivensystem is modified by including terms in the matrix Mto account for incoherent population transfer.

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

R(coh)sca =Γ

∑q

|〈Σq〉|2,

R(tot)sca =Γ

∑q

〈Σ†qΣq〉.(10)

In our method, these rates are obtained by solvingthe MLS master equation in the steady-state by settingdρ/dt = 0 either with symbolic computations for purepolarizations to determine the exact analytical formulasor numerically otherwise. We checked that the resultsof both methods are totally consistent. From the den-sity matrix solution, we compute the exact coherent andtotal scattering rates using Eq. (10). The parametersαeff and βeff are then obtained by fitting the exact rateswith the effective TLS model from Eq. (6). For sim-plicity, in the following, these parameters will be notedas α and β.

IV. EFFECTIVE TLS OF THE D2-LINE OFRUBIDIUM 87

To study the role of experimental imperfections suchas polarization orientation, DC magnetic fields and in-coherent background on the scattering rates, we restrictour model to multiple degenerate closed states. As anexample, we choose to simulate our model on all Zee-man states of the transition |Fg = 2〉 → |Fe = 3〉 of the87Rb D2-line and restrict ourselves to situations wherethe power broadening is much smaller than the hy-perfine energy splitting. Also, the ratio of scatteringrates between the |Fe = 3〉 and |Fe = 2〉 states (44 Γdetuned from |Fe = 3〉) is about 1000 for sc = 60 onthe closed transition. As a result, the transition from

4

(a) Θ = 0

y

x

ζ = 0

ζ = 1

0 < ζ < 1

z

(b) Θ = π2

z

x

ζ = 0

ζ = 1

0 < ζ < 1

y

Figure 1. Two standard polarization cases and their de-pendence on ellipticity ζ and quantization axis (z): a) ForΘ = 0, the polarization lies in the (x,y) plane and is circu-lar for ζ = 1 with a perpendicular quantization axis, b) ForΘ = π/2, the polarization lies in the (x,z) plane and is linearfor ζ = 0 with a parallel quantization axis.

|Fg = 2〉 to |Fe = 3〉 is considered to be closed in therange sc ∈ [0.1, 30]. In the following, we therefore ne-glect the residual coupling to other hyperfine excitedand ground states. In the steady-state regime, thesecoupling would lead to depumping out of the consid-ered transition. Our study is therefore valid only beforedepumping occurs and is robust for closed transitions.

The role of the driving field polarization (Sec. IVA),DC magnetic field (Sec. IVB) and isotropic incoher-ent field (Sec. IVC) are studied independently. Thequantization axis is taken as εz.

A. Role of polarization: σ±, π and elliptical

In this section, only the coherent drive field polar-ization is being changed at zero magnetic field and zeroincoherent field. We use the spherical basis with respectto the Cartesian as follows:

ε± =∓ (εx ± iεy)√2

, (11)

ε0 =εz. (12)

This polarization ε = (ε−, ε+, ε0) is parametrized inthe spherical basis (ε−, ε+, ε0) by an ellipticity ζ =E0x/E0y and π polarization projection angle Θ (Fig.1) as:

ε± =∓ 1√2‖ε‖

(ζ ∓ cos Θ),

ε0 =i

‖ε‖sin Θ,

(13)

where ‖ε‖ =√

1 + ζ2. Using the TLS ansatz of Eq.(3), we evaluate the parameters α ≡ αε and β ≡ βεthat best match the exact scattering rates. For a givenζ, both scattering rates R(coh)

sca , R(tot)sca are computed nu-

merically as a function of the coherent saturation pa-rameter sc. αε and βε are the best fitting parameter for

Figure 2. a) α and b) β as a function of the ellipticity ζfor a polarization in the (x,y) plane (Θ = 0), c) input andoutput intensities per polarization, d) scalar product of theinput and output fields.

sc ∈ [0.1, 30].In the limit case of a σ− circular polarization ε =(1, 0, 0) parametrized by (ζ,Θ) = (1, 0), the atom ispumped in a perfect two-level cycling transition and isexpected to reach the maximal scattering cross sectionσ0 = 3λ2/2π with ασ = βσ = 1. In the opposite limitof a π linear polarization ε = (0, 0, 1), the populationsand coherences have been computed analytically [20].The system formed by five π-transitions for |Fg = 2〉 to|Fe = 3〉 is equivalent to a TLS with a reduced crosssection for which απ = 461/252 = 1.829 and βπ = 1.

In Fig. 2, intermediate ellipticities are obtained byvarying ζ from 1 (circular) to 0 (linear) at Θ = 0.The values of αε and βε are exact (error bars go to 0)which means that the scattering rates are also exactlydescribed by Eq. (3). It does not necessarily mean thatthis situation is exactly equivalent to a TLS, that is ascalar scattering problem, since the MLS is vectorial.The circular and linear polarizations yield the expectedvalues ασ = 1 and απ = 1.829. Note that the lin-ear polarization is at a 45° angle of the x and y axis.In absence of magnetic field bias, the electric field setsthe quantization axis. The same curves would be ob-tained for Θ = π/2. An imperfect polarization as couldoccur in experiments induces little changes for the cir-cular polarization (ζ = 0) -below 10% variation on theparameters- even up to ζ = 0.5, while it has more effectfor a linear polarization. This is due to optical pumpingin the closed transition that protects the atomic state.

5

For the same reason, the polarization of the radiatedfield differs from the input polarization at maximum by8% for an ellipticity about ζ = 0.2.

B. Role of a DC magnetic field

Under a constant magnetic field, magneto-optical ef-fects occur. We refer to the review [24] for more detail.The Faraday effect, for example, results in the opticalrotation and ellipticity change of the output scatter-ing. The scattering process is therefore a truly vectorialproblem that cannot be exactly mapped onto the TLSsolution due to the output polarization.

Figure 3. a) α and b) β as a function of the ellipticityζ in the case of Θ = 0. ζ = 1 is circular and ζ = 0 islinear along εx and does not correspond to a π polarization.For any ellipticity, B is perpendicular to the electric field.The maximum and the standard deviation of the relativeerror between the approximated TLS solution and the exactcalculation of the c) excited state population and d) densitymatrix coherence.

In this section, we focus the analysis on the TLS pa-rameters α and β that best mimic the scattering rateamplitude when the Zeeman degeneracy is lifted by amagnetic field bias such that B = Bεz. The driv-ing field frequency is kept constant and is equal tothe unshifted transition. The magnetic Zeeman shiftbetween two states |Fe,me〉 and |Fg,mg〉 is ∆(B) =µbB/~

(gFemg − gFgmg

)where gFe/g are the Landé fac-

tors and µb is the Bohr magneton (see Appendix VI).The strength of magnetic field considered in this studyare up to 2G which is well below the restriction to firstorder Zeeman perturbation and results in a frequencyshift smaller than Γ. The state |Fg = 2〉 (resp. |Fe = 2〉)has a frequency sensibility to magnetic shift of 0.12Γ/G

Figure 4. a) α and b) β as a function of ε in the case ofΘ = π/2. ζ = 1 is circular and ζ = 0 is linear along εzand does correspond to a π polarization. The maximumand the standard deviation of the relative error between theapproximated TLS solution and the exact calculation of thec) excited state population and d) density matrix coherence.

(resp. 0.15Γ/G).In a σ− polarization case, α = β = ασ(1 +

4 (δω + δB)2) where δB = µbB/~Γ. It simply corre-

sponds to a TLS probed off-resonantly due to the Zee-man shift and the driving field detuning. Due to op-tical pumping, only the two states |Fg = 2,mg = −2〉and |Fe = 3,me = −3〉 are occupied in the steady-state.The atomic response is the one of a TLS and the resultsof Eq. (2) are exactly recovered. In this situation, theZeeman shift can be experimentally compensated by thedriving field detuning. Interestingly, for a linear polar-ization aligned with a magnetic bias (π polarization),the scattering rates are also exactly given by the effec-tive TLS Eqs. (2) with α = απ

(1 + 4δ2

B41

1008απ+ 4δ2

ω

)and β = 1. These expressions can be derived bysolving the linear system of Eqs. (9) containing allπ−transitions. In this situation, the sensitivity of thescattering rates to the detuning is reduced by a factor

411008απ

≈ 0.02 with respect to the σ polarization case.As a result, for pure σ or π polarizations, α and β havewell known lower and upper limits.

We consider now deviations from these ideal cases bystudying the influence of the ellipticity ζ in the (x, y)plane corresponding to Θ = 0, and in the (x, z) planecorresponding to Θ = π/2. α and β from Eq. (6) aretogether fitted from the scattering rates obtained bynumerically solving Eq. (9) and plotted with error barswithin 95% confidence interval of the fits parameters.

As expected, Fig. 3 (Θ = 0) shows that the scatter-ing rates are sensitive to the polarization. The growing

6

error bars for an increased ellipticity indicate a growingdeviation from the effective TLS behavior. The offsetbetween the curves is merely due to the Zeeman shift.The relative error between the simulated scattering rateand the model of Eq. (3) is plotted in terms of the max-imum and the standard deviation of the relative errorover the range of the saturation parameter sc used forthe fitting. It shows that the model has a maximumrelative error which does not exceed 6% for the totalscattering rate, which happens only for an ellipticity of1. The coherent scattering rate is more sensitive to theellipticity with up to 50% relative error. Nevertheless,for quasi-circular polarizations (ζ ≈ 1) as typically usedin σ absorption imaging, the MLS reduces to a TLS.

Fig. 4 presents the influence of the ellipticity in the(x, z) plane. For ζ = 0, corresponding to a π polariza-tion, the TLS is exact with little dependence on mag-netic field. As the ellipticity is increased, this TLS be-havior becomes less accurate with increasing error barson the fitting parameters. Consequently, π polarizationimaging will be less accurate to define the absorptioncross section and the atom numbers.

C. Role of an incoherent drive

The scattering cross section is also modified in thepresence of an incoherent field. It could be for examplegenerated by a thermal lamp, or by the temporally in-coherent response of the surrounding atomic gas to thecoherent excitation. Here, we consider the influence of apartially polarized incoherent field on the coherent andincoherent scattering response of a MLS probed by acoherent σ− polarized light under zero magnetic fieldoffset. Electromagnetic fields are here considered astemporally incoherent if their coherence time is smallerthan the Rabi period, meaning that the density matrixcoherences are zero on average while the excited statepopulations are non-zero. An incoherent field yields op-tical pumping without coherence.

We consider a partially σ− polarized incoherent fieldthat could for example be generated by the incoherentscattering of a σ− polarized coherent field. This in-coherent field is parametrized by a polarization degreer ∈ [0, 1] in the form sσi = rsi and siso

i = (1 − r)siwhere the sσi /siso

i /si are respectively the σ− polarized/ isotropic / total saturation intensities expressed rela-tively to the saturation intensity of σ− polarized light.With such definition, r = 1 describes a σ− polarizedincoherent field while r = 0 describes an isotropic inco-herent field.

In the limit r = 1, both the coherent and incoher-ent fields are σ− polarized. The atomic population arepumped in the closed transition and the situation isexactly described as in Sec. II B with scattering ratesgiven by Eq. (6) where δ = 0 and α = 1. In the oppo-

site limit r = 0, a purely isotropic incoherent field drivewill redistribute the ground state population. In theabsence of the coherent field drive (sc = 0), the pop-ulations are equally redistributed among the Zeemanstates. Interestingly, we notice that for large saturationparameters si, the total population in the excited statecan be higher than 1/2. It is indeed bounded by thenumber of excited Zeeman states over the total num-ber of states which is 7/12 for the considered transitionof 87Rb. The total excited state population ρee in theabsence of coherent field sc = 0 can be analytically de-rived from the master equation (Appendix VI) and isgiven by:

ρee =7

12

si/3012

1 + si/3012

. (14)

In the low saturation regime i.e. si 1, Eq. (14)reduces to ρee = 1

2siαiso

where αiso = 157 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 ∈ [0, 1]) and by a coherent drive sc. In the AppendixVI, the master equation given by Eq. (19) is expressedin the form of Eq. (24) and the steady-state solutionsof all elements of the density matrix are obtained by analgebraic solver. The algebraic solution has the form ofa ratio of polynomials containing few 100 terms. Thisexact algebraic solution of the scattering rates is com-pared to a phenomenological model of a TLS:

R(coh)sca /Γ =

β

2α

sc/α(1 + siso

i / 3012 + sσi /α+ sc/α

)2 ,R(tot)sca /Γ = η(r, si, sc) +

1

2

sc/α

1 + sisoi / 30

12 + sσi /α+ sc/α,

η(r, si, sc) =η0 − η∞

1 + scα(1+sisoi / 30

12 +sσi /α)+ η∞, (15)

where α, β, η0 and η∞ are fitting parameters. Tominimize the number of fitted parameters, the satu-ration intensity correction of the isotropic incoherentfield was fixed to 30/12. In this model, α quantifiesthe reduction of the coherent absorption cross section,β < 1 quantifies the reduction of coherent field emis-sion with respect to a perfect TLS. η(r, si, sc) is thepart of the total excited state population induced bythe incoherent field drive. This contribution of the ex-cited state population includes states which are coupledto all three fields (e.g. |Fe = 3,mF = −3,−2,−1, 0, 1〉)and states coupled only to the isotropic polarization(e.g. |Fe = 3,mF = 2, 3〉). This excited state popula-tion η(r, si, sc) has a complex dynamic that dependson the polarization ratio of the incoherent field r, thevalue of the incoherent si and coherent sc saturation

7

Figure 5. a) Variations of the total excited state populationand b) σ−-polarized coherences with r = 0.5 and si = 5 inblue and the fits in red using Eq. (15).

intensities. For a given couple of parameters r and si,we observe that this contribution varies monotonicallyfrom η0 in absence of coherent drive and saturates atη∞ for large coherent drive sc si. The model givenin Eq. (15) reproduces the saturation behavior with acrossover at sc/α = 1 + siso

i / 3012 + sσi /α.

As an example, the total scattering rate and coherentscattering rate are given as a function of sc for r = 0.5and si = 5 in Fig. 5. We observe that the equivalentTLS given by Eqs. (15) describes well the dynamicswith a standard deviation of the relative error below1% on this example. The excited state population off-set at sc = 0 is due to the incoherent drive.

We compare the TLS model and the exact solutionby fitting the parameters α(si, r), β(si, r) η0(si, r) andη∞(si, r). The fits are realized at a fixed incoherent in-tensity si ∈ [0, 10] and polarization degree r ∈ [0, 1] andfor a coherent intensity sc varying between 0 and 10si.In this parameter range (sc ≤ 100) the closed transitionapproximation holds at least for 500 scattering events.The results are shown in Fig. 6. As a comparison, theexpected parameters for an exact TLS are α = 1, β = 1and η0 = η∞ given by the second term in Eq. (6) whichare indeed the exact solutions found for a polarized in-coherent field (r = 1). For r = 1, the MLS is opticallypumped in an exact TLS. As the incoherent drive po-larization is randomized r → 0, the fitted parametersslightly deviate from these initial values with a com-plex behavior that is mostly driven by optical pumpingmechanisms. In absence of coherent drive, the excitedstate population η0 increases as a function of the inco-herent saturation intensity. For an isotropic incoherentdrive, 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 dependson α and β. For large and unpolarized incoherent drive,the coherent scattering rate can be reduced by up to afactor 1/2 with respect to the pure TLS. This reductionis essentially due to Clebsch-Gordon coefficients enter-ing in the calculation of β. As checked numerically, the

Figure 6. Model parameters a) α, b) β, c) η0 and d) η∞as a function of the incoherent intensity si for polarizationdegrees r ∈ [0, 1] by step of 0.1. The shaded area representsthe 95% confidence interval of the fits and is very small onthe plots. In c), the limit case r = 0 and r = 1 exactlycoincide with the analytical solutions given in Eq. (14) andthe second term of Eq. (6) respectively.

polarization of the coherently scattered field is exactlyaligned with the input field (σ-polarized). This wasexpected given that the incoherent field cannot drivecoherence between the Zeeman sub-levels and thereforedoes not alter the scattered polarization (Eq. (23) inthe Appendix).

V. DISCUSSION

The combined coherent and incoherent response ofatoms in the saturated regime has a strong impacton the interpretation of fluorescence and absorptionimaging of ensembles. Experimental imperfections arethe common explanations [25] for the reduction of theatomic cross section. With this argument, the abso-lute determination of atom numbers is subject to a pre-cise calibration of the experimental conditions whichis often questioned. In Fig. 2, we have shown thatthe effective TLS description of a multi-level atom isvery robust to polarization imperfections especially forthe case of σ polarized light. We attribute this robust-ness to optical pumping mechanisms that protects thestretched-state hyperfine transition (|Fg = 2,mF = −2〉to |Fe = 3,mF = −3〉 in σ−). This pumping was ad-ditionally observed in the steady-state solutions of theOBE via a strong imbalance of the repartition of the

8

population that favored the |Fg = 2,mF = −2〉 stateeven for ζ ∈ [0.5, 1]. This optical pumping protectionof σ transitions is not specific to the imaging transitionof the D2 line of Rubidium and will be applicable toany Zeeman degenerated closed transition. In the ab-sence of Zeeman degeneracy, it was additionally shownthat MLS behaves exactly as an effective TLS for anydrive polarization. As expected for 87Rb, the cross sec-tion reduction factor α varies from 1 (σ) to 1.829 (π)depending on the driving field polarization.

On the other hand, stray magnetic field strongly im-pact the calibration of the scattering cross section of aσ polarized probe (Fig. 3(a)). Nevertheless, this correc-tion which is solely given by detuning of the coherentdrive from the extreme TL transition can be exactlycompensated for by tuning the drive frequency on res-onance with the extreme TL transition in experiments.It is therefore not a concern for cross section calibra-tions. In addition, due to symmetry of the first orderZeeman splitting, the cross section correction of a π po-larized probe is independent of the stray magnetic fieldamplitude if it is aligned along the linear polarizationaxis.

As mentioned earlier, in the presence of a stray mag-netic field, light scattering is a vectorial process, andthe coherent scattered field polarization is not alignedwith the drive. Nevertheless, in the specific case of σand π polarized light with a magnetic bias well-definedwith respect to light polarization, both atom-light andmagnetic interaction are diagonalized in the same ba-sis leading to an aligned output field. The scatteringprocess is therefore scalar in these two situations withadditional robustness to imperfections for the σ polar-ized drive (Fig. 2(d)). Polarization and stray magneticfield imperfections have therefore little influence on thereduction of the cross section.

On the other hand, we have observed that in thepresence of an incoherent background that would mimicthe temporally and spatially incoherent scattering fromother atoms in the ensemble, the cross section is no-tably reduced which results in systematics errors onabsorption imaging measurements of atom number inthe saturation regime and/or at large optical thickness[3, 26]. For large optical depth, most of the coherentdrive is converted to an incoherent field via multiplescattering mechanism. A total conversion of field leadsto si = sc which gives an upper bound for the modifi-cation of α = 1 + si.

For large saturations, the atom is driven in the Mol-low triplet regime, and we expect this large value of αto be mitigated by a reduction of the reabsorption crosssection [27]. In other words, the converted light will bepartly off-resonant with the atomic transition.

VI. CONCLUSION

In this paper, we have thoroughly studied the to-tal and coherent scattering rates of a MLS atoms illu-minated by different configurations of the electromag-netic field that correspond to situations often encoun-tered in experimental realization. We proposed to mapthese scattering properties to the one of an effectiveTLS model which is particularly relevant to reduce thecomplexity of multiple scattering simulations. We haveshown that, at zero magnetic field, our effective TLSmodel describes exactly the scattering rates of a MLSfor any saturation parameter. In the presence of straymagnetic fields that lift the Zeeman degeneracy, the am-plitude of the scattered fields is well described by theTLS model, with an exact mapping for the specific caseof σ and π polarizations. For other polarizations, vecto-rial scattering (magneto-optical effects) can occur. Ourscalar model cannot exactly render such rotations butproved to be robust for σ polarized scattering. In thelimit of strong saturation, incoherent scattering dom-inates. In dense ensembles, the dynamic of a singleatom will be affected by this incoherent electromagneticbackground. We have shown that, while the MLS dy-namic becomes complex, an equivalent TLS model canbe adapted to enlighten the general behavior and mainscattering response. In particular, we noticed that eventhough the intrinsic reduction α of the saturation inten-sity stays close to 1, the effective reduction αeff is pro-portional to the incoherent background intensity thusreducing the coherent scattered field. This work givesan upper bound to situations in which a TLS model ap-plies and can be extended to any atom having a cyclingtransition.

ACKNOWLEDGEMENTS

The authors thank W. Guerin, R. Bachelard and K.Vinck for helpful discussions. R.V acknowledge PhDsupport from the university of Bordeaux, J-B.G andV.M. acknowledge support from the French State, man-aged by the FrenchNational Research Agency (ANR) inthe frame of the Investments for the future ProgrammeIdEx Bordeaux-LAPHIA (ANR-10-IDEX-03-02). Thiswork was also supported by the ANR contract ANR-18-CE47-0001-01.

APPENDIX

In the following, we briefly give the formalism usedto calculate the steady-states solutions of the densitymatrix that were used to evaluate the atomic scatteringcross section. A detailed derivation of the formalism

9

can be found in [18].

The lowering operator between two hyperfine statesFg and Fe is given by:

Σq =∑

Fg,mg,Fe,me

λ(Fg,mg, Fe,me, q) |Fg,mg〉 〈Fe,me| ,

(16)where

λ(Fg,mg, Fe,me, q) =(−1)Fe+Jg+1+I√

(2Fe + 1)(2Jg + 1)

〈Fg,mg|Fe,me; 1, q〉Je Jg 1Fg Fe I

,

(17)

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

〈Σq〉 =∑

Fg,mg,Fe,me

λ(Fg,mg, Fe,me, q)ρFe,me,Fg,mg

∑q

〈Σ†qΣq〉 =(2Jg + 1)/(2Je + 1)∑Fe,me

ρFe,me,Fe,me .

(18)

So the coherent scattering is given by the optical coher-ences and the total scattering by the population of theexcited state.Projecting Eq. (8) onto the general states 〈α,mα| and|β,mβ〉, the full master equation for an arbitrary num-ber of hyperfine and Zeeman states reads:

ρα,mα,β,mβ =− i

2[∑Fe,me

Ω∗(α,mα, Fe,me)δαgρFe,me,β,mβ +∑Fg,mg

Ω(Fg,mg, α,mα)δαeρFg,mg,β,mβ

−∑Fg,mg

Ω∗(Fg,mg, β,mβ)δeβρα,mα,Fg,mg −∑Fe,me

Ω(β,mβ , Fe,me)δgβρα,mα,Fe,me ]

− i (δαgδeβ − δgβδαe) ∆FgFeρα,mα,β,mβ − i∆α,mα,β,mβ (Bz)ρα,mα,β,mβ

− Γ

2δαeρα,mα,βmβ −

Γ

2δβeρα,mα,βmβ

+ δαgδgβ∑

q,Fe,F ′e

Γ(−1)−α−β(2Je + 1)

Jg Je 1Fe α I

Jg Je 1F ′e β I

√2α+ 1

√2β + 1

〈Fe,mα − q|α,mα; 1,−q〉 〈F ′e,mβ − q|β,mβ ; 1,−q〉 ρFe,mα−q,F ′e,mβ−q.

(19)

The first 4 terms in Eq. (19) are the field terms, fol-lowed by the laser detuning, the Zeeman splitting, andfinally 3 decay terms proportional to Γ. ρi,mi,j,mj arethe density matrix elements, δi,j is the Kronecker sym-bol, ∆Fg,Fe = ω − ωFg,Fe is the detuning of the lasercompared to the atomic hyperfine transition from Fg toFe. The magnetic field is along εz, the Landé factorsare gi. The Zeeman shift is given by ∆α,mα,β,mβ (Bz) =µbBz

~ (gαmα − gβmβ). The Rabi frequency depends onClebsch-Gordan coefficients and the field complex am-plitude:

Ω(Fe,me, Fg,mg) =(−1)Fe+Jg+1+I√

(2Fe + 1)(2Jg + 1)

〈Fg,mg|Fe,me; 1,mg −me〉Jg Je 1Fe Fg I

Ω

(Jg,Je)mg−me , (20)

where Ω(Jg,Je)q = −2 〈Jg| | er | |Je〉E(+)

0q /~ with〈Jg| |er| |Je〉 being the reduced dipole matrix element

between the states Jg and Je and E+0q the positive ro-

tating amplitude of the electric field. The coupling be-tween two Zeeman states is expressed with the Wigner-6j symbol and Clebsch-Gordan coefficients expressedwith the Wigner-3j symbols:

〈Fg,mg|Fe, 1;me, q〉 =(−1)Fe−1+mg√

2Fg + 1(Fe 1 Fgme q −mg

), (21)

〈Fe,me|Fg, 1;mg,−q〉 =(−1)Fg−me−1√

2Fe + 1(Fe 1 Fgme q −mg

). (22)

The optical coherence, under the adiabatic approxima-tion where the coherences are always in equilibriumwith respect to the population evolution (ρeg ≈ 0), fora transition between me and mg on resonance is givenby (23). It depends on the population difference and onZeeman coherences. In the case of a pure polarization(aligned with only one element of the spherical basis), if

10

the Zeeman coherences start at zero then they remainzero at steady-state, so the optical coherences are di-rectly given by the population difference. Otherwise,Zeeman coherences can be driven and contribute to thescattering rates.

ρe,me,g,mg =− i

Γ

∑m′g

Ω(m′g,me)ρg,m′g,g,mg

− i

Γ

∑m′e

Ω(m′g,m′e)ρe,me,e,m′

e.

(23)

To include an incoherent field background in the mas-ter equation, Zeeman coherences are neglected in Eq.(23). Writing a master equation only for incoherentfield terms leads to the following rate equations with

saturation si = 2Ω2i /Γ

2:

Γρα,mα,α,mα =δαe∑i

Ω2i (mα, i)ρg,i,g,i

− δαe

(∑i

Ω2i (mα, i)

)ρα,mα,α,mα

δαg∑i

Ω2i (i,mα)ρe,i,e,i

− δαg

(∑i

Ω2i (i,mα)

)ρα,mα,α,mα .

(24)

Finally, the full master equation including all effectsis obtained by adding the terms of Eq. (24) to Eq. (19).Also, solving the full master equation at zero magneticfield, without coherent driving field (sc = 0) and keep-ing the decay terms, one obtains the total excited statepopulation given in the main text in Eq. (14).

[1] S. Jennewein, L. Brossard, Y. R. P. Sortais,A. Browaeys, P. Cheinet, J. Robert, and P. Pillet, Co-herent scattering of near-resonant light by a dense, mi-croscopic cloud of cold two-level atoms: Experimentversus theory, Phys. Rev. A 97, 053816 (2018).

[2] A. Glicenstein, G. Ferioli, N. Šibalić, L. Brossard,I. Ferrier-Barbut, and A. Browaeys, Collective shift inresonant light scattering by a one-dimensional atomicchain, Phys. Rev. Lett. 124, 253602 (2020).

[3] L. Corman, J. L. Ville, R. Saint-Jalm, M. Aidels-burger, T. Bienaimé, S. Nascimbène, J. Dalibard,and J. Beugnon, Transmission of near-resonant lightthrough a dense slab of cold atoms, Phys. Rev. A 96,053629 (2017).

[4] W. Guerin, M. O. Araújo, and R. Kaiser, Subradiancein a large cloud of cold atoms, Phys. Rev. Lett. 116,083601 (2016).

[5] F. Cottier, R. Kaiser, and R. Bachelard, Role of disorderin super- and subradiance of cold atomic clouds, Phys.Rev. A 98, 013622 (2018).

[6] S. D. Jenkins and J. Ruostekoski, Controlled manipu-lation of light by cooperative response of atoms in anoptical lattice, Phys. Rev. A 86, 031602 (2012).

[7] J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zei-her, D. M. Stamper-Kurn, C. Gross, and I. Bloch, Asubradiant optical mirror formed by a single structuredatomic layer, Nature 583, 369 (2020).

[8] F. Cottier, A. Cipris, R. Bachelard, and R. Kaiser, Mi-croscopic and macroscopic signatures of 3d anderson lo-calization of light, Phys. Rev. Lett. 123, 083401 (2019).

[9] D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righ-ini, Localization of light in a disordered medium, Nature

390, 671 (1997).[10] J. Chabé, M.-T. Rouabah, L. Bellando, T. Bienaimé,

N. Piovella, R. Bachelard, and R. Kaiser, Coherent andincoherent multiple scattering, Phys. Rev. A 89, 043833(2014).

[11] I. M. Sokolov and W. Guerin, Comparison of three ap-proaches to light scattering by dilute cold atomic en-sembles, J. Opt. Soc. Am. B 36, 2030 (2019).

[12] G. Labeyrie, D. Delande, C. A. Müller, C. Miniatura,and R. Kaiser, Coherent backscattering of light by aninhomogeneous cloud of cold atoms, Phys. Rev. A 67,033814 (2003).

[13] G. Labeyrie, C. A. Müller, D. S. Wiersma,C. Miniatura, and R. Kaiser, Observation of coherentbackscattering of light by cold atoms, J. Opt. B: Quan-tum Semiclass. Opt 2, 672 (2000).

[14] T. S. do Espirito Santo, P. Weiss, A. Cipris, R. Kaiser,W. Guerin, R. Bachelard, and J. Schachenmayer, Col-lective excitation dynamics of a cold atom cloud, Phys.Rev. A 101, 013617 (2020).

[15] C. A. M. ller and C. Miniatura, Multiple scattering oflight by atoms with internal degeneracy, J. Phys. A:Math. Gen. 35, 10163 (2002).

[16] M. D. Lee, S. D. Jenkins, and J. Ruostekoski, Stochasticmethods for light propagation and recurrent scatteringin saturated and nonsaturated atomic ensembles, Phys.Rev. A 93, 063803 (2016).

[17] 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).

[18] D. A. Steck, Quantum and atom optics (2019), revision

11

0.12.6.[19] D. Steck, Rubidium 87 D line data (2001).[20] 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).

[21] A. Dodin, T. Tscherbul, R. Alicki, A. Vutha, andP. Brumer, Secular versus nonsecular redfield dynamicsand fano coherences in incoherent excitation: An ex-perimental proposal, Phys. Rev. A 97, 013421 (2018).

[22] A. Dodin, T. V. Tscherbul, and P. Brumer, Quantumdynamics of incoherently driven v-type systems: An-alytic solutions beyond the secular approximation, J.Chem. Phys. 144, 244108 (2016).

[23] 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).

[24] D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester,V. V. Yashchuk, and A. Weis, Resonant nonlinearmagneto-optical effects in atoms, Rev. Mod. Phys. 74,1153 (2002).

[25] G. Reinaudi, T. Lahaye, Z. Wang, and D. Guéry-Odelin,Strong saturation absorption imaging of dense clouds ofultracold atoms, Opt. Lett. 32, 3143 (2007).

[26] R. Veyron, J.-B. Gerent, V. Mancois, P. Bouyer, andS. Bernon, Quantitative absorption imaging of denseclouds at high saturation intensity and short timescales, In preparation.

[27] B. R. Mollow, Stimulated emission and absorption nearresonance for driven systems, Phys. Rev. A 5, 2217(1972).