Transmission of near-resonant light through a dense slab of cold atoms
Laura Corman, Jean-Loup Ville, Raphaël Saint-Jalm, Monika Aidelsburger, Tom Bienaimé, Sylvain Nascimbène, Jean Dalibard, Jérôme Beugnon
TTransmission of near-resonant light through a dense slab of cold atoms
L. Corman † , J.L. Ville, R. Saint-Jalm, M. Aidelsburger ‡ , T. Bienaim´e, S. Nascimb`ene, J. Dalibard, and J. Beugnon ∗ Laboratoire Kastler Brossel, Coll`ege de France, CNRS,ENS-PSL Research University, UPMC-Sorbonne Universit´es,11 place Marcelin-Berthelot, 75005 Paris, France (Dated: December 6, 2017)The optical properties of randomly positioned, resonant scatterers is a fundamentally difficultproblem to address across a wide range of densities and geometries. We investigate it experimentallyusing a dense cloud of rubidium atoms probed with near-resonant light. The atoms are confined ina slab geometry with a sub-wavelength thickness. We probe the optical response of the cloud as itsdensity and hence the strength of the light-induced dipole-dipole interactions are increased. We alsodescribe a theoretical study based on a coupled dipole simulation which is further complementedby a perturbative approach. This model reproduces qualitatively the experimental observation of asaturation of the optical depth, a broadening of the transition and a blue shift of the resonance.
I. INTRODUCTION
The interaction of light with matter is a fundamentalproblem which is relevant for simple systems, such as anatom strongly coupled to photons [1–3], as well as forcomplex materials, whose optical properties provide in-formation on their electronic structure and geometry [4].This interaction can also be harnessed to create materi-als and devices with tailored properties, from quantuminformation systems such as memories [5] and nanopho-tonic optical isolators [6] to solar cells combining highlyabsorptive materials with transparent electrodes [7].The slab geometry is especially appropriate to studylight-matter interaction [8, 9]. In the limit of a mono-layer, two-dimensional (2D) materials exhibit fascinat-ing optical properties. For simple direct band gap 2Dsemi-conductors, the single particle band structure im-plies that the transmission coefficient takes a universalvalue [10, 11]. This was first measured for single layergraphene samples [12], which have an optical transmis-sion independent of the light frequency in the eV range, |T | = 1 − πα where α is the fine structure constant[13, 14]. The same value was recovered in InAs semicon-ductors [15]. This universality does not hold for morecomplex 2D materials, for instance when the Coulombinteraction plays a more important role [16].Atomic gases represent in many respects an ideal testbed for investigating light-matter interaction. First, theycan be arranged in regular arrays [17, 18] or randomlyplaced [19] to tailor the optical properties of the system.Second, an atom always scatters light, in contrast withsolid-state materials where the optical excitation can beabsorbed and dissipated in a non-radiative way. Evenfor thin and much more dilute samples than solid-statesystems, strong attenuation of the transmission can be ∗ [email protected]; † Present address: Institute for QuantumElectronics, ETH Zurich, 8093 Zurich, Switzerland; ‡ Presentaddress: Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨atM¨unchen, Schellingstr. 4, 80799 Munich, Germany observed at resonance. Third, inhomogeneous Dopplerbroadening can be made negligible using ultracold atomicclouds. Finally, the geometry and the density of the gasescan be varied over a broad range.In the dilute limit, such that the three-dimensional(3D) atomic density ρ and the light wavenumber k verify ρk − (cid:28)
1, and for low optical depths, a photon enter-ing the atomic medium does not recurrently interact withthe same atom. Then, for a two-level atom, the transmis-sion of a resonant probe beam propagating along the z axis is given by the Beer-Lambert law: |T | = e − σ (cid:82) ρdz ,where σ = 6 πk − is the light cross section at the opti-cal resonance [20]. At larger densities the transmission isstrongly affected by the light-induced dipole-dipole cou-pling between neighboring atoms.Modification of the atomic resonance lineshape orsuper- and sub-radiance in dilute (but usually opti-cally dense) and cold atomic samples have been largelyinvestigated experimentally [21–30]. Recently, experi-ments have been performed in the dense regime study-ing nanometer-thick hot vapors [31] and mesoscopic coldclouds [32–35]. Interestingly, it has been found that themean-field Lorentz-Lorenz shift is absent in cold systemswhere the scatterers remain fixed during the measure-ment. A small redshift is still observed for dense cloudsin Refs.[32, 34] but could be specific to the geometry ofthe system.Achieving large densities is concomitant with a van-ishingly small transmission T . It is therefore desirableto switch to a 2D or thin slab geometry in order to in-vestigate the physical consequences of these resonant in-teraction effects at the macroscopic level. Using a 2Dgeometry also raises a fundamental question inspired bythe monolayer semiconductor case: Can the light extinc-tion through a plane of randomly positioned atoms bemade arbitrarily large when increasing the atom den-sity or does it remain finite, potentially introducing amaximum of light extinction through 2D random atomicsamples independent of the atomic species of identicalelectronic spin?In this article, we study the transmission of nearly a r X i v : . [ phy s i c s . a t o m - ph ] D ec resonant light through uniform slabs of atoms. We re-port experiments realized on a dense layer of atoms witha tunable density and thickness. For dense clouds, thetransmission is strongly enhanced compared to the oneexpected from the single-atom response. We also observea broadening and a blue shift of the resonance line onthe order of the natural linewidth. This blue shift con-trasts with the mean-field Lorentz-Lorenz red shift andis a signature of the strongly-correlated regime reachedin our system because of dipole-dipole interactions [36].To our knowledge, it is the first time that a blue shiftis reported. In addition, we observe deviations of theresonance lineshape from the single-atom Lorentzian be-havior, especially in the wings where the transmissiondecays more slowly. We model this system with cou-pled dipole simulations complemented by a perturbativeapproach which qualitatively supports our observations.After describing our experimental system in Sec. II, weinvestigate theoretically light scattering for the geome-try explored in the experiment in Sec. III. In Sec. IV wepresent our experimental results and compare them withtheory. We conclude in Sec. V. II. EXPERIMENTAL METHODSA. Cloud preparation
We prepare a cloud of Rb atoms with typically N = 1 . × atoms in the | F = 1 , m F = − (cid:105) state. The atoms are confined in an all-optical trap, de-scribed in more detail in [37], with a strong harmonicconfinement in the vertical direction z with frequency ω z / π = 2 . x − and y − directions is produced by a flat-bottomdisk-shaped potential of diameter 2 R = 40 µ m. For ourinitial cloud temperature (cid:39)
300 nK, there is no extendedphase coherence in the cloud [38]. Taking into accountthis finite temperature, we compute for an ideal Bosegas an r.m.s. thickness ∆ z = 0 . µ m, or equivalently k ∆ z = 2 . nk − ≈ . n = N/ ( πR ) is the surface density and to a maxi-mum density ρk − ≈ . z where ρ is the volume density. We tune the number of atomsthat interact with light by partially transferring them tothe | F = 2 , m F = − (cid:105) state using a resonant microwavetransition. Atoms in this state are sensitive to the probeexcitation, contrary to the ones in the | F = 1 , m F = − (cid:105) state. In this temperature range the Doppler broadeningis about 3 orders of magnitude smaller than the naturallinewidth of the atomic transition.The cloud thickness is varied in a controlled way us-ing mainly two techniques: (i) Varying the vertical har-monic confinement by modifying the laser power in theblue-detuned lattice that traps the atoms, thus chang-ing its frequency from ω z / π = 1 . ω z / π =2 . N = 1 . × atoms ata temperature of T (cid:39)
300 nK, this corresponds to r.m.s.thicknesses between 0.3 µ m and 0.6 µ m. (ii) Allowing theatoms to expand for a short time after all traps have beenswitched off. The extent of the gas in the xy –directiondoes not vary significantly during the time of flight (ToF)(duration between 0.7 ms and 4.7 ms). In that case, ther.m.s. thickness varies between 3 µ m and 25 µ m. Forthe densest clouds, the thickness is also influenced bythe measurement itself. Indeed, the light-induced dipole-dipole forces between atoms lead to an increase of the sizeof the cloud during the probing. In the densest case, weestimate from measurements of the velocity distributionafter an excitation with a duration of τ = 10 µ s that thethickness averaged over the pulse duration is increasedby ∼
20 %. In some experiments, in which the signal islarge enough, we limit this effect by reducing the probeduration τ to 3 µ s. (a) (b) FIG. 1. (a) Schematic representation of the imaging setup.The atoms are confined by a single, disk-shaped potentialwhich is imaged using a microscope objective onto a back-illuminated CCD camera. The numerical aperture of the sys-tem is limited to ≈ . n = 25 µ m − . We extract a region of interest with uniformdensity for our analysis with a typical area of 200 µ m . B. Transmission measurement
We probe the response of the cloud by measuringthe transmission of a laser beam propagating along the z − direction (See Fig. 1). The light is linearly polarizedalong the x − axis and tuned close to the | F = 2 (cid:105) →| F (cid:48) = 3 (cid:105) D transition. The duration of the light pulseis fixed to 10 µ s for most experiments and we limit theimaging intensity I to the weakly saturating regime with0 . < I/I sat < .
2, where I sat (cid:39) .
67 mW/cm is theresonant saturation intensity. We define ∆ ν as the de-tuning of the laser beam with respect to the single-atomresonance. The cloud intensity transmission |T | is ex-tracted by comparing images with and without atomsand we compute the optical depth D = − ln |T | (seeSec. II.C). The numerical aperture of the optical systemis limited on purpose to minimize the collection of fluores-cence light from directions different from the propagationdirection of the light beam. − − − ∆ ν/ Γ . . . . . . . . . D FIG. 2. Example of resonance curves. Symbols represent theexperimental data, and the corresponding dashed lines areLorentzian fits. All curves are taken with the cloud thickness k ∆ z = 2 . nk − = 0 . . . C. Computation of the optical depth
We extract the optical depth (D) of the clouds by com-paring pictures with and without atoms. The read-outnoise on the count number N count is d N count ∼ M with and M without . Thetypical noise on the count number per pixel is thusd N = √ N count ∼ µ m. The typical mean number of counts per pixelaccumulated during the 10 µ s imaging pulse is 80 on thepicture without atoms. We optimize the signal-to-noiseratio by summing all the pixels in the region of inter-est for M with and M without . This yields a total countnumber in the picture with atoms N with and withoutatoms N without from which we compute the optical depth:D = − ln( N with /N without ). The region of interest varieswith the time-of-flight of the cloud. This region is a diskthat ensures that we consider a part of the cloud withapproximately constant density (with 15% rms fluctua-tions), comprising typically 200 pixels. With these imag- ing parameters we can reliably measure optical depths upto 4 but we conservatively fit only data for which D < .
01. At D ∼
3, it reaches 0.12.
D. Atom number calibration
As demonstrated in this article, dipole-dipole interac-tions strongly modify the response of the atomic cloudto resonant light and make an atom number calibra-tion difficult. In this work, we measure the atom num-ber with absorption imaging for different amounts ofatoms transferred by a coherent microwave field from the | F = 1 , m F = − (cid:105) “dark” state to the | F = 2 , m F = − (cid:105) state in which the atoms are resonant with the linearlypolarized probe light. We perform resonant Rabi oscilla-tions for this coherent transfer and fit the measured atomnumber as a function of time by a sinus square function.We select points with an optical depth below 1, to limitthe influence of dipole-dipole interactions. This corre-sponds to small microwave pulse area or to an area closeto a 2 π pulse, to make the fit more robust. From the mea-sured optical depth D, we extract nk − = (15 /
7) D / (6 π ).The factor 7/15 corresponds to the average of equally-weighted squared Clebsch-Gordan coefficients for linearlypolarized light resonant with the F = 2 to F (cid:48) = 3 tran-sition. This model does not take into account possibleoptical pumping effects that could lead to an unequalcontribution from the different transitions and hence asystematic error on the determination of the atom num-ber. E. Experimental protocol
Our basic transmission measurements consist in scan-ning the detuning ∆ ν close to the F = 2 to F (cid:48) = 3resonance ( | ∆ ν | <
30 MHz) and in measuring the opti-cal depth at a fixed density. The other hyperfine levels F (cid:48) = 2 , , P / level play a negligiblerole for this detuning range. The position of the single-atom resonance is independently calibrated using a dilutecloud. The precision on this calibration is of 0 .
03 Γ ,where Γ / π = 6 . ν (cid:55)→ D max / [1 + 4 (∆ ν − ν ) / Γ ] . (1)This function captures well the central shape of the curvefor thin gases, as seen in the examples of Fig. 2. Whenincreasing the atomic density we observe a broadening ofthe line Γ > Γ , a non-linear increase of the maximal op-tical depth D max and a blue shift ν >
0. In Sec. IV wepresent the evolution of these fitted parameters for differ-ent densities and thicknesses. Note that in our analysisall points with values of D above 3 are discarded to avoidpotential systematic errors. Whereas this threshold haslittle influence for thin clouds (as shown in Fig. 2) forwhich the maximal optical depths are not large comparedto the threshold, for thick gases this typically removes themeasurements at detunings smaller than 1.5 Γ . Hence,in this case, we consider the amplitude and the width ofthe fits to be not reliable and we use the position of themaximum of the resonance ν with caution.We investigate the dependence of the fit parametersD max , Γ and ν for different atomic clouds in Sec IV.These results are compared to the prediction from a the-oretical model that we describe in the following Section. III. THEORETICAL DESCRIPTION
Light scattering by a dense sample of emitters is acomplex many-body problem and it is quite challengingto describe. The slab geometry is a textbook situationwhich has been largely explored. A recent detailed studyof the slab geometry can be found in Ref. [39]. We fo-cus in this section first on a perturbative approach whichis valid for low enough densities. We then report cou-pled dipole simulations following the method presentedin [19] but extended with a finite-size scaling approachto address the situation of large slabs. We also discussthe regime of validity for these two approaches.
A. Perturbative approach
We describe here a semi-analytical model accountingfor the multiple scattering of light by a dilute atom sam-ple, inspired from reference [40]. By taking into accountmultiple scattering processes between atom pairs, it pro-vides the first correction to the Beer-Lambert law whendecreasing the mean distance l between nearest neighborstowards k − .
1. Index of refraction of a homogeneous system
In reference [40], the index of refraction of a homo-geneous dilute atomic gas was calculated, taking intoaccount the first non-linear effects occurring when in-creasing the volume atom density. The small parametergoverning the perturbative expansion is ρk − , where ρ isthe atom density. At second order in ρk − two physi-cal effects contribute to the refraction index, namely theeffect of the quantum statistics of atoms on their posi-tion distribution, and the dipole-dipole interactions oc-curring between nearby atoms after one photon absorp-tion. Here we expect the effect of quantum statistics toremain small, and thus neglect it hereafter (see Ref. [41]for a recent measurement of this effect). Including the ef-fect of multiple scattering processes between atom pairs,one obtains the following expression for the refractive in- dex: n r = 1 + αρ − αρ/ βρ (2) β = − (cid:90) d r (cid:20) α G (cid:48) + α G (cid:48) e − ikz − α G (cid:48) (cid:21) xx ( r ) (3)where we introduced the atom polarisability α =6 πik − / (1 − iδ/ Γ) and the Green function [ G ] of anoscillating dipole G αβ ( r ) = − δ ( r ) δ αβ + G (cid:48) αβ ( r ) , G (cid:48) αβ ( r ) = − k π e ikr kr (cid:20) (cid:18) ikr − kr ) (cid:19) r α r β r − (cid:18) ikr − kr ) (cid:19) δ αβ (cid:21) , (4)in which retardation effects are neglected [42]. Note thatfor a thermal atomic sample of Doppler width larger thanΓ, we expect an averaging of the coherent term β to zerodue to the random Doppler shifts. When setting β = 0in Eq. 2 we recover the common Lorentz-Lorenz shift ofthe atomic resonance [43]. We plot in Fig. 3 the imagi-nary part of the index of refraction as a function of thedetuning δ , for a typical atom density used in the ex-periment (solid line) and compare with the single-atomresponse with (dotted line) and without (dashed line)Lorentz-Lorenz correction. The resonance line is modi-fied by dipole-dipole interactions and we observe a blueshift of the position of the maximum of the resonance[44]. − − / Γ . . . . . . . I m ( n r ) FIG. 3. Imaginary part of the index of refraction of anhomogeneous atomic sample of density ρk − (cid:39) .
2. Transmission through an infinite slab with a gaussiandensity profile
In order to account more precisely for the light ab-sorption occurring in the experiment, we extend the per-turbative analysis of light scattering to inhomogeneousatom distributions, for which the notion of index of re-fraction may not be well-defined. The atom distributionis modeled by an average density distribution ρ ( z ) of in-finite extent along x and y , and depending on z only, as ρ ( z ) = ρ exp[ − z / (2∆ z )]. We describe the propaga-tion of light along z in the atomic sample. The incomingelectric field is denoted as E e i ( kz − ωt ) e x . The total elec-tric field, written as E ( z ) e − iωt , is given by the sum ofthe incoming field and the field radiated by the excitedatomic dipoles: E ( z ) = E e ikz e x + (cid:90) d r (cid:48) ρ ( z (cid:48) ) [ G ( r − r (cid:48) )] (cid:15) d ( z (cid:48) ) , (5)where d ( z ) is the dipole amplitude of an atom located at z and (cid:15) is the vacuum permittivity. The integral over x and y can be performed analytically, leading to theexpression E ( z ) = E e ikz e x + ik (cid:15) (cid:90) d z (cid:48) ρ ( z (cid:48) ) e ik | z − z (cid:48) | d ⊥ ( z (cid:48) ) , (6)where d ⊥ ( z ) is the dipole amplitude projected in the x, y plane.The dipole amplitude can be calculated from the atompolarisability α and the electric field at the atom posi-tion. Taking into account multiple light scattering be-tween atom pairs, we obtain a self-consistent expressionfor the dipole amplitude, valid up to first order in atomdensity, as d ( z ) = d ( z ) e x , with d ( z ) = α(cid:15) E e ikz + (cid:90) d r (cid:48) ρ ( z (cid:48) ) (cid:26) (cid:20) α G − α G (cid:21) xx ( r − r (cid:48) ) d ( z (cid:48) )+ (cid:20) α G − α G (cid:21) xx ( r − r (cid:48) ) d ( z ) (cid:27) . (7)Note that the dipole amplitude also features a compo-nent along z , but it would appear in the perturbativeexpansion in the atom density at higher orders.The electric field and dipole amplitude are numericallycomputed by solving the linear system (6)-(7). The op-tical depth is then calculated as D = − ln( | E ( z ) | / | E | )for z (cid:29) ∆ z . The results of this approach will be dis-played and quantitatively compared to coupled dipolesimulations in the next subsection. B. Coupled dipole simulations
1. Methods
Our second approach to simulate the experiments fol-lows the description in Ref. [19] and uses a coupled dipole model. We consider atoms with a J = 0 to J = 1 tran-sition. For a given surface density n and thickness ∆ z we draw the positions of the N atoms with a uniformdistribution in the xy plane and a Gaussian distributionalong the z direction. The number of atoms and hencethe disk radius is varied to perform finite-size scaling. Fora given detuning and a linear polarization along x of theincoming field, we compute the steady-state value of eachdipole d j which is induced by the sum of the contribu-tions from the laser field and from all the other dipoles inthe system. The second contribution is obtained thanksto the tensor Green function G giving the field radiatedat position r by a dipole located at origin.Practically, the values of the N dipoles are obtained bynumerically solving a set of 3 N linear equations, whichlimits the atom number to a few thousands, a much lowervalue than in the experiment (where we have up to 10 atoms). From the values of the dipoles we obtain thetransmission T of the sample: T = 1 − i σ nk − N (cid:88) j k π(cid:15) E L d j,x e − ikz j (8)where z j is the vertical coordinate of the j -th atom, E L the incoming electric field, and d j,x is the x componentof the dipole of the j -th atom. From the transmission,we compute the optical depth D = − ln |T | and fit theresonance line with a Lorentzian line shape to extract, asfor the experimental results, the maximum, the positionand the width of the line. .
00 0 .
04 0 .
08 0 . / √ N sim . . . . . . ν / Γ (a) .
00 0 .
04 0 .
08 0 . / √ N sim . . . . . Γ / Γ (b) FIG. 4. Example of finite-size scaling to determine (a) theposition of the maximum of the resonance ν and (b) thewidth of the resonance. Here k ∆ z = 1 . nk − = 0 .
05, 0.11 and 0.21. Simulations are repeatedfor different atom number N sim . The number of averagesranges from 75 (left points, N sim = 2000) to 25 000 (rightpoints, N sim = 100). When plotting the shift as a functionof 1 / √ N sim ∝ /R , and for low enough densities, data pointsare aligned and allow for a finite-size scaling. Vertical errorbars represent the standard error obtained when averagingthe results over many random atomic distributions. As the number of atoms used in the simulations is lim-ited, it is important to verify the result of the simula-tions is independent of the atom number. In this work,we are mostly interested in the response of an infinitelylarge system in the xy − plane. It is indeed the situation .
00 0 .
04 0 .
08 0 . / √ N sim − . − . − . − . − . . ν / Γ (a) .
00 0 .
04 0 .
08 0 . / √ N sim . . . . . . Γ / Γ (b) FIG. 5. Example of finite-size scaling to determine (a) theposition of the maximum of the resonance ν and (b) thewidth Γ of the resonance. Here, k ∆ z = 80 and (from topto bottom) nk − = 0 . N sim . The number ofaverages ranges from 75 (left points, N sim = 2000) to 25 000(right points, N sim = 100). Vertical error bars represent thestandard error obtained when averaging the result over manyrandom atomic distributions. considered in the perturbative approach and in the ex-perimental system for which the diameter is larger than300 k − and where finite-size effects should be small. Theatom number in the simulations is typically two orderof magnitudes lower than in the experiment and finite-size effects could become important. For instance, somediffraction effects due to the sharp edge of the disk couldplay a role [39]. Consequently, we varied the atom num-ber in the simulations and observed, for simulated cloudswith small radii, a significant dependence of the simula-tion results on the atom number. We have developed afinite-size scaling approach to circumvent this limitation.We focus in the following on transmission measurementsas in the experiment.We show two examples of this finite-size scaling ap-proach for k ∆ z = 1 . k ∆ z = 80 in Fig. 5.For low enough surface densities, the results of the sim-ulations (maximal optical depth, width, shift,...) for dif-ferent atom numbers in the simulation are aligned, whenplotted as a function of 1 / √ N sim , and allow for the de-sired finite-size scaling. All the results presented in thissection and in Sec. IV [45] are obtained by taking the ex-trapolation to an infinite system size, which correspondsto the offset of the linear fit in Figs. 4 and 5.Interestingly, we observe in Fig. 4 for a thin cloud thatconsidering a finite-size system only leads to a small un-derestimate of the blue shift of the resonance. How-ever, for thicker slabs, such as in Fig. 5, we get, for fi-nite systems, a small red shift and a narrowing of theline. Considering our experimental system, we have1 / √ N ≈ . nk − , typi-cally 0.1, whereas we can reach densities 15 times largerin the experiment, which could enhance finite-size effects.Simulation of thick and optically dense slabs is thus chal- lenging and the crossover between the thin slab situationexplored in this article and the thick regime is an inter-esting perspective of this work.
2. Role of the thickness and density of the cloud
We now investigate the results of coupled dipole sim-ulations for different densities and thicknesses of theatomic cloud. We limit the study to low densities, forwhich the finite-size scaling approach works. It is impor-tant to note that the computed line shapes deviate signif-icantly from a Lorentzian shape and become asymmetric.Consequently there is not a unique definition for the cen-ter of the line and for its width. In our analysis, we fitthe resonance lines around their maximum with a typicalrange of ± . ∼ . k ∆ z <
1) it fails for thick and opticallydense systems [46].
3. Comparison with the perturbative model
The perturbative approach is limited to low densities ρk − (cid:28) nk − ∼ .
1. We investigate theshift of the position of the maximum in Fig. 7(b). Wereport, as a function of the inverse thickness (1 /k ∆ z ),the slope γ of the shift with density, ν = γnk − , com-puted for surface densities below 0.1. The dotted line .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . nk − . . . . . . . . . D m a x (a) .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . nk − . . . . . . . . ν / Γ (b) .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . nk − . . . . . . . Γ / Γ (c) FIG. 6. Coupled dipole simulations for different thicknesses. (a) Maximal optical depth, (b) Position of the maximum of theline, (c) Width of the resonance line. We report results for k ∆ z =0, 1.6 and 8, the darkest lines corresponding to the smallestthicknesses. The black dashed lines correspond to the single-atom response. .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 . nk − . . . . . . . . . D (a) /k ∆ z . . . . . . γ (b) FIG. 7. Comparison between the coupled dipole simula-tions and the perturbative model. (a) Behavior of the opticaldepth at the single-atom resonance D with surface densityfor different thicknesses ( k ∆ z =0, 1.6 and 3.2, from bottomto top). Coupled dipole simulations are shown as solid lines,perturbative approach as dotted lines and the dashed line isthe Beer-Lambert prediction. (b) Slope γ of the blue shift ν = γnk − as a function of the inverse thickness 1 /k ∆ z .The solid line is the result of the coupled dipole model, thedash-dotted line is the zero-thickness coupled dipole result(1 /k ∆ z → ∞ ), the dotted line is the perturbative model. is the result from the perturbative approach, the solidline corresponds to coupled dipole simulations and thedash-dotted line to the result for zero thickness. Theperturbative approach approximates well coupled dipole simulations. This result also confirms that the finite-sizescaling approach provides a good determination of theresponse of an infinite system in the xy − direction.We have identified in this Section the specific featuresof the transmission of light trough a dense slab of atoms.We focus here on the transmission coefficient to show thatwe observe the same features in the experiment and wewill make a quantitative comparison between our exper-imental findings and the results obtained with coupleddipole simulations. Our theoretical analysis is comple-mented by a study of the reflection coefficient of a strictly2D gas detailed in Appendix A. IV. EXPERIMENTAL RESULTS
We show in Fig. 8 the results of the experiments in-troduced in Sec. II. The fitted D max for different sur-face densities is shown in Fig. 8(a). We compare theseresults to the Beer-Lambert prediction (narrow dashes)D BL = nσ and to the same prediction corrected by afactor 7/15 (large dashes). This factor is the average ofthe Clebsch-Gordan coefficients relevant for π -polarizedlight tuned close to the | F = 2 (cid:105) → | F (cid:48) = 3 (cid:105) transitionand, as discussed previously, it is included in the cali-bration of the atom number. At large surface densities,we observe an important deviation from this correctedBeer-Lambert prediction: we measure that D max seemsto saturate around D max ≈ . BL ≈
13 [47].We also show the prediction of the coupled dipole model,as a solid line for the full range of surface densities at k ∆ z = 0 and as a dotted line for the numerically accessi-ble range of surface densities at k ∆ z = 2 .
4. The coupleddipole simulation at k ∆ z = 0 shows the same trend asin the experiment but with D max now bounded by 2. Areason could be the non-zero thickness of the atomic slab.In order to test this hypothesis, we investigated the in-fluence of probing duration for the largest density. Forsuch a density we could decrease the pulse duration whilekeeping a good enough signal to noise ratio (see inset inFig. 2(b)). For a shorter probing duration, hence for a . . . . . nk − . . . . . . . . Γ / Γ (b) . . . . . nk − − . . . . . . ν / Γ (c) . . . . . nk − D m a x (a) τ [ µ s] D m a x FIG. 8. Maximum optical depth (a), broadening (b) and fre-quency shift (c) of the resonance line for our thinnest sam-ples with k ∆ z = 2 . k ∆ z = 30(8) (squares). In (a) the shaded area represents theuncertainty in the frequency calibration of the single-atomresonance. In (a) and (b), the dark black solid (resp. lightblue dotted) line is the prediction of the coupled dipole modelfor k ∆ z = 0 (resp. k ∆ z = 2 .
4) in its accessible range of den-sities. The dashed lines represent the single-atom response. smaller expansion of the cloud, D max decreases, in qual-itative agreement with the expected effect of the finitethickness.The saturation of the optical depth with density is acounterintuitive feature. It shows that increasing thesurface density of an atomic layer does not lead to anincrease of its optical depth. Coupled dipole simula-tions at k ∆ z = 0 even show that the system becomesslightly more transparent as the surface density is in-creased. These behavior may be explained qualitatively by the broadening of the distribution of resonance fre-quencies of the eigenmodes of this many-body system. Adense system scatters light for a large range of detuningsbut the cross section at a given detuning saturates orbecomes lower as the surface density is increased.We display in Fig. 8(b) the width Γ of the Lorentzianfits for k ∆ z = 2 . . This broadening isconfirmed by the simulation results for k ∆ z = 0 (solidline). Note that the exact agreement with the experimen-tal data should be considered as coincidental. The rangeon which we can compute the broadening for k ∆ z = 2 . ν with density.A blue shift, reaching 0 . for the largest density, isobserved. At the largest density, an even larger shift isobserved when decreasing the pulse duration ( ≈ . ,not shown here). We also display the result of the coupleddipole model for the cases k ∆ z = 2 . k ∆ z = 0. Bothsimulations confirm the blue shift but predict a differentbehavior and a larger effect. In addition, we show thevariation of ν for a thick cloud with k ∆ z = 30(8). Inthat case we observe a marginally significant red shift[48].The experimental observation of a blue shift has neverbeen reported experimentally to our knowledge. It isin stark contrast, both in amplitude and in sign, withthe mean-field prediction of the Lorentz-Lorenz red shift ν MF0 / Γ = − πρk − = − (cid:112) π/ nk − / ( k ∆ z ), written hereat the center of the cloud along z . The failure of theLorentz-Lorenz prediction for cold atom systems has al-ready been observed and discussed for instance in Refs[30, 34, 36]. As discussed with the perturbative approachin Sec. III, the Lorentz-Lorenz contribution is still presentbut it is (over)compensated by multiple scattering effectsfor a set of fixed scatterers. In hot vapors, where theDoppler effect is large, the contribution of multiple scat-tering vanishes and thus the Lorentz-Lorenz contributionalone is observed. The related Cooperative Lamb shifthas been recently demonstrated in hot vapor of atomsconfined in a thin slab in Ref. [31]. In the cold regimewhere scatterers are fixed, such effects are not expected[39]. However, in these recent studies with dense and coldsamples a small red shift is still observed [30, 34, 36]. Thisdifference on the sign of the frequency shift with respectto the results obtained in this work may be explained byresidual inhomogeneous broadening induced by the finitetemperature or the diluteness of the sample in Ref. [30]and by the specific geometry in Ref. [34], where the sizeof the atomic cloud is comparable to λ and where diffrac-tion effects may play an important role. As discussed inSec. III, our observation of a blue shift is a general resultwhich applies to the infinite slab. It is robust to a widerange of thicknesses and density, and while we computedit theoretically for a two-level system, it also shows upexperimentally in a more complex atomic level structure. ν/ Γ . . . . D ν/ Γ . . . . D k ∆ z − . − . − . − . − . . η e x p (a) (b)(c) FIG. 9. Non-Lorentzian wings of the resonance line. (a) Twoexamples of the scaling of optical depth with ∆ ν (blue side), inlog-log scale, for k ∆ z = 2 . k ∆ z = 350(90)(squares) and their power-law fit. (b) Coupled dipole simu-lations at zero thickness and for nk − = 1 . nk − = 1 . η r (resp. η b ) to the far-detunedregions of the resonance line on the red and blue side, respec-tively. The fit function is ∆ ν (cid:55)→ D(∆ ν ) = A (∆ ν − ν ) η . Theerror on the fitted exponents is also determined using a boot-strap analysis. The horizontal dashed black line ( η = − It was also predicted in Ref. [39] but for a uniform dis-tribution along the z axis instead of the Gaussian profileconsidered in this work, and also discussed in [44]. Con-sequently, we believe that it is an important and genericfeature of light scattering in a extended cloud of fixedrandomly distributed scatterers.Finally, we compare the lineshape of the resonancewith the Lorentzian shape expected for a single atom. Wemeasure for nk − = 1 . η r (resp. η b ) as shown,for two examples, in Fig. 9(a). If the behavior were in-deed Lorentzian, the exponents should be − − .
3, show- ing the strong influence of dipole-dipole interactions inour system. We show the result of coupled dipole simu-lations for k ∆ z = 0 in Fig. 4(b) along with their power-law fit. We extract the exponents η r = − . η b = − . V. CONCLUSION
In summary we have studied the transmission of amacroscopic dense slab of atoms with uniform in-planedensity and a transverse gaussian density distribution.We observed a strong reduction of the maximum opticaldensity and a broadening of the resonance line. Moresurprisingly, we showed the presence of a large blue shiftof the resonance line and a deviation from Lorentzian be-havior in the wings of the resonance line. These resultsare qualitatively confirmed by coupled dipole simulationsand a perturbative approach of this scattering problem.We also confirm the difficulty already observed to obtaina quantitative agreement between coupled dipole sim-ulations and experimental results in the dense regime[32, 34]. Possible explanations for this discrepancy are(i) residual motion of the atoms during the probing dueto the strong light-induced dipole-dipole interactions, (ii)a too large intensity used in the experiment which goesbeyond the validity of the coupled dipole approach, (iii)the influence of the complex atomic level structure. Wewere careful in this work to limit the influence of the twofirst explanations and the last possibility is likely to bethe main limitation. The complex level structure leads tooptical pumping effects during the probing and thus thescattering cross-section of the sample is not well-defined.A simple way to take into account the level structure is,as discussed in Sec. IV, to renormalize the scattering crosssection by the average of the Clebsch-Gordan coefficientsinvolved in the process. For Rb atoms this amounts forthe factor 7/15 already discussed earlier. However thisis a crude approximation which neglects optical pump-ing effects during scattering and whose validity in thedense regime is not clear. Two approaches can be con-sidered to remove this limitation. First, one can useanother atomic species such as strontium or ytterbiumbosonic isotopes which have a spin singlet ground stateand in which almost exact two-level systems are avail-able for some optical transitions. Scattering experimentson strontium clouds have been reported [25, 30, 49] butthey did not explore the dense regime tackled in thiswork. The comparison with theory thus relies on model-ing their inhomogeneous density distribution accurately.Second, an effective two-level system can be created inthe widely used alkali atoms by imposing a strong mag-netic field which could separate the different transitionsby several times the natural linewidth as demonstrated insome recent experiments on three-level systems [50, 51].This method could be in principle applied on our setup0 . . . . . . . nk − . . . . . . | R | . . . . . D FIG. 10. Intensity reflection coefficient as a function of surfacedensity for k ∆ z = 0 (solid line). For comparison we show thecorresponding optical depth D (dotted line, right axis) andthe lower bound for the reflection coefficient deduced fromthis optical depth (dashed line). to create an effective two-level system and could help tounderstand the aforementioned discrepancies.Finally, we note that this article focuses on the steady-state transmission of a cloud illuminated by a uniformmonochromatic beam. The slab geometry that we havedeveloped here is of great interest for comparison betweentheory and experiments and our work opens interestingperspectives for extending this study to time-resolved ex-periments, to fluorescence measurements or to spatiallyresolved propagation of light studies. ACKNOWLEDGMENTS
We thank Vitaly Kresin, Klaus Mølmer, Janne Ru-ostekoski, Markus Greiner, Zoran Hadzibabic, WilhelmZwerger for fruitful discussions. This work is supportedby DIM NanoK, ERC (Synergy UQUAM). L.C. ac-knowledges the support from DGA. This project has re- ceived funding from the European Union’s Horizon 2020research and innovation programme under the MarieSk(cid:32)lodowska-Curie grant agreement N ◦ Appendix A: Reflection coefficient of a 2D gas
Thanks to their large scattering cross section at res-onance, array of atoms can be used to emit light witha controlled spatial pattern [52]. A single-atom mirrorhas been demonstrated [53] and, more generally, regulartwo-dimensional arrays of atoms have been consideredfor realizing controllable light absorbers [17] or mirrors[18] with atomic-sized thicknesses. For the disorderedatomic samples considered in this article the strong de-crease of the transmission because of dipole-dipole inter-actions could lead to a large reflection coefficient. Fora strictly two-dimensional gas we show as a solid linein Fig. 10 the result of the coupled dipole model for theintensity reflection coefficient |R| at resonance and atnormal incidence as a function of density. This intensityreflection coefficient has a behavior with density similarto the optical depth D (dotted line). The relation be-tween these two quantities depends on the relative phasebetween the incoming and the reflected field. For a trans-mitted field in phase with the incident field we find, usingthe boundary condition R + T = 1, a lower bound forthis reflection coefficient, |R| ≥ (1 − |T | ) , shown as adashed line in Fig. 10. The intensity reflection coefficientis close to this lower bound in the regime explored in thiswork. The maximum computed value for the reflectioncoefficient is close to 40 % which shows that a single dis-ordered layer of individual atoms can significantly reflectan incoming light beam [54]. Note that for a non-2Dsample light can be diffused at any angle. For our exper-imental thickness and the relevant densities the reflectioncoefficient is in practice much lower than the above pre-diction. [1] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, andP. Thickstun, Atom-photon interactions: basic processesand applications (Wiley Online Library, 1992).[2] J.-M. Raimond and S. Haroche,
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