Transmission of Near-Resonant Light Through a Dense Slab of Cold Atoms

PHYSICAL REVIEW A 96, 053629 (2017)

Transmission of near-resonant light through a dense slab of cold atoms

L. Corman,* J. L. Ville, R. Saint-Jalm, M. Aidelsburger,† T. Bienaimé, S. Nascimbène, J. Dalibard, and J. Beugnon‡ Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL Research University, UPMC-Sorbonne Universités, 11 place Marcelin-Berthelot, 75005 Paris, France (Received 30 June 2017; revised manuscript received 4 October 2017; published 27 November 2017)

The optical properties of randomly positioned, resonant scatterers is a fundamentally difﬁcult problem to address across a wide range of densities and geometries. We investigate it experimentally using a dense cloud of rubidium atoms probed with near-resonant light. The atoms are conﬁned in a slab geometry with a subwavelength thickness. We probe the optical response of the cloud as its density and hence the strength of the light-induced dipole-dipole interactions are increased. We also describe a theoretical study based on a coupled dipole simulation which is further complemented by a perturbative approach. This model reproduces qualitatively the experimental observation of a saturation of the optical depth, a broadening of the transition, and a blueshift of the resonance.

DOI: 10.1103/PhysRevA.96.053629

I. INTRODUCTION than solid-state systems, strong attenuation of the transmission can be observed at resonance. Third, inhomogeneous Doppler The interaction of light with matter is a fundamental broadening can be made negligible using ultracold atomic problem which is relevant for simple systems, such as an atom clouds. Finally, the geometry and the density of the gases strongly coupled to photons [1–3], as well as for complex can be varied over a broad range. materials, whose optical properties provide information on In the dilute limit, such that the three-dimensional (3D) their electronic structure and geometry [4]. This interaction − atomic density ρ and the light wave number k verify ρk 3 1, can also be harnessed to create materials and devices with and for low optical depths, a photon entering the atomic tailored properties, from quantum information systems such medium does not recurrently interact with the same atom. as memories [5] and nanophotonic optical isolators [6] to solar Then, for a two-level atom, the transmission of a resonant cells combining highly absorptive materials with transparent probe beam propagating along the z axis is given by the electrodes [7]. − − Beer-Lambert law: |T |2 = e σ0 ρdz, where σ = 6πk 2 is The slab geometry is especially appropriate to study 0 the light cross section at the optical resonance [20]. At larger light-matter interaction [8,9]. In the limit of a monolayer, densities the transmission is strongly affected by the light- two-dimensional (2D) materials exhibit fascinating optical induced dipole-dipole coupling between neighboring atoms. properties. For simple direct band gap 2D semiconductors, Modification of the atomic resonance line shape or super- the single-particle band structure implies that the transmission and subradiance in dilute (but usually optically dense) and cold coefficient takes a universal value [10,11]. This was first atomic samples have been largely investigated experimentally measured for single-layer graphene samples [12], which have [21–30]. Recently, experiments have been performed in the an optical transmission independent of the light frequency in dense regime studying nanometer-thick hot vapors [31] and the eV range, |T |2 = 1 − πα, where α is the fine-structure mesoscopic cold clouds [32–35]. Interestingly, it has been constant [13,14]. The same value was recovered in InAs found that the mean-field Lorentz-Lorenz shift is absent in semiconductors [15]. This universality does not hold for cold systems where the scatterers remain fixed during the more complex 2D materials, for instance when the Coulomb measurement. A small redshift is still observed for dense interaction plays a more important role [16]. clouds in Refs. [32,34] but could be specific to the geometry Atomic gases represent in many respects an ideal test bed of the system. for investigating light-matter interaction. First, they can be Achieving large densities is concomitant with a vanishingly arranged in regular arrays [17,18] or randomly placed [19]to small transmission T . It is therefore desirable to switch tailor the optical properties of the system. Second, an atom to a 2D or thin slab geometry in order to investigate the always scatters light, in contrast with solid-state materials physical consequences of these resonant interaction effects where the optical excitation can be absorbed and dissipated in a at the macroscopic level. Using a 2D geometry also raises nonradiative way. Even for thin and much more dilute samples a fundamental question inspired by the monolayer semi- conductor case: can the light extinction through a plane of randomly positioned atoms be made arbitrarily large when *Also at Institute for Quantum Electronics, ETH Zurich, 8093 increasing the atom density or does it remain finite, potentially Zurich, Switzerland. †Also at Fakultät für Physik, Ludwig-Maximilians-Universität introducing a maximum of light extinction through 2D random München, Schellingstr. 4, 80799 Munich, Germany. atomic samples independent of the atomic species of identical ‡[email protected] electronic spin? In this article, we study the transmission of nearly resonant Published by the American Physical Society under the terms of the light through uniform slabs of atoms. We report experiments Creative Commons Attribution 4.0 International license. Further realized on a dense layer of atoms with a tunable density distribution of this work must maintain attribution to the author(s) and thickness. For dense clouds, the transmission is strongly and the published article’s title, journal citation, and DOI. enhanced compared to the one expected from the single-atom

2469-9926/2017/96(5)/053629(11) 053629-1 Published by the American Physical Society L. CORMAN et al. PHYSICAL REVIEW A 96, 053629 (2017) response. We also observe a broadening and a blueshift of (a) the resonance line on the order of the natural linewidth. This blueshift contrasts with the mean-ﬁeld Lorentz-Lorenz redshift and is a signature of the stronglycorrelated regime reached in our system because of dipole-dipole interactions [36]. In addition, we observe deviations of the resonance line shape from the single-atom Lorentzian behavior, especially in the wings where the transmission decays more slowly. We model (b) this system with coupled dipole simulations complemented by a perturbative approach which qualitatively supports our observations. After describing our experimental system in Sec. II, we investigate theoretically light scattering for the geometry explored in the experiment in Sec. III. In Sec. IV we present our experimental results and compare them with theory. We conclude in Sec. V.

II. EXPERIMENTAL METHODS FIG. 1. (a) Schematic representation of the imaging setup. The atoms are confined by a single, disk-shaped potential which is imaged A. Cloud preparation using a microscope objective onto a back-illuminated CCD camera. We prepare a cloud of 87Rb atoms with typically N = The numerical aperture of the system is limited to ≈0.2 using an iris in 5 the Fourier plane of the atoms to limit the collected fluorescence light. 1.3(2) × 10 atoms in the |F = 1,mF =−1 state. The atoms are confined in an all-optical trap, described in more detail (b) Typical in situ image obtained on a back illuminated CCD camera in [37], with a strong harmonic confinement in the vertical of the in-plane density distribution averaged over three individual = measurements. For this example, the atom surface density is n = direction z with frequency ωz/2π 2.3(2) kHz leading to a −2 Gaussian density profile along this direction. The transverse 25 μm . We extract a region of interest with uniform density for our analysis with a typical area of 200 μm2. confinement along the x and y directions is produced by a flat-bottom disk-shaped potential of diameter 2R = 40 μm. For our initial cloud temperature 300 nK, there is no τ = 10 μs that the thickness averaged over the pulse duration extended phase coherence in the cloud [38]. Taking into is increased by ∼20%. In some experiments, in which the account this finite temperature, we compute for an ideal Bose signal is large enough, we limit this effect by reducing the gas a rms thickness z = 0.25(1) μm, or equivalently kz = probe duration τ to 3 μs. 2.0(1). This situation corresponds to nk−2 ≈ 1.5, where n = N/(πR2) is the surface density and to a maximum density ρk−3 ≈ 0.3 at the trap center along z where ρ is the volume B. Transmission measurement density. We tune the number of atoms that interact with light We probe the response of the cloud by measuring the trans- by partially transferring them to the |F = 2,mF =−2 state mission of a laser beam propagating along the z direction (See using a resonant microwave transition. Atoms in this state Fig. 1). The light is linearly polarized along the x axis and tuned are sensitive to the probe excitation, contrary to the ones in the close to the |F = 2→|F = 3 D2 transition. The duration |F = 1,mF =−1 state. In this temperature range the Doppler of the light pulse is fixed to 10 μs for most experiments and we broadening is about three orders of magnitude smaller than the limit the imaging intensity I to the weakly saturating regime 2 natural linewidth of the atomic transition. with 0.075

053629-2 TRANSMISSION OF NEAR-RESONANT LIGHT THROUGH A . . . PHYSICAL REVIEW A 96, 053629 (2017)

The magnification of the optical system is 11.25, leading to 4.0 an effective pixel size in the plane of the atoms of 1.16 μm. The 3.5 typical mean number of counts per pixel accumulated during 3.0 the 10 μs imaging pulse is 80 on the picture without atoms. We optimize the signal-to-noise ratio by summing all the pixels 2.5 in the region of interest for Mwith and Mwithout. This yields a 2.0 D total count number in the picture with atoms Nwith and without 1.5 atoms Nwithout from which we compute the optical depth: D = 1.0 − ln(Nwith/Nwithout). The region of interest varies with the time of flight of the cloud. This region is a disk that ensures that 0.5 we consider a part of the cloud with approximately constant 0.0 density (with 15% rms fluctuations), comprising typically 200 −6 −4 −2 0 2 4 6 pixels. With these imaging parameters we can reliably measure Δν/Γ0 optical depths up to 4 but we conservatively fit only data for which D<3. At low densities, the statistical error on D due FIG. 2. Example of resonance curves. Symbols represent the to the readout noise is about 0.01. At D ∼ 3, it reaches 0.12. experimental data and the corresponding dashed lines are Lorentzian fits. All curves are taken with the cloud thickness kz = 2.4(6) and for surface densities of nk−2 = 0.06(1) (circles), 0.38(6) (squares), D. Atom number calibration and 1.5(2) (diamonds). The errors on the fitted parameters are As demonstrated in this article, dipole-dipole interactions determined using a basic bootstrap analysis, repeating the fitting strongly modify the response of the atomic cloud to resonant procedure 100 times on a set of random points drawn from the original light and make an atom number calibration difficult. In set of data, of the same length as this original set. this work, we measure the atom number with absorption imaging for different amounts of atoms transferred by a a blueshift ν > 0. In Sec. IV we present the evolution of coherent microwave field from the |F = 1,m =−1 “dark” 0 F these fitted parameters for different densities and thicknesses. statetothe|F = 2,m =−2 state in which the atoms are F Note that in our analysis all points with values of D above 3 resonant with the linearly polarized probe light. We perform are discarded to avoid potential systematic errors. Whereas this resonant Rabi oscillations for this coherent transfer and fit threshold has little influence for thin clouds (as shown in Fig. 2) the measured atom number as a function of time by a sinus for which the maximal optical depths are not large compared square function. We select points with an optical depth below to the threshold, for thick gases this typically removes the 1, to limit the influence of dipole-dipole interactions. This measurements at detunings smaller than 1.5 . Hence, in this corresponds to small microwave pulse area or to an area 0 case, we consider the amplitude and the width of the fits to be close to a 2π pulse, to make the fit more robust. From the not reliable and we use the position of the maximum of the measured optical depth D, we extract nk−2 = (15/7)D/(6π). resonance ν with caution. The factor 7/15 corresponds to the average of equally weighted 0 We investigate the dependence of the fit parameters D , squared Clebsch-Gordan coefficients for linearly polarized max , and ν for different atomic clouds in Sec. IV. These results light resonant with the F = 2toF = 3 transition. This model 0 are compared to the prediction from a theoretical model that does not take into account possible optical pumping effects we describe in the following section. that could lead to an unequal contribution from the different transitions and hence a systematic error on the determination of the atom number. III. THEORETICAL DESCRIPTION Light scattering by a dense sample of emitters is a complex E. Experimental protocol many-body problem and it is quite challenging to describe. Our basic transmission measurements consist in scanning The slab geometry is a textbook situation which has been the detuning ν close to the F = 2toF = 3 resonance largely explored. A recent detailed study of the slab geometry (|ν| < 30 MHz) and in measuring the optical depth at a fixed can be found in Ref. [39]. We focus in this section first density. The other hyperfine levels F = 2,1,0 of the excited on a perturbative approach which is valid for low enough densities. We then report coupled dipole simulations following 5P3/2 level play a negligible role for this detuning range. The position of the single-atom resonance is independently the method presented in [19] but extended with a finite-size calibrated using a dilute cloud. The precision on this calibration scaling approach to address the situation of large slabs. We also discuss the regime of validity for these two approaches. is of 0.03 0, where 0/2π = 6.1 MHz is the atomic linewidth. The measured resonance curves are fitted with a Lorentzian function: A. Perturbative approach 2 2 We describe here a semianalytical model accounting for the ν → Dmax/[1 + 4(ν − ν0) / ]. (1) multiple scattering of light by a dilute atom sample, inspired This function captures well the central shape of the curve for from Ref. [40]. By taking into account multiple scattering thin gases, as seen in the examples of Fig. 2. When increasing processes between atom pairs, it provides the first correction the atomic density we observe a broadening of the line > 0, to the Beer-Lambert law when decreasing the mean distance l −3 a nonlinear increase of the maximal optical depth Dmax, and between nearest neighbors towards k .

053629-3 L. CORMAN et al. PHYSICAL REVIEW A 96, 053629 (2017)

1. Index of refraction of a homogeneous system 0.30 In Ref. [40], the index of refraction of a homogeneous 0.25 dilute atomic gas was calculated, taking into account the first nonlinear effects occurring when increasing the volume 0.20 atom density. The small parameter governing the perturbative ) − r expansion is ρk 3, where ρ is the atom density. At second n 0.15 order in ρk−3 two physical effects contribute to the refraction Im( index, namely the effect of the quantum statistics of atoms on 0.10 their position distribution and the dipole-dipole interactions 0.05 occurring between nearby atoms after one photon absorption. Here we expect the effect of quantum statistics to remain 0.00 small, and thus neglect it hereafter (see Ref. [41]fora −2 −1 0 1 2 2Δ/Γ recent measurement of this effect). Including the effect of 0 multiple scattering processes between atom pairs, one obtains the following expression for the refractive index: FIG. 3. Imaginary part of the index of refraction of a homogeneous atomic sample of density ρk−3 0.026. The three curves αρ n = 1 + , (2) correspond to the absorption of independent atoms (dashed black r 1 − αρ/3 + βρ line), to the resonance line taking into account the Lorentz-Lorenz correction (dotted blue line), and to the perturbative analysis discussed α2G2 + α3G3e−ikz β =− dr (r), (3) in the text (solid black line), which takes into account multiple − 2 2 1 α G xx scattering of photons between pairs of atoms [40]. where we introduced the atom polarizability α = 6πik−3/(1 − 2iδ/ ) and the Green function [G]ofan dipoles: oscillating dipole, − ikz 3 [G(r r )] E(z) = E0e ex + d r ρ(z ) d(z ), (5) 1 G (r) =− δ(r)δ + G (r), 0 αβ 3 αβ αβ where d(z) is the dipole amplitude of an atom located at z and k3 eikr 3i 3 r r is the vacuum permittivity. The integral over x and y can be G (r) =− 1 + − α β 0 αβ 4π kr kr (kr)2 r2 performed analytically, leading to the expression i 1 ikz ik ik|z−z| = + ⊥ − 1 + − δαβ , (4) E(z) E0e ex dz ρ(z )e d (z ), (6) kr (kr)2 2 0 in which retardation effects are neglected [42]. Note that for where d⊥(z) is the dipole amplitude projected in the x,y plane. a thermal atomic sample of Doppler width larger than ,we The dipole amplitude can be calculated from the atom expect an averaging of the coherent term β to zero due to polarizability α and the electric field at the atom position. the random Doppler shifts. When setting β = 0inEq.(2) Taking into account multiple light scattering between atom we recover the common Lorentz-Lorenz shift of the atomic pairs, we obtain a self-consistent expression for the dipole = resonance [43]. We plot in Fig. 3 the imaginary part of the amplitude, valid up to first order in atom density, as d(z) d(z)ex, with index of refraction as a function of the detuning δ, for a typical atom density used in the experiment (solid line), and compare αG d(z) = α E eikz + dr ρ(z) (r − r)d(z) with the single-atom response with (dotted line) and without 0 0 1 − α2G2 (dashed line) Lorentz-Lorenz correction. The resonance line xx 2 2 is modified by dipole-dipole interactions and we observe a α G + (r − r )d(z) . (7) − 2 2 blueshift of the position of the maximum of the resonance 1 α G xx [44]. Note that the dipole amplitude also features a component along z, but it would appear in the perturbative expansion in the atom 2. Transmission through an infinite slab with a Gaussian density at higher orders. density profile The electric field and dipole amplitude are numerically In order to account more precisely for the light absorption computed by solving the linear system (6) and (7). The 2 2 occurring in the experiment, we extend the perturbative anal- optical depth is then calculated as D =−ln(|E(z)| /|E0| ) ysis of light scattering to inhomogeneous atom distributions, for z z. The results of this approach will be displayed and for which the notion of index of refraction may not be well quantitatively compared to coupled dipole simulations in the defined. The atom distribution is modeled by an average next subsection. density distribution ρ(z) of infinite extent along x and y, = − 2 2 and depending on z only, as ρ(z) ρ0 exp[ z /(2z )]. We B. Coupled dipole simulations describe the propagation of light along z in the atomic sample. i(kz−ωt) 1. Methods The incoming electric field is denoted as E0e ex.The total electric field, written as E(z)e−iωt, is given by the sum of Our second approach to simulate the experiments follows the incoming field and the field radiated by the excited atomic the description in Ref. [19] and uses a coupled dipole model.

053629-4 TRANSMISSION OF NEAR-RESONANT LIGHT THROUGH A . . . PHYSICAL REVIEW A 96, 053629 (2017)

We consider atoms with a J = 0toJ = 1 transition. For a (a) (b) 1.15 given surface density n and thickness z we draw the positions 0.25 1.10 of the N atoms with a uniform distribution in the xy plane and 0.20 0 a Gaussian distribution along the z direction. The number of 0.15 0 1.05 Γ Γ / atoms and hence the disk radius is varied to perform finite-size / 0 Γ scaling. For a given detuning and a linear polarization along ν 0.10 1.00 0.05 x of the incoming field, we compute the steady-state value of 0.95 each dipole dj which is induced by the sum of the contributions 0.00 from the laser field and from all the other dipoles in the system. 0.00 0.04√ 0.08 0.12 0.00 0.04√ 0.08 0.12 The second contribution is obtained thanks to the tensor Green 1/ Nsim 1/ Nsim function G giving the field radiated at position r by a dipole located at origin. FIG. 4. Example of finite-size scaling to determine (a) the Practically, the values of the N dipoles are obtained by position of the maximum of the resonance and (b) the width of the − numerically solving a set of 3N linear equations, which limits resonance. Here kz = 1.6 and (from bottom to top) nk 2 = 0.05, the atom number to a few thousands, a much lower value than 0.11, and 0.21. Simulations are repeated for different atom number = in the experiment (where we have up to 105 atoms). From Nsim. The number of averages ranges from 75 (left points, Nsim = the values of the dipoles we obtain the transmission T of the 2000) to 25 000 (right√ points, Nsim 100). When plotting the shift ∝ sample: as a function of 1/ Nsim 1/R, and for low enough densities, data points are aligned and allow for a finite-size scaling. Vertical error −2 3 i nk k − bars represent the standard error obtained when averaging the results T = 1 − σ d e ikzj , (8) 2 N 6π E j,x over many random atomic distributions. j 0 L √ where zj is the vertical coordinate of the jth atom, EL the experimental system, we have 1/ N ≈ 0.003, leading to a incoming electric field, and dj,x is the x component of the small correction according to the fits in Fig. 5. However, for dipole of the jth atom. From the transmission, we compute such thick systems we are able to simulate only systems with the optical depth D =−ln |T |2 and fit the resonance line with low nk−2, typically 0.1, whereas we can reach densities 15 a Lorentzian line shape to extract, as for the experimental times larger in the experiment, which could enhance finite-size results, the maximum, the position, and the width of the line. effects. Simulation of thick and optically dense slabs is thus As the number of atoms used in the simulations is limited, challenging and the crossover between the thin slab situation it is important to verify the result of the simulations is explored in this article and the thick regime is an interesting independent of the atom number. In this work, we are mostly perspective of this work. interested in the response of an infinitely large system in the xy plane. It is indeed the situation considered in the 2. Role of the thickness and density of the cloud perturbative approach and in the experimental system for − We now investigate the results of coupled dipole which the diameter is larger than 300k 1 and where finite-size simulations for different densities and thicknesses of the effects should be small. The atom number in the simulations is atomic cloud. We limit the study to low densities, for which typically two orders of magnitude lower than in the experiment the finite-size scaling approach works. It is important to note and finite-size effects could become important. For instance, that the computed line shapes deviate significantly from a some diffraction effects due to the sharp edge of the disk could Lorentzian shape and become asymmetric. Consequently play a role [39]. Consequently, we varied the atom number in the simulations and observed, for simulated clouds with small radii, a significant dependence of the simulation results on the (a) (b) 1.0 atom number. We have developed a finite-size scaling approach 0.00 to circumvent this limitation. We focus in the following on −0.05 0.9 transmission measurements as in the experiment. −0.10 0.8 0 0 Γ Γ

We show two examples of this finite-size scaling approach / −0.15 / 0.7 0 Γ for kz = 1.6inFig.4 and kz = 80 in Fig. 5.For ν 0.6 low enough surface densities, the results of the simulations −0.20 (maximal optical depth, width, shift, etc.) for different atom −0.25 0.5 numbers in the√ simulation are aligned, when plotted as a 0.00 0.04√ 0.08 0.12 0.00 0.04√ 0.08 0.12 function of 1/ Nsim, and allow for the desired finite-size 1/ Nsim 1/ Nsim scaling. All the results presented in this section and in Sec. IV [45] are obtained by taking the extrapolation to an infinite FIG. 5. Example of finite-size scaling to determine (a) the system size, which corresponds to the offset of the linear fit in position of the maximum of the resonance ν0 and (b) the width Figs. 4 and 5. of the resonance. Here, kz = 80 and (from top to bottom) Interestingly, we observe in Fig. 4 for a thin cloud nk−2 = 0.027, 0.08, and 0.13. Simulations are repeated for different that considering a finite-size system only leads to a small atom number Nsim. The number of averages ranges from 75 (left underestimate of the blueshift of the resonance. However, for points, Nsim = 2000) to 25 000 (right points, Nsim = 100). Vertical thicker slabs, such as in Fig. 5, we get, for finite systems, a error bars represent the standard error obtained when averaging the small redshift and a narrowing of the line. Considering our result over many random atomic distributions.

053629-5 L. CORMAN et al. PHYSICAL REVIEW A 96, 053629 (2017)

(a) 4.0 (b) 0.35 (c) 1.25 3.5 0.30 1.20 3.0 0.25 2.5 1.15 0 0.20 0 Γ 2.0 Γ 1.10 / max 0.15 / 0 Γ D

1.5 ν 1.05 1.0 0.10 0.5 0.05 1.00 0.0 0.00 0.95 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25 nk−2 nk−2 nk−2

FIG. 6. Coupled dipole simulations for different thicknesses. (a) Maximal optical depth; (b) position of the maximum of the line; (c) width of the resonance line. We report results for kz = 0, 1.6, and 8, the darkest lines corresponding to the smallest thicknesses. The black dashed lines correspond to the single-atom response. there is not a unique definition for the center of the line and simulations. This result also confirms that the finite-size for its width. In our analysis, we fit the resonance lines around scaling approach provides a good determination of the their maximum with a typical range of ±0.5 . The shift thus response of an infinite system in the xy direction. corresponds to a variation of the position of the maximum of We have identified in this section the specific features of the line and the “width” characterizes the curvature of the line the transmission of light through a dense slab of atoms. We around its maximum. The results of these fits are reported in focus here on the transmission coefficient to show that we Fig. 6 as a function of surface density for different thicknesses. In these plots, we observe the same features as qualitatively (a) 4.0 described in Sec. II: a decrease of the maximal optical depth with respect to the single-atom response [Fig. 6(a)], a 3.5 blueshift of the position of the maximum [Fig. 6(b)], and a 3.0 broadening of the line [Fig. 6(c)]. For a fixed thickness, these 2.5 effects increase with surface density and for a fixed surface 0 2.0 density they are more pronounced for lower thicknesses. Note D that we only explore here surface densities lower than 0.25 1.5 which is quite lower than the maximum experimental value 1.0 (∼1.5). Whereas our finite-size scaling approach can be well extended for very thin systems (kz < 1) it fails for thick 0.5 and optically dense systems [46]. 0.0 0.00 0.05 0.10 0.15 0.20 0.25 nk−2 3. Comparison with the perturbative model (b) 2.5 The perturbative approach is limited to low densities − ρk 3 1 but it gives the response of an infinitely expanded 2.0 cloud in the transverse direction. Coupled dipole simulations can in principle address arbitrarily large densities but the 1.5 number of atoms considered in a simulation is limited, and thus for a given density the size of the system is limited. γ 1.0 Coupled dipole simulations are thus more relevant for thin and dense samples and the perturbative approach more suited for 0.5 nonzero thickness samples. In Fig. 7 we choose two illustrative examples to confirm, 0.0 in the regime where both models could be used, that these 0 1 2 3 4 5 two approaches are in quantitative agreement. In Fig. 7(a) we 1/kΔz compare the maximum optical depth as a function of surface density for three different thicknesses. The perturbative FIG. 7. Comparison between the coupled dipole simulations approach is typically valid, for this set of thicknesses, up to and the perturbative model. (a) Behavior of the optical depth at nk−2 ∼ 0.1. We investigate the shift of the position of the the single-atom resonance D0 with surface density for different maximum in Fig. 7(b). We report, as a function of the inverse thicknesses (kz = 0, 1.6, and 3.2, from bottom to top). Coupled thickness (1/kz), the slope γ of the shift with density, dipole simulations are shown as solid lines, perturbative approach −2 ν0 = γnk , computed for surface densities below 0.1. The as dotted lines, and the dashed line is the Beer-Lambert prediction. −2 dotted line is the result from the perturbative approach, the (b) Slope γ of the blueshift ν0 = γnk as a function of the inverse solid line corresponds to coupled dipole simulations, and thickness 1/kz. The solid line is the result of the coupled dipole the dash-dotted line to the result for zero thickness. The model, the dash-dotted line is the zero-thickness coupled dipole result perturbative approach approximates well coupled dipole (1/kz →∞), and the dotted line is the perturbative model.

053629-6 TRANSMISSION OF NEAR-RESONANT LIGHT THROUGH A . . . PHYSICAL REVIEW A 96, 053629 (2017)

= (a) 8 Beer-Lambert prediction (narrow dashes) DBL nσ0 and 4 to the same prediction corrected by a factor 7/15 (large 7 3 dashes). This factor is the average of the Clebsch-Gordan 2 6 max D coefﬁcients relevant for π-polarized light tuned close to the 1 | = →| = 5 0 F 2 F 3 transition and, as discussed previously, 0 2 4 6 8 10 12 4 τ[μs] it is included in the calibration of the atom number. At large max

D 3 surface densities, we observe an important deviation from this corrected Beer-Lambert prediction: we measure that Dmax 2 seems to saturate around Dmax ≈ 3.5, whereas DBL ≈ 13 [47]. 1 We also show the prediction of the coupled dipole model, as 0 a solid line for the full range of surface densities at kz = 0 0.0 0.5 1.0 1.5 2.0 and as a dotted line for the numerically accessible range of −2 nk surface densities at kz = 2.4. The coupled dipole simulation = (b) 4.0 at kz 0 shows the same trend as in the experiment but with Dmax now bounded by 2. A reason could be the nonzero 3.5 thickness of the atomic slab. In order to test this hypothesis, 3.0 we investigated the inﬂuence of probing duration for the largest density. For such a density we could decrease the pulse

0 2.5

Γ duration while keeping a good enough signal-to-noise ratio / 2.0 [see inset in Fig. 2(b)]. For a shorter probing duration, hence for Γ a smaller expansion of the cloud, D decreases, in qualitative 1.5 max agreement with the expected effect of the ﬁnite thickness. 1.0 The saturation of the optical depth with density is a 0.5 counterintuitive feature. It shows that increasing the surface 0.0 0.5 1.0 1.5 2.0 density of an atomic layer does not lead to an increase of its nk−2 optical depth. Coupled dipole simulations at kz = 0even show that the system becomes slightly more transparent as the (c) 0.4 surface density is increased. This behavior may be explained 0.3 qualitatively by the broadening of the distribution of resonance frequencies of the eigenmodes of this many-body system. A 0.2 dense system scatters light for a large range of detunings but 0 the cross section at a given detuning saturates or becomes Γ /

0 0.1 lower as the surface density is increased. ν We display in Fig. 8(b) the width of the Lorentzian 0.0 fits for kz = 2.4(6) along with coupled dipole simulation results [46]. We observe a broadening of the resonance line −0.1 up to more than 3 0. This broadening is confirmed by the 0.0 0.5 1.0 1.5 2.0 simulation results for kz = 0 (solid line). Note that the exact nk−2 agreement with the experimental data should be considered as coincidental. The range on which we can compute the FIG. 8. Maximum optical depth (a), broadening (b), and fre- broadening for kz = 2.4 (light dotted line) is too small to quency shift (c) of the resonance line for our thinnest samples with discuss a possible agreement. kz = 2.4(6) (circles) and for thicker samples with kz = 30(8) We show in Fig. 8(c) the evolution of ν0 with density. A (squares). In (a) the shaded area represents the uncertainty in the blueshift, reaching 0.2 0 for the largest density, is observed. frequency calibration of the single-atom resonance. In (a) and (b), At the largest density, an even larger shift is observed when the dark black solid (light blue dotted) line is the prediction of the decreasing the pulse duration (≈0.4 0, not shown here). We coupled dipole model for kz = 0(kz = 2.4) in its accessible range also display the result of the coupled dipole model for the of densities. The dashed lines represent the single-atom response. cases kz = 2.4 and kz = 0. Both simulations confirm the blueshift but predict a different behavior and a larger effect. observe the same features in the experiment and we will make In addition, we show the variation of ν0 for a thick cloud with a quantitative comparison between our experimental findings kz = 30(8). In that case we observe a marginally significant and the results obtained with coupled dipole simulations. redshift [48]. Our theoretical analysis is complemented by a study of the The experimental observation of a blueshift is in stark reflection coefficient of a strictly 2D gas detailed in Appendix. contrast, both in amplitude and in sign, with the mean- MF = field prediction√ of the Lorentz-Lorenz redshift ν0 / 0 −πρk−3 =− π/2 nk−2/(kz), written here at the center IV. EXPERIMENTAL RESULTS of the cloud along z. The failure of the Lorentz-Lorenz We show in Fig. 8 the results of the experiments introduced prediction for cold atom systems has already been observed in Sec. II. The fitted Dmax for different surface densities and discussed for instance in Refs. [30,34,36]. As discussed is shown in Fig. 8(a). We compare these results to the with the perturbative approach in Sec. III, the Lorentz-Lorenz

053629-7 L. CORMAN et al. PHYSICAL REVIEW A 96, 053629 (2017) contribution is still present but it is (over)compensated by (a)3.0 (b) 1.5 multiple scattering effects for a set of fixed scatterers. In hot vapors, where the Doppler effect is large, the contribution 1.0 of multiple scattering vanishes and thus the Lorentz-Lorenz 1.0 contribution alone is observed. The related cooperative Lamb 0.3 0.8 shift has been recently demonstrated in hot vapor of atoms D D confined in a thin slab in Ref. [31]. In the cold regime where scatterers are fixed, such effects are not expected 0.1 [39]. However, in these recent studies with dense and cold 0.5 samples a small redshift is still observed [30,34,36]. This 2 3 4 5 6 1 2 3 4 5 6 difference on the sign of the frequency shift with respect to Δν/Γ Δν/Γ the results obtained in this work may be explained by residual 0 0 inhomogeneous broadening induced by the finite temperature (c) 0.0 or the diluteness of the sample in Ref. [30] and by the specific geometry in Ref. [34], where the size of the atomic cloud is −0.5 comparable to λ and where diffraction effects may play an important role. As discussed in Sec. III, our observation of a −1.0

blueshift is a general result which applies to the infinite slab. exp It is robust to a wide range of thicknesses and density, and η −1.5 while we computed it theoretically for a two-level system, it also shows up experimentally in a more complex atomic −2.0 level structure. It was also predicted in Ref. [39]butfora −2.5 uniform distribution along the z axis instead of the Gaussian 100 101 102 profile considered in this work, and also discussed in [44]. kΔz Consequently, we believe that it is an important and generic feature of light scattering in a extended cloud of fixed randomly FIG. 9. Non-Lorentzian wings of the resonance line. (a) Two distributed scatterers. examples of the scaling of optical depth with ν (blue side), in Finally, we compare the line shape of the resonance with log-log scale, for kz = 2.4(6) (circles) and for kz = 350(90) the Lorentzian shape expected for a single atom. We measure (squares) and their power-law fit. (b) Coupled dipole simulations at − −2 = for nk 2 = 1.5(2), the optical depth at large detunings, and for zero thickness and for nk 1.5(2). The optical depth is plotted as a various cloud thicknesses. We fit it with a power law on the function of detuning (minus the detuning) from the resonance line, for red-detuned (blue-detuned) frequency interval with exponent the blue (circles) [red (squares)] side. The solid lines are power-law −2 = η (η ) as shown, for two examples, in Fig. 9(a). If the behavior fits. (c) Experimental results for nk 1.5(2). Circles (squares) r b represent the fitted exponents η (η ) to the far-detuned regions of the were indeed Lorentzian, the exponents should be −2inthe r b resonance line on the red and blue side, respectively. The fit function limit of large detuning. As seen in Fig. 4(c), for the thinnest is ν → D(ν) = A (ν − ν )η. The error on the fitted exponents gases, the fitted exponents are significantly different from the 0 − is also determined using a bootstrap analysis. The horizontal dashed expected value and can reach values up to 1.3, showing the black line (η =−2) emphasizes the expected asymptotic value for strong influence of dipole-dipole interactions in our system. low densities for a Lorentzian line (at large detunings). We show the result of coupled dipole simulations for kz = 0 in Fig. 4(b) along with their power-law fit. We extract the exponents ηr =−0.36(1) and ηb =−0.70(1) that are reported strong light-induced dipole-dipole interactions, (ii) a too large as solid lines in Fig. 4(c). Our experimental results interpolate intensity used in the experiment which goes beyond the validity between the single-atom case and the simulated 2D situation. of the coupled dipole approach, and (iii) the influence of the complex atomic level structure. We were careful in this work to limit the influence of the two first explanations V. CONCLUSION and the last possibility is likely to be the main limitation. In summary we have studied the transmission of a macro- The complex level structure leads to optical pumping effects scopic dense slab of atoms with uniform in-plane density and a during the probing and thus the scattering cross section of transverse Gaussian density distribution. We observed a strong the sample is not well defined. A simple way to take into reduction of the maximum optical density and a broadening of account the level structure is, as discussed in Sec. IV,to the resonance line. More surprisingly, we showed the presence renormalize the scattering cross section by the average of the of a large blueshift of the resonance line and a deviation Clebsch-Gordan coefficients involved in the process. For 87Rb from Lorentzian behavior in the wings of the resonance line. atoms this amounts to the factor 7/15 already discussed earlier. These results are qualitatively confirmed by coupled dipole However, this is a crude approximation which neglects optical simulations and a perturbative approach of this scattering pumping effects during scattering and whose validity in the problem. We also confirm the difficulty already observed dense regime is not clear. Two approaches can be considered to obtain a quantitative agreement between coupled dipole to remove this limitation. First, one can use another atomic simulations and experimental results in the dense regime species such as strontium or ytterbium bosonic isotopes which [32,34]. Possible explanations for this discrepancy are (i) have a spin singlet ground state and in which almost exact residual motion of the atoms during the probing due to the two-level systems are available for some optical transitions.

053629-8 TRANSMISSION OF NEAR-RESONANT LIGHT THROUGH A . . . PHYSICAL REVIEW A 96, 053629 (2017)

Scattering experiments on strontium clouds have been reported 1.0 2.0 [25,30,49] but they did not explore the dense regime tackled in 0.8 this work. The comparison with theory thus relies on modeling 1.5 their inhomogeneous density distribution accurately. Second, 0.6 an effective two-level system can be created in the widely used 2 1.0 alkali-metal atoms by imposing a strong magnetic field which D |R| 0.4 could separate the different transitions by several times the 0.5 natural linewidth as demonstrated in some recent experiments 0.2 on three-level systems [50,51]. This method could be in principle applied on our setup to create an effective two-level 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.01.2 system and could help one to understand the aforementioned −2 discrepancies. nk Finally, we note that this article focuses on the steady-state transmission of a cloud illuminated by a uniform monochro- FIG. 10. Intensity reflection coefficient as a function of surface density for kz = 0 (solid line). For comparison we show the matic beam. The slab geometry that we have developed corresponding optical depth D (dotted line, right axis) and the lower here is of great interest for comparison between theory and bound for the reflection coefficient deduced from this optical depth experiments and our work opens interesting perspectives for (dashed line). extending this study to time-resolved experiments, to fluorescence measurements, or to spatially resolved propagation of light studies. strong decrease of the transmission because of dipole-dipole interactions could lead to a large reflection coefficient. For ACKNOWLEDGMENTS a strictly two-dimensional gas we show as a solid line in We thank Vitaly Kresin, Klaus Mølmer, Janne Ruostekoski, Fig. 10 the result of the coupled dipole model for the intensity |R|2 Markus Greiner, Zoran Hadzibabic, and Wilhelm Zwerger for reflection coefficient at resonance and at normal incidence fruitful discussions. This work is supported by DIM NanoK, as a function of density. This intensity reflection coefficient has ERC (Synergy UQUAM). L.C. acknowledges the support from a behavior with density similar to the optical depth D (dotted DGA. This project has received funding from the European line). The relation between these two quantities depends on the Union’s Horizon 2020 research and innovation programme relative phase between the incoming and the reflected field. under the Marie Skłodowska-Curie grant agreement No. For a transmitted field in phase with the incident field we R + T = 703926. find, using the boundary condition 1, a lower bound for this reflection coefficient, |R|2 (1 −|T |)2, shown as a dashed line in Fig. 10. The intensity reflection coefficient is APPENDIX: REFLECTION COEFFICIENT OF A 2D GAS close to this lower bound in the regime explored in this work. Thanks to their large scattering cross section at resonance, The maximum computed value for the reflection coefficient arrays of atoms can be used to emit light with a controlled spa- is close to 40%, which shows that a single disordered layer tial pattern [52]. A single-atom mirror has been demonstrated of individual atoms can significantly reflect an incoming light [53] and, more generally, regular two-dimensional arrays of beam [54]. Note that for a non-2D sample light can be diffused atoms have been considered for realizing controllable light at any angle. For our experimental thickness and the relevant absorbers [17] or mirrors [18] with atomic-sized thicknesses. densities the reflection coefficient is in practice much lower For the disordered atomic samples considered in this article the than the above prediction.

[1] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, and P. Optical Isolator Controlled by the Internal State of Cold Atoms, Thickstun, Atom-photon Interactions: Basic Processes and Phys. Rev. X 5, 041036 (2015). Applications (Wiley Online Library, New York, 1992). [7] L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, [2] J.-M. Raimond and S. Haroche, Exploring the Quantum (Oxford A. Mishchenko, Y.-J. Kim, R. V. Gorbachev, T. Georgiou, University Press, Oxford, 2006). S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, [3] P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Trapping A. H. Castro Neto, and K. S. Novoselov, Strong light-matter an atom with single photons, Nature (London) 404, 365 interactions in heterostructures of atomically thin ﬁlms, Science (2000). 340, 1311 (2013). [4] X. Ling, H. Wang, S. Huang, F. Xia, and M.S. Dresselhaus, The [8] A. Damascelli, Z. Hussain, and Z.-X. Shen, Angle-resolved renaissance of black phosphorus, Proc. Natl. Acad. Sci. U.S.A. photoemission studies of the cuprate superconductors, Rev. 112, 4523 (2015). Mod. Phys. 75, 473 (2003). [5] B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. [9] J. Braun, The theory of angle-resolved ultraviolet photoemission Laurat, Demonstration of a Memory for Tightly Guided and its applications to ordered materials, Rep. Prog. Phys. 59, Light in an Optical Nanoﬁber, Phys. Rev Lett. 114, 180503 1267 (1996). (2015). [10] T. Stauber, D. Noriega-Pérez, and J. Schliemann, Universal [6] C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. absorption of two-dimensional systems, Phys.Rev.B91, 115407 Schneeweiss, J. Volz, and A. Rauschenbeutel, Nanophotonic (2015).

053629-9 L. CORMAN et al. PHYSICAL REVIEW A 96, 053629 (2017)

[11] D. J. Merthe and V. V. Kresin, Transparency of graphene and [30] S. L. Bromley, B. Zhu, M. Bishof, X. Zhang, T. Bothwell, J. other direct-gap two-dimensional materials, Phys. Rev. B 94, Schachenmayer, T. L. Nicholson, R. Kaiser, S. F. Yelin, M. D. 205439 (2016). Lukin, A. M. Rey, and J. Ye, Collective atomic scattering and [12] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. motional effects in a dense coherent medium, Nat. Commun. 7 Zhang, S. V.Dubonos, I. V.Grigorieva, and A. A. Firsov, Electric 11039 (2016). field effect in atomically thin carbon films, Science 306, 666 [31] J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D. Sarkisyan, (2004). and C. S. Adams, Cooperative Lamb Shift in an Atomic Vapor [13] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Layer of Nanometer Thickness, Phys. Rev. Lett. 108, 173601 Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, Fine structure (2012). constant defines visual transparency of graphene, Science 320, [32] J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sortais, 1308 (2008). A. Browaeys, S. D. Jenkins, and J. Ruostekoski, Observation [14] K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and of Suppression of Light Scattering Induced by Dipole-Dipole T. F. Heinz, Measurement of the Optical Conductivity of Interactions in a Cold-Atom Ensemble, Phys. Rev. Lett. 113, Graphene, Phys. Rev. Lett. 101, 196405 (2008). 133602 (2014). [15] H. Fang, H. A. Bechtel, E. Plis, M. C. Martin, S. Krishna, E. [33] S. D. Jenkins, J. Ruostekoski, J. Javanainen, R. Bourgain, S. Yablonovitch, and A. Javey, Quantum of optical absorption in Jennewein, Y.R. P.Sortais, and A. Browaeys, Optical Resonance two-dimensional semiconductors, Proc. Natl. Acad. Sci. U.S.A. Shifts in the Fluorescence of Thermal and Cold Atomic Gases, 110, 11688 (2013). Phys. Rev. Lett. 116, 183601 (2016). [16] G. Eda and S. A. Maier, Two-dimensional crystals: managing [34] S. Jennewein, M. Besbes, N. J. Schilder, S. D. Jenkins, C. light for optoelectronics, Acs Nano 7, 5660 (2013). Sauvan, J. Ruostekoski, J.-J. Greffet, Y. R. P. Sortais, and A. [17] R. J. Bettles, S. A. Gardiner, and C. S. Adams, Enhanced Optical Browaeys, Coherent Scattering of Near-Resonant Light by a Cross Section Via Collective Coupling of Atomic Dipoles in a Dense Microscopic Cold Atomic Cloud, Phys.Rev.Lett.116, 2D Array, Phys.Rev.Lett.116, 103602 (2016). 233601 (2016). [18] E. Shahmoon, D. S. Wild, M. D. Lukin, and S. F. Yelin, Coop- [35] S. D. Jenkins, J. Ruostekoski, J. Javanainen, S. Jennewein, R. erative Resonances in Light Scattering from Two-Dimensional Bourgain, J. Pellegrino, Y. R. P. Sortais, and A. Browaeys, Atomic Arrays, Phys.Rev.Lett.118, 113601 (2017). Collective resonance fluorescence in small and dense atom [19] L. Chomaz, L. Corman, T. Yefsah, R. Desbuquois, and J. clouds: Comparison between theory and experiment, Phys. Rev. Dalibard, Absorption imaging of a quasi-two-dimensional gas: A 94, 023842 (2016). a multiple scattering analysis, New J. Phys. 14, 055001 (2012). [36] J. Javanainen, J. Ruostekoski, Y. Li, and S.-M. Yoo, Shifts of [20] T. Yefsah, R. Desbuquois, L. Chomaz, K. J. Günter, and J. Dal- a Resonance Line in a Dense Atomic Sample, Phys.Rev.Lett. ibard, Exploring the Thermodynamics of a Two-Dimensional 112, 113603 (2014). Bose Gas, Phys.Rev.Lett.107, 130401 (2011). [37] J.-L. Ville, T. Bienaimé, R. Saint-Jalm, L. Corman, M. Aidels- [21] T. Bienaimé, S. Bux, E. Lucioni, P. W. Courteille, N. Piovella, burger, L. Chomaz, K. Kleinlein, D. Perconte, S. Nascimbène, J. and R. Kaiser, Observation of a Cooperative Radiation Force in Dalibard, and J. Beugnon, Loading and compression of a single the Presence of Disorder, Phys.Rev.Lett.104, 183602 (2010). two-dimensional Bose gas in an optical accordion, Phys. Rev. A [22]S.Balik,A.L.Win,M.D.Havey,I.M.Sokolov,andD.V. 95, 013632 (2017). Kupriyanov, Near-resonance light scattering from a high-density [38] The 2D phase-space density of the cloud is below the transverse ultracold atomic 87Rb gas, Phys.Rev.A87, 053817 (2013). condensation threshold as defined in [55]. [23] Z. Meir, O. Schwartz, E. Shahmoon, D. Oron, and R. Ozeri, [39] J. Javanainen, J. Ruostekoski, Y. Li, and S.-M. Yoo, Exact Cooperative Lamb Shift in a Mesoscopic Atomic Array, Phys. electrodynamics versus standard optics for a slab of cold dense Rev. Lett. 113, 193002 (2014). gas, Phys.Rev.A96, 033835 (2017). [24] K. Kemp, S. J. Roof, M. D. Havey, I. M. Sokolov, and D. V. [40] O. Morice, Y. Castin, and J. Dalibard, Refractive index of a Kupriyanov, Cooperatively enhanced light transmission in cold dilute Bose gas, Phys.Rev.A51, 3896 (1995). atomic matter, arXiv:1410.2497. [41] P. C. Bons, R. de Haas, D. de Jong, A. Groot, and P. van [25] C. C. Kwong, T. Yang, M. S. Pramod, K. Pandey, D. Delande, R. der Straten, Quantum Enhancement of the Index of Refraction Pierrat, and D. Wilkowski, Cooperative Emission of a Coherent in a Bose-Einstein Condensate, Phys. Rev. Lett. 116, 173602 Superflash of Light, Phys.Rev.Lett.113, 223601 (2014). (2016). [26] A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. [42] P. W. Milonni and P. L. Knight, Retardation in the resonant Painter, and H. J. Kimble, Superradiance for Atoms Trapped interaction of two identical atoms, Phys.Rev.A10, 1096 (1974). Along a Photonic Crystal Waveguide, Phys.Rev.Lett.115, [43] J. Ruostekoski and J. Javanainen, Lorentz-Lorenz shift in a Bose- 063601 (2015). Einstein condensate, Phys.Rev.A56, 2056 (1997). [27] M. O. Araújo, I. Krešic,´ R. Kaiser, and W. Guerin, Superradiance [44] N. Cherroret, D. Delande, and B. A. van Tiggelen, Induced in a Large and Dilute Cloud of Cold Atoms in the Linear-Optics dipole-dipole interactions in light diffusion from point dipoles, Regime, Phys. Rev. Lett. 117, 073002 (2016). Phys. Rev. A 94, 012702 (2016). [28] S. J. Roof, K. J. Kemp, M. D. Havey, and I. M. Sokolov, Ob- [45] Except for Fig. 9(b) for which simulations are performed for a servation of Single-Photon Superradiance and the Cooperative fixed atom number of 2000. Lamb Shift in an Extended Sample of Cold Atoms, Phys. Rev. [46] We guess that this is due to the limited atom number used in Lett. 117, 073003 (2016). the simulation which prevents one from investigating a regime [29] W. Guerin, M. O. Araújo, and R. Kaiser, Subradiance in a Large where the geometry of the simulated cloud has an aspect ratio Cloud of Cold Atoms, Phys.Rev.Lett.116, 083601 (2016). similar to the experimental one.

053629-10 TRANSMISSION OF NEAR-RESONANT LIGHT THROUGH A . . . PHYSICAL REVIEW A 96, 053629 (2017)

[47] The observed saturation is not due to our detection procedure. [51] D. J. Whiting, J. Keaveney, C. S. Adams, and I. G. Hughes, In the fitting procedure, we only select points with a measured Direct measurement of excited-state dipole matrix elements optical depth below 3 to avoid any bias. using electromagnetically induced transparency in the hyperfine [48] For thick clouds the expected optical depth of the cloud becomes Paschen-Back regime, Phys. Rev. A 93, 043854 (2016). larger than the maximal value we can detect. We then only fit the [52] S. D. Jenkins and J. Ruostekoski, Controlled manipulation of points with a measured optical depth below 3. From this fit we light by cooperative response of atoms in an optical lattice, cannot estimate reliably the value of the maximum and width of Phys. Rev. A 86, 031602 (2012). the resonance line, but we can get an estimate of the “center” of [53] G. Hétet, L. Slodicka,ˇ M. Hennrich, and R. Blatt, Single Atom the line which may deviate from the position of the maximum as a Mirror of an Optical Cavity, Phys. Rev. Lett. 107, 133002 of the resonance for an asymmetric line shape. (2011). [49] Y. Bidel, B. Klappauf, J. C. Bernard, D. Delande, G. Labeyrie, [54] We have investigated here only the behavior at normal incidence. C. Miniatura, D. Wilkowski, and R. Kaiser, Coherent Light A full characterization of such an atomic mirror is beyond the Transport in a Cold Strontium Cloud, Phys.Rev.Lett.88, scope of this work . 203902 (2002). [55] L. Chomaz, L. Corman, T. Bienaimé, R. Desbuquois, C. [50] D. J. Whiting, E. Bimbard, J. Keaveney, M. A. Zentile, Weitenberg, S. Nascimbene, J. Beugnon, and J. Dalibard, C. S. Adams, and I. G. Hughes, Electromagnetically induced Emergence of coherence via transverse condensation in a absorption in a nondegenerate three-level ladder system, Opt. uniform quasi-two-dimensional Bose gas, Nat. Commun. 6, Lett. 40, 4289 (2015). 6162 (2015).

053629-11