Homogenization of an ensemble of interacting resonant scatterers

TL;DR

This paper investigates the limits of optical homogenization in dense atomic ensembles, revealing that collective interactions prevent homogenization near resonance unless spatial correlations or non-radiative losses are introduced.

Contribution

It demonstrates that the standard density criterion for homogenization is insufficient near resonance and highlights the role of collective modes and correlations in achieving homogenization.

Findings

Homogenization criteria are not fulfilled near resonance despite high density.

Light-induced dipole-dipole interactions lead to collective scattering modes.

Spatial correlations or non-radiative losses enable homogenization in dense atomic ensembles.

Abstract

We study theoretically the concept of homogenization in optics using an ensemble of randomly distributed resonant stationary atoms with density $\rho$. The ensemble is dense enough for the usual condition for homogenization, viz. $\rho\lambda^3 \gg 1$, to be reached. Introducing the coherent and incoherent scattered powers, we define two criteria to define the homogenization regime. We find that when the excitation field is tuned in a broad frequency range around the resonance, none of the criteria for homogenization is fulfilled, meaning that the condition $\rho\lambda^3\gg 1$ is not sufficient to characterize the homogenized regime around the atomic resonance. We interpret these results as a consequence of the light-induced dipole-dipole interactions between the atoms, which implies a description of scattering in terms of collective modes rather than as a sequence of individual scattering events. Finally, we show that, although homogenization can never be reached for a dense ensemble of randomly positioned laser-cooled atoms around resonance, it becomes possible if one introduces spatial correlations in the positions of the atoms or non-radiative losses, such as would be the case for organic molecules or quantum dots coupled to a phonon bath.

Equations

$$\nabla \times \nabla \times ⟨{\mathbf{E}}_{\mathrm{sc}}⟩-{ϵ}_{\mathrm{eff}}(\omega ,\mathbf{r})\frac{{\omega }^2}{c^2}⟨{\mathbf{E}}_{\mathrm{sc}}⟩=0.$$

$$\alpha (\omega )=\frac{3\pi {\Gamma }_0/k^3}{{\omega }_0-\omega -i\frac{{\Gamma }_0+{\Gamma }_{\text{nr}}}{2}},$$

$$P_{\text{coh}}=\frac{{ϵ}_{0}c}{2}{\oint }_{\Sigma }|⟨{\text{E}}_{\text{sc}}⟩{|}^{2}dS$$

$$P_{\text{incoh}}=\frac{{ϵ}_{0}c}{2}{\oint }_{\Sigma }⟨|\delta {\text{E}}_{\text{sc}}{|}^2⟩dS.$$

$$P_{\mathrm{abs}}^{\mathrm{macro}}$$

$$\frac{{ϵ}_{0}c}{2}{\oint }_{\Sigma }⟨|\delta {\text{E}}_{\mathrm{sc}}{|}^2⟩dS=\frac{\omega }{2}{ϵ}_0{ϵ}_{\mathrm{eff}}^{\prime \prime }(\omega ){\int }_V|⟨\mathbf{E}(\mathbf{r})⟩{|}^{2}dV.$$

$$P_{\mathrm{coh}}$$

$$|{\mathbf{E}}_{\mathrm{sc}}(\mathbf{k}){|}^2=|{\mathbf{E}}_{0,\mathrm{sc}}(\mathbf{k}){|}^{2}S(\mathbf{k}-{\mathbf{k}}_{\mathrm{L}}),$$

$$S(\mathbf{q})=|\sum _{j=1}^N{e}^{i\mathbf{q}\cdot {\mathbf{r}}_j}{|}^2$$

$$⟨|{\mathbf{E}}_{\mathrm{sc}}(\mathbf{k}){|}^2⟩\propto N(1-⟨e^{i\mathbf{q}\cdot (\mathbf{r}-{\mathbf{r}}^{\prime })}⟩)+N^2⟨e^{i\mathbf{q}\cdot (\mathbf{r}-{\mathbf{r}}^{\prime })}⟩.$$

$$⟨e^{i\mathbf{q}\cdot (\mathbf{r}-{\mathbf{r}}^{\prime })}⟩={\int }_V{d}^3\mathbf{r}{d}^3{\mathbf{r}}^{\prime }P(\mathbf{r},{\mathbf{r}}^{\prime })e^{i\mathbf{q}\cdot (\mathbf{r}-{\mathbf{r}}^{\prime })},$$

$$⟨e^{i\mathbf{q}\cdot (\mathbf{r}-{\mathbf{r}}^{\prime })}⟩={\left|\frac{1}{N}{\int }_V{d}^3\mathbf{r}\rho (\mathbf{r})e^{i\mathbf{q}\cdot \mathbf{r}}\right|}^2.$$

$${1\over 2}{\rm Re}\Big{[}\oint_{\Sigma}{\bf E}\times{\bf H}^{*}\cdot d{\bf S}+\int_{V}{\bf J}\cdot{\bf E}^{*}\,dV\Big{]}=0~\ ,$$

$$\displaystyle P_{\rm ext}=-{1\over 2}{\rm Re}\Big{[}\oint_{\Sigma}({\bf E}_{\rm i}\times\langle{\bf H}_{\rm sc}^{*}\rangle+\langle{\bf E}_{\rm sc}\rangle\times{\bf H}_{\rm i}^{*})\cdot d{\bf S}\Big{]}$$

$${\int }_V⟨\mathbf{J}\cdot {\mathbf{E}}^{\ast }⟩dV={\int }_V⟨\delta \mathbf{J}\cdot \delta {\mathbf{E}}^{\ast }⟩dV+{\int }_V⟨\mathbf{J}⟩\cdot ⟨{\mathbf{E}}^{\ast }⟩dV=0.$$

$$\displaystyle P_{\rm ext}=-{1\over 2}{\rm Re}\Big{[}\oint_{\Sigma}({\bf E}_{\rm i}\times\langle{\bf H}_{\rm sc}^{*}\rangle+\langle{\bf E}_{\rm sc}\rangle\times{\bf H}_{\rm i}^{*})\cdot d{\bf S}\Big{]}$$

$$P_{\rm incoh}={1\over 2}{\rm Re}\Big{[}\int_{V}\langle{\bf J}\rangle\cdot\langle{\bf E}^{*}\rangle\,dV\Big{]}\ ,$$

Full text

I Introduction

Homogenization is the procedure by which one replaces a discrete distribution of particles by a continuous density distribution. In the framework of the electrodynamics of continuous media, the standard procedure of homogenization, described in many textbooks Jackson ; Ashcroft ; Zangwill2012 , supposes that the inter-particle distance $\rho^{-{1\over 3}}$ ($\rho$ is the spatial density) is much smaller than the characteristic length scale associated with the phenomenon under study, usually the propagation of a wave through the medium. The characteristic scale being the wavelength $\lambda$, the condition for homogenization is thus assumed to be $\rho\lambda^{3}\gg 1$. When this condition is satisfied, one derives the macroscopic properties of an ensemble of scatterers from the microscopic properties of each of them by means of an effective medium theory. In the context of optics, for example, several models exist that relate the (microscopic) atomic polarizability Frisch1968 or the dielectric constants of spherical nano-particles in a composite dielectric random medium  Maxwell1904 ; Bruggeman1935 ; Sheng1995 ; Lagendijk1996 ; Greffet2007 ; Sentenac2005 to the (macroscopic or effective) dielectric constant of the system.

In order to derive criteria for homogenization in optics, one usually decomposes the electric field scattered by an ensemble of scatterers (e.g. atoms) into coherent and incoherent (or diffuse) components, $\langle\textbf{E}_{\rm sc}\rangle$ and $\delta{\bf E}_{\rm sc}$ respectively: $\textbf{E}_{\rm sc}=\langle\textbf{E}_{\rm sc}\rangle+\delta\textbf{E}_{\rm sc}$, where $\langle\delta\textbf{E}_{\rm sc}\rangle=0$. Here $\langle.\rangle$ denotes an ensemble average over many different spatial realizations. The coherent (or average) monochromatic field $\langle\textbf{E}_{\rm sc}\rangle$ follows the Helmholtz equation with an effective (i.e. ensemble-averaged) dielectric constant $\epsilon_{\rm eff}(\omega,{\bf r})$ describing the medium Frisch1968 ; Sheng1995 ; Lagendijk1996 ; Greffet2007 :

[TABLE]

Importantly one can always associate an effective dielectric constant (and therefore an effective index of refraction) to this coherent component, no matter whether the medium is homogeneous or not, and even in the presence of a strong incoherent field. The incoherent scattered field $\delta\textbf{E}_{\rm sc}$ originates from the random positions of the scatterers in the ensemble. These two components lead to the coherent and incoherent scattered powers, $P_{\rm coh}$ and $P_{\rm incoh}$. Daily life experience indicates that a gas of atoms or molecules, such as the atmosphere, scatters light efficiently away from the direction of the incoming light beam. However, most of the scattered power is coherent and in the direction of the incoming beam. This suggests a first, and weak criterion for homogenization: $P_{\rm incoh}/P_{\rm coh}\rightarrow 0$ when the density of scatterers increases. In this case, the question is therefore whether the effective dielectric constant is enough to describe accurately the propagation of light in the ensemble of scatterers, since coherent light scattering dominates. A second, stronger criterion comes from the observation that pure water or an amorphous glass, which are dense materials with $\rho\lambda^{3}\gg 1$, do not scatter light: there is therefore no incoherent scattering ($P_{\rm incoh}=0$) in this homogenized situation. From these examples, it would seem that the condition $\rho\lambda^{3}\gg 1$ leads to homogenization according to at least one of the two criteria described above.

In this work, however, we show theoretically that this condition $\rho\lambda^{3}\gg 1$ is not sufficient in the case of dense ensembles of randomly positioned resonant scatterers. Our study is motivated by recent experimental developments, which now make it possible to prepare ensembles of resonant scatterers in volumes comparable or smaller than the wavelength of an optical transition, such as ensembles of quantum dots Novotny2006 ; Belacel2013 or clouds of laser-cooled atoms  Pellegrino2014 ; Jennewein2016 . In particular we have recently measured both the incoherent Pellegrino2014 and coherent Jennewein2016 response of a wavelength-sized cold atomic cloud for which the atomic density can be varied. The main result of the present work is the finding that this ensemble of atoms submitted to a near-resonant light field can never be homogenized according to both criteria presented above. The situation is all the more striking that, as we have shown in a previous work Nick2016 , the effective index of refraction of the cloud can be as large as 2, a value even larger than for many condensed matter systems for which homogenization has been extensively proven to work. We show that the origin of this feature lies in the light-induced dipolar interactions between the atoms that are very strong when the light is tuned near-resonance. They lead to a collective response of the medium that has to be described in terms of modes rather than in term of individual atoms.

The paper is organized as follows. We first detail the model used to calculate the coherent and incoherent scattered powers. In Sec. III, we apply our formalism to the far-off resonance case, and check that the weak criterion for homogenization applies. Section IV presents the transition from the far-off to the near-resonant case and the fact that a dense cloud of laser-cooled atoms can never be homogenized in the latter regime according to both criteria. We then give a first interpretation in terms of collective modes (Sec. V) and a second one, which considers the cloud as an effective medium described by a dielectric constant (Sec. VI). In Sec. VII, we discuss the influence of correlations in the positions of the scatterers and observe that they lead to a recovery of homogenization according to the two criteria. Finally (Sec.VIII), we show that introducing a non-radiative decay channel also leads to a recovery of homogenization.

II Description of the system

We study theoretically the scattering of a plane wave (frequency $\omega$, wavevector $k=\omega/c=2\pi/\lambda$) from a disordered, wavelength-size cloud of cold atoms, as illustrated in Fig. 1. We idealize the experimental situation of our previous works Pellegrino2014 ; Jennewein2016 by assimilating the cloud to a parallelepiped with dimensions $V=4.8\lambda_{0}\times 0.6\lambda_{0}\times 0.6\lambda_{0}$, where $\lambda_{0}=780$ nm is the resonance wavelength of the D2 transition of rubidium-87 atoms used in the experiment. The atoms are uniformly distributed (spatial density $\rho=N/V$) and modeled as point-like and identical scatterers characterized by an isotropic polarizability:

[TABLE]

with $\omega_{0}=2\pi c/\lambda_{0}$ the transition frequency, $c$ the speed of light in vacuum, $\Gamma_{0}$ and $\Gamma_{\text{nr}}$ respectively the radiative and non-radiative decay rate ($\Gamma_{0}=2\pi\times 6\text{ MHz}$ for Rb). When $\Gamma_{\text{nr}}=0$, this polarizability model corresponds to a classical $J=0\rightarrow J=1$ atom, where $J$ is the angular momentum 111Here we neglect the complex internal structure of the rubidium atom used in the experiment.. This model can also include non-radiative decay channels ($\Gamma_{\text{nr}}$), as would be necessary if we were discussing e.g., systems of quantum dots. However, unless stated differently, we assume $\Gamma_{\text{nr}}=0$, which is a good model for a cold atomic gas. Finally, the scattering cross section is given by $\sigma_{\rm sc}(\omega)=k^{4}|\alpha(\omega)|^{2}/(6\pi)$ Jackson .

As we discuss dense atomic systems, i.e. $\rho/k^{3}\gtrsim 1$, we include the resonant dipole-dipole interactions between the atoms. As explained, e.g. in Refs. Pellegrino2014 ; Jennewein2016 ; Nick2016 , the dipole of atom $j$ is driven by the laser field and the field radiated by all the other atoms. This approach leads to a set of coupled dipole equations, in steady-state: ${\bf p}_{j}=\epsilon_{0}\alpha({\bf E}_{{\rm L}j}+\mu_{0}\omega^{2}\sum_{i\neq j}[{\bf G({\bf r}_{i}-{\bf r}_{j})}]{\bf p}_{i})$, with ${\bf E}_{{\rm L}j}$ the field of the laser at the position of atom $j$. Here, the Green’s tensor $[{\bf G({\bf r}_{i}-{\bf r}_{j})}]$ describes the resonant dipole-dipole interactions between atoms $i$ and $j$ including the $1/r$, $1/r^{2}$ and $1/r^{3}$ terms. After solving the set of equations to get each dipole moment, we calculate the scattered electric field $\textbf{E}_{\text{sc}}({\bf r})=\mu_{0}\omega^{2}\sum_{i}[{\bf G({\bf r}-{\bf r}_{i})}]{\bf p}_{i}$ (for more details, see  Pellegrino2014 ; Jennewein2016 ; Nick2016 ).

After few hundreds of realizations for which the atomic positions are changed according to a uniform probability distribution, we calculate the scattered field $\textbf{E}_{\rm sc}$, the coherent, ensemble-average field $\langle\textbf{E}_{\rm sc}\rangle$ and the incoherent, fluctuating field $\delta\textbf{E}_{\rm sc}=\textbf{E}_{\rm sc}-\langle\textbf{E}_{\rm sc}\rangle$. The ensemble-averaged scattering pattern is then decomposed in its coherent and incoherent parts: $\langle|\textbf{E}_{\text{sc}}|^{2}\rangle=|\langle\textbf{E}_{\text{sc}}\rangle|^{2}+\langle|\delta\textbf{E}_{\text{sc}}|^{2}\rangle$. The coherent scattering pattern corresponds to the diffraction pattern of a homogeneous object described by an effective dielectric constant and the incoherent scattering pattern is a quasi-isotropic speckle originating from the random positions of the atoms in the cloud (see Fig. 1).

In order to characterize quantitatively the level of homogenization, we define the integrated scattered powers corresponding respectively to the ensemble-averaged fields $|\langle\textbf{E}_{\text{sc}}\rangle|^{2}$ and $\langle|\delta\textbf{E}_{\text{sc}}|^{2}\rangle$ evaluated on a spherical surface $\Sigma$ in the far field:

[TABLE]

and

[TABLE]

The total scattered power is then $P_{\text{tot}}=P_{\text{coh}}+P_{\text{incoh}}$. It does not include the incident field.

III Far-off resonance light scattering

Figure 2 presents the coherent and incoherent scattered powers as a function of the number of atoms $N$ inside the fixed volume of the cloud, normalized by the scattered power of a single atom, for different detunings $\Delta=\omega-\omega_{0}$ of the incoming plane wave with respect to the atomic transition. This figure allows us to explore the transition between far-off to near-resonance scattering.

When the laser is very far-off resonance ($\Delta=-10^{4}\Gamma_{0}$), we observe the scalings $P_{\text{incoh}}\propto N$, and $P_{\text{coh}}\propto N^{2}$. As reminded in Appendix A, this result is expected as the wavelength-size atomic cloud is in the single scattering regime Berkeley : the mean free path $\ell_{\text{sc}}=1/[\rho\sigma_{\text{sc}}(\omega)]=3\text{ m}$ for $N=450$ atoms is much larger than the size of the atomic cloud. In this regime of large detuning, the weak criterion for homogenization applies, as $P_{\rm incoh}/P_{\rm coh}\propto 1/N\rightarrow 0$ when $N$ increases. We note from Fig. 2 that the cloud enters the homogenization regime for $N\gtrsim 20$, i.e., $\rho\lambda^{3}>1$.

The fact that, in the single scattering limit, $P_{\rm incoh}\propto N$ is extensively used to calibrate the number of atoms in a cloud of e.g. cold atoms in experiments on laser cooling or quantum degenerate gases Ketterle_Varenna_1998 . It is also common in these experiments to measure the index of refraction of the atomic sample by measuring the coherent optical response of the cloud using e.g. absorption or phase contrast imaging. This again emphasizes the fact that one can always define an index of refraction (or a dielectric constant) to characterize the coherent response of the cloud, even in the presence of incoherent scattering.

IV From far-off to near-resonance scattering: failure of homogenization

Coming back to Fig. 2, we observe that when the detuning gets closer to resonance, the scaling laws for both the coherent and incoherent scattering as a function of the atom number are strongly modified with respect to the far-off resonance case.

When $\Delta=-500\Gamma_{0}$, the coherent power still follows $P_{\rm coh}\propto N^{2}$. However, and quite unexpectedly, the incoherent power also features the same $N^{2}$ dependence for $N\gtrsim 30$, despite the fact that the cloud still operates in the single-scattering limit ($\ell_{\text{sc}}=3$ mm for $N=1000$). When the detuning is close to resonance ($\Delta=-5\Gamma_{0}$), the coherent and incoherent powers are nearly identical for $N\gtrsim 30$, and saturate when $N\gtrsim 100$. Finally, at resonance ($\Delta=0$), in stark contrast with the off-resonance scattering case, the two powers become rapidly independent of $N$ and saturate to the approximately same value when the number of atoms increases. This saturation of both the incoherent and coherent scattered powers was actually observed in our recent experimental works Pellegrino2014 ; Jennewein2016 , although there we could only measure a fraction of the powers in a given solid angle. As a consequence, as far as the homogenization of the ensemble of atoms is concerned, neither the weak ($P_{\rm incoh}/P_{\rm coh}\rightarrow 0$) nor the strong criterion ($P_{\rm incoh}=0$) apply for detunings in the range $|\Delta/\Gamma_{0}|\lesssim 500$, although the condition $\rho\lambda^{3}\gg 1$ is fulfilled. It thus appears that a dense cloud of cold atoms can never be homogenized in a broad frequency range around the resonance! In other words: a dense cloud of cold atoms keeps scattering a lot of incoherent light, and homogenization is reached very far from resonance, and only according to the weak criterion.

Before we give a consistent interpretation of this fact in the next sections, we further explore the resonant case. Figure 1(b) shows the ensemble-averaged scattering pattern of the cloud. We observe that the amplitude of the speckle is very low in the forward direction compared to the coherent scattering. In other directions, incoherent scattering dominates. From the figure, the coherent scattered power seems to dominate the total power. However as it is scattered in a limited solid angle as opposed to the incoherent power, after integration over all directions it turns out that the coherent and incoherent powers have similar values in our case (see Fig. 2).

Finally, we calculate the powers scattered coherently and incoherently as a function of the detuning of the laser, near the atomic resonance. We plot in Fig. 3 four curves corresponding to (1) the ensemble-averaged coherent scattered light, (2) the ensemble-averaged incoherent scattered light, (3) the sum of the ensemble-averaged coherent and incoherent scattered light, and (4) the scattered light for a single realization of the distribution of atoms in the cloud. Firstly, we observe that the line shapes are significantly different from a Lorentzian, contrarily to what it would be in the single scattering regime. Secondly, the coherent and incoherent powers exhibit similar shapes, in particular a double structure with a peak for a negative detuning.

All the features presented in this section indicate that the interpretation of scattering as a sequence of individual scattering events breaks down in a broad frequency range around resonance, despite the fact that for all the atom numbers used in this work, the mean-free path $\ell_{\text{sc}}$ is always at least 10 times larger than the cloud largest size, even on resonance. We would otherwise observe a lorentzian line shape only. This emphasizes that the definition of the mean free path by $\ell_{\text{sc}}=1/[\rho\sigma_{\text{sc}}(\omega)]$ is not appropriate to describe the scattering of light in our situation. Instead the correct length scale is associated with the decay of the field in the medium and is given by $1/(n^{\prime\prime}k)$, with $n^{\prime\prime}$ the imaginary part of the effective refractive index. As $n^{\prime\prime}$ reaches values as high as 2 Nick2016 , the length scale is 100 nm, smaller than the size of the cloud.

In the next two sections we interpret the above observations using two different, but complementary points of view.

V Interpretation in terms of collective modes

As discussed by many authors (for recent works, see e.g. Ruostekoski1997 ; Scully2010 ; Chomaz2012 ; Kupriyanov2013 ; Zhu2013 ; Kaiser2014 ; Bettles2015 ; Bettles2016 ; Nick2016 ), the coupling between atoms resulting from the light-induced dipole-dipole interaction leads to a set of collective modes $\beta$ of the atomic dipoles. These modes are eigenstates of the set of coupled dipole equations. Each of these $3N$, non-orthogonal modes has its own (complex) eigenfrequency $\tilde{\omega}_{\beta}=\omega_{0}+\Omega_{\beta}-i\frac{\Gamma_{\beta}}{2}$, with $\Omega_{\beta}$ the collective shift and $\Gamma_{\beta}$ the collective decay rate. Some of these modes, featuring $\Gamma_{\beta}<\Gamma_{0}$, are sub-radiant, while others with $\Gamma_{\beta}>\Gamma_{0}$ are super-radiant. In Ref. Nick2016 , we studied in detail the modes corresponding to the situation analyzed here and depicted in Fig. 1. We found that the modes fall in three categories (see Fig. 2 of Ref. Nick2016 ). The first one consists of modes with collective linewidth $\Gamma_{\beta}\approx 2\Gamma_{0}$ (super-radiant) and $\Gamma_{\beta}\ll\Gamma_{0}$ (sub-radiant). These so-called dimer modes are made of pairs of atoms and have large collective frequency shifts $\Omega_{\beta}\sim{\Gamma_{0}/(kr)^{3}}$, with $r$ the interatomic distance. The second category consists of few polaritonic modes that have four key features: (i) all atoms have significant contributions to the modes (ii) they are robust against disorder (i.e., they depend on density and geometry but not on the precise positions of the scatterers), (iii) they are super-radiant with $\Gamma_{\beta}\gtrsim 10\Gamma_{0}$ and (iv) although representing less than 1% of the total number of modes, we calculated that they contribute for a large ($>50\%$) fraction of the coherent scattering of the cloud. Finally, the last category consists of modes with the excitation delocalized over many atoms, but with no regular spatial structure. They can be super- or sub-radiant and their frequency shift is on the order of a few $\Gamma_{0}$.

We can now interpret the transition between far-off to near-resonance scattering observed in Fig. 2 in terms of collective modes. First, when the incoming light is very far-detuned, with $|\Delta|$ larger than the largest shift corresponding to the pair of closest atoms in the cloud (see Fig. 2 of Ref. Nick2016 ), the light cannot excite any mode. In this regime, the cloud scatters the light as a collection of independent atoms, with the usual scaling laws $P_{\text{incoh}}\propto N$ and $P_{\text{coh}}\propto N^{2}$.

When the detuning becomes small enough that dimer modes are excited (see Fig. 2 of Ref. Nick2016 ), the incoherent scattering is dominated by the scattering from the super-radiant pairs of atoms, which is essentially isotropic for atoms closer than $1/k$. The incoherent scattered power is therefore proportional to the number of excited pairs, hence $\propto N^{2}$. As a consequence, a description in terms of dimer modes naturally leads to a $N^{2}$ scaling for the incoherent scattered power. A semi-classical model, presented in Appendix B, predicts a critical number of atoms $N_{\rm c}\approx 25$ where the transition between scattering by individual atoms to scattering by dimers occurs and is in good agreement with the value observed in Fig. 2. As for the coherent power, we show in Appendix B that it is still dominated by the coherent scattering from individual atoms, with a negligible contribution from the dimers. Therefore, the coherent power varies as $N^{2}$ as in the far-off resonance case (see Sec. III and Appendix A).

Finally, when the detuning becomes very close to resonance, many delocalized (non-polaritonic) modes are excited. Incoherent scattering comes from many of these modes, which have no regular spatial structure. In our intermediate regime $V\sim\lambda^{3}$, we could not derive any simple scaling law for the incoherent scattering as a function of the atom number. However, qualitatively, the saturation of the incoherent power observed in Fig. 2 indicates that, when the number of atoms increases beyond $\rho/k^{3}\gtrsim 1$, the product of the number of modes excited in a bandwidth of the order of $\Gamma_{0}$ by the power they radiate has to become independent of the number of atoms. As for the coherent scattered power, few polaritonic modes have important contributions. Their collective shifts and widths are mainly set by the geometry of the cloud and hardly depend on the number of atoms Nick2016 , leading to a saturation of the coherent power. Finally, we have checked that the saturation of the coherent and incoherent powers at comparable values (see Fig. 2) is a consequence of the particular dimensions of cloud.

VI Interpretation in terms of effective medium

In this second approach, we forget about the discrete atomic distribution and consider instead the cloud as an effective homogeneous medium with a dielectric constant $\epsilon_{\rm eff}(\omega)=\epsilon_{\rm eff}^{\prime}(\omega)+i\epsilon_{\rm eff}^{\prime\prime}(\omega)$. In this macroscopic description the absorbed power is given by the imaginary part of the dielectric constant and the macroscopic, ensemble-averaged field $\langle{\bf E}({\bf r})\rangle$ inside the medium Jackson :

[TABLE]

with $\langle{\bf J}\rangle=-i\omega\epsilon_{0}[\epsilon_{\rm eff}(\omega)-1]\langle{\bf E}\rangle$. A detailed balance derived in Appendix C shows that the total power $P_{\rm ext}$ taken from the incident field is the sum of the power absorbed by the cloud and of the scattered power $P_{\rm sc}^{\rm macro}$, i.e., $P_{\rm ext}=P_{\rm abs}^{\rm macro}+P_{\rm sc}^{\rm macro}$. Now, in a microscopic description, this same power $P_{\rm ext}$ is also the sum of the coherent and incoherent scattered powers, defined after ensemble averaging by Eq. (3) and (4): $P_{\rm ext}=P_{\text{coh}}+P_{\text{incoh}}$ 222We have assumed here that the atoms are elastic scatterers that do not absorb light as e.g. particles in China ink would.. Finally, the coherent scattered power defined by Eq. (3) is the power scattered in the macroscopic approach, $P_{\text{coh}}=P_{\rm sc}^{\text{macro}}$, leading to the identification of the absorbed power $P_{\rm abs}^{\rm macro}$ with the power $P_{\rm incoh}$ incoherently scattered by the cloud, i.e.:

[TABLE]

As a consequence, the homogenization criteria $P_{\rm incoh}=0$ requires that the imaginary part of the dielectric constant is negligible 333Care must be taken that the fact that we can describe the incoherent power by the imaginary part of the effective (i.e. ensemble-averaged) dielectric constant does not imply that the incoherent field itself is described by an effective dielectric constant..

We can now discuss the behavior of the coherent and incoherent scattered powers as a function of the detuning shown in Fig. 3. As we studied in Nick2016 , once we know the effective dielectric constant $\epsilon_{\rm eff}(\omega)$, we can calculate the electric field inside the cloud considered as a continuous medium. We find resonances for certain values of the frequency $\omega$, precisely corresponding to the polaritonic modes. When hitting a resonance, the electric field $\langle{\bf E}\rangle$ inside the medium is large. Consequently the scattered field, and therefore the coherent power, is large as well. The variation of the coherent scattered power shown in Fig. 3 thus reflects the mode structure of the effective homogeneous particle equivalent to the cloud. We have for example checked numerically that the frequency corresponding to the maximum of the broad peak on the red side of the resonance is proportional to the length of the cloud, a signature of a shape resonance in an object of finite size.

The fact that the incoherent and coherent scattered powers show similar behavior as a function of $\omega$ is a direct consequence of Eq. (6): if $\epsilon_{\rm eff}^{\prime\prime}(\omega)$ were independent of $\omega$ the two powers would vary identically with $\omega$. As we showed in Ref. Nick2016 , $\epsilon_{\rm eff}^{\prime\prime}(\omega)$ is in fact a broad, resonant function and the incoherent scattered power combines the shape resonance of $\langle\bf E\rangle$ with the resonant behavior of $\epsilon_{\rm eff}^{\prime\prime}(\omega)$.

Finally, we can also understand why the coherent and incoherent scattered powers saturate when the number of atoms increases. For large atom numbers, the cloud behaves like a sharp object. The coherent scattering corresponds to the diffraction pattern of this object Zangwill2012 ; Bohren , and is only dependent on its shape 444As an example, the scattering cross section of an homogeneous particle with a size larger than the wavelength of the scattered light is twice the geometrical cross section perpendicular to the incoming wave direction, and is therefore independent of the number of atoms in the particle.. To understand why the incoherent power also saturates at large atom numbers, we rely on the expression of the coherent power scattered by an object of volume $V$ and dielectric constant $\epsilon_{\rm eff}(\omega)$ Zangwill2012 :

[TABLE]

with ${\bf e}_{r}$ the unit vector in a given scattering direction. If the coherent scattered power saturates for large atom numbers, as it should based on the diffraction argument above, this formula indicates that the coherent field inside the object and the dielectric constant must saturate. Therefore, according to Eq. (6) the incoherent power must saturate as well.

VII Influence of the correlations in the positions of the atoms

The impossibility to reach homogenization for an ensemble of cold atoms presented in Fig. 2 makes us wonder if there exists any resonant system, which can reach the homogenization regime. In this Section, we show that a way to make this ensemble of scatterers homogenized consists in introducing spatial correlations in the positions of the atoms in order to reduce the fluctuations of the spatial distribution Frisch1968 ; Lagendijk1996 ; Greffet2007 . We are guided here by the well-known fact that a pure liquid scatters less light than a gas, although being much denser: the presence of position correlations in the liquid inhibits incoherent scattering. Another situation is the transparency of the cornea, which is due to spatial correlations Hart1969 .

Here we implement spatial correlations by introducing a spherical exclusion volume with diameter $d$ around each scatterer 555In this calculation, the number of atoms $N$ is kept constant. If for a given distribution we find that two atoms are distant by less than $d$, we draw a new configuration with the same number of atoms.. This diameter sets the minimum distance between nearest-neighbor scatterers. Figure 4 presents the coherent and incoherent scattered powers as a function of the diameter $d$ of the exclusion volume for $N=100$ and various detunings. We observe that for $\Delta$ ranging from $-10^{4}\Gamma_{0}$ to $\sim-10\Gamma_{0}$ incoherent light scattering is reduced by introducing spatial order in the system, while the coherent scattering is weakly affected. As a consequence, this procedure leads to homogenization according to the stringent criterion $P_{\rm incoh}=0$. Qualitatively, this suppression comes from the fact that the exclusion volume thwarts the formation of dimer modes.

Closer to resonance, the behavior is more complex. For detunings in the range $-5\leq\Delta/\Gamma_{0}<0$, we observe an increase of the coherent power, while the incoherent decreases. At resonance ($\Delta=0$), coherent power still increases with $d$, while the incoherent power remains approximatively constant. These findings seem to indicate that the spatial correlations lead to at least the weak homogenization criterion, with a decrease of the ratio $P_{\rm incoh}/P_{\rm coh}$. We could not explore exclusion diameters larger than $0.2\lambda_{0}$, while maintaining the number of atoms constant 666When increasing the diameter $d$, the probability to find an exclusion volume with only one atom becomes too small. , and therefore could not check the point when the stringent criterion starts to be valid.

VIII Effect of non-radiative losses on near-resonance light scattering

In this last Section, we discuss the influence of non-radiative losses on light scattering. We know that, for example, a suspension of non-resonant absorbing particles, such as a droplet of China ink consisting of a suspension of colloidal carbon nanoparticles, does not scatter light if the absorption cross section is much larger than the scattering cross section of each particle. In this case, all the energy gets absorbed by the particles, converted into heat, and scattering can be neglected. Guided by this example of a non-resonant situation, we introduce here non-radiative losses and study if homogenization according to the strong criterion ($P_{\rm incoh}=0$) is reached in the resonant case as well. Although this procedure would not apply to a cold atomic ensemble, it would be relevant for ensembles of e.g. molecules or quantum dots coupled to phonons.

We report in Fig. 5 the coherent and incoherent powers calculated for resonant light scattering for different values of the ratio of the non-radiative loss rate $\Gamma_{\rm nr}$ relative to the radiative loss rate $\Gamma_{0}$. We observe that as the amount of non-radiative losses increases the incoherent power gets significantly reduced, whereas the coherent power is only slightly affected.

This observation can be understood by using the collective mode picture. By introducing non-radiative losses characterized by the rate $\Gamma_{\rm nr}$, the only change to the (complex) eigenfrequency of a collective eigenmode $\beta$ is to replace its value $\tilde{\omega}_{\beta}=\omega_{0}+\Omega_{\beta}-i\frac{\Gamma_{\beta}}{2}$ by $\tilde{\omega}_{\beta}-i\frac{\Gamma_{\text{nr}}}{2}$. As a consequence, the modes with radiative decay rates $\Gamma_{\beta}<\Gamma_{\rm nr}$ damp in a time $1/\Gamma_{\rm nr}$, irrespective of their radiative damping rate. Therefore, when $\Gamma_{\rm nr}>\Gamma_{0}$, the sub-radiant modes ($\Gamma_{\beta}<\Gamma_{0}$) decay non-radiatively very quickly and therefore hardly contribute to the scattering in steady state. As they are in particular responsible for the incoherent scattering, this one is suppressed. By contrast, light scattered by a polaritonic (super-radiant) mode is not affected by the non-radiative decay as long as $\Gamma_{\beta}>\Gamma_{\rm nr}>\Gamma_{0}$. These polaritonic modes lead to coherent scattering. Finally, when the radiative decay rate of the most super-radiant mode gets smaller than $\Gamma_{\rm nr}$, even the coherent scattering is suppressed. Therefore, the introduction of non-radiative losses appears as an efficient way to achieve homogenization in the sense of suppressing incoherent light scattering.

IX Conclusion

As a conclusion, we have studied theoretically the concept of homogenization in optics using an ensemble of resonant scatterers dense enough for the usual condition for homogenization, viz. $\rho\lambda^{3}\gg 1$, to be reached. We have introduced two criteria to define the homogenization regime in terms of incoherent and coherent scattered powers. When the excitation field is tuned very far from the resonance of the scatterers, we recovered the well-known scaling laws as a function of the atom number $N$, leading to a suppression of the ratio $P_{\rm incoh}/P_{\rm coh}$ at large $N$. However, when excited in a broad frequency range around the resonance, we observed that none of the criteria for homogenization apply, meaning that the condition $\rho\lambda^{3}\gg 1$ is not sufficient to characterize the homogenized regime. We interpreted this result as an effect of the dipole-dipole interactions between the atoms, which implies a description of scattering in terms of collective modes rather than as a sequence of individual scattering events. Finally, we showed that, although homogenization can never be reached for a dense ensemble of randomly positioned laser-cooled atoms, it becomes possible if one introduces spatial correlations in the positions of the atoms or non-radiative losses, such as would be the case for organic molecules or quantum dots coupled to a phonon bath.

Acknowledgements.

We thank R. Carminati for fruitful discussions. We acknowledge support from the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (Labex PALM, ANR-10-LABX-0039), and Région Ile-de-France (LISCOLEM project). N.J. S. is supported by Triangle de la Physique and Université Paris-Sud. J.-J. G. acknowledges support from Institut Universitaire de France and the SAFRAN-IOGS chair on Ultimate Photonics.

Appendix A Scattering by an ensemble of atoms in the single-scattering limit

In this Appendix, we summarize textbook arguments about scattering from ensemble of scatterers in the single-scattering limit, valid when the mean-free path $\ell_{\text{sc}}=1/[\rho\sigma_{\text{sc}}(\omega)]$ is much larger than the size of the ensemble. The power scattered at a distance $r$ in the direction ${\bf k}$ by the ensemble of $N$ atoms placed at positions ${\rm r}_{j}$ is then proportional to

[TABLE]

where ${\bf k}_{\rm L}$ is the wave vector of the incident plane wave, ${\bf E}_{0,\rm sc}({\bf k})$ the field scattered by a single atom, and

[TABLE]

the structure factor of the cloud. The ensemble-average power in the direction ${\bf k}={\bf k}_{\rm L}+{\bf q}$ is then proportional to $\langle S({\bf q})\rangle$ and thus:

[TABLE]

Here, the average phase factor s

[TABLE]

with $P({\bf r},{\bf r}^{\prime})$ the joint probability distribution to find a particle at position ${\bf r}$ and ${\bf r}^{\prime}$. When the positions are not correlated, $P({\bf r},{\bf r}^{\prime})=\rho({\bf r})\rho({\bf r}^{\prime})/N^{2}$, with $\rho({\bf r})$ the spatial density distribution, and the term $\langle e^{i{\bf q}\cdot({\bf r}-{\bf r}^{\prime})}\rangle$ is the diffraction pattern of the cloud:

[TABLE]

The $N^{2}$ term in Eq. (10) is the coherent component, and dominates in the solid angle corresponding to diffraction. In the other directions, the power is proportional to $N$ and corresponds to the incoherent scattering. It is almost isotropic.

Appendix B Contribution of the dimer modes to incoherent scattering in the intermediate detuning regime

Here we derive a semi-classical model to calculate the critical number of atoms $N_{\rm c}$ where the transition between scattering by individual atoms to scattering by dimers occurs (Sec. V). This model is inspired by the one used to calculate the rate of light-assisted collisions in cold atomic samples GallagherPritchard89 ; WeinerRMP .

As explained in Sec. V, in the intermediate detuning regime the incoherent scattering is due to the super-radiant pairs of atoms. We first note that when two atoms are separated by $r\ll 1/k$, their interaction energy is dominated by the dipolar (near-field) term: $U(r)\sim\hbar\Gamma_{0}/(kr)^{3}$. To estimate the incoherent scattered power, we start by calculating the number of excited pairs as follows: for a given detuning $|\Delta|$, the light is resonant with the excitation of a pair of atoms located at a relative distance $r_{\rm ex}$ such that $|\Delta|\sim\Gamma_{0}/(kr_{\rm ex})^{3}$. The number of excited dimers is then the product of the atom number $N$ with the number of atoms in a shell of radius $r_{\rm ex}$, $(N/V)4\pi r_{\rm ex}^{2}\Delta r$, with $V$ the cloud volume, $N$ the atom number and $\Delta r$ the thickness of the shell. The latter one is determined by writing that the collective width $2\Gamma_{0}$ of a super-radiant pair is $U^{\prime}(r_{\rm ex})\Delta r/\hbar=3\Gamma_{0}/(kr_{\rm ex})^{3}(\Delta r/r_{\rm ex})$. As a consequence, assuming that approximatively half the dimers are super-radiant, the number of super-radiant pairs is $N_{\rm p}\sim(N^{2}/2)[8\pi/(3k^{3}V)](\Gamma_{0}/\Delta)^{2}$. The power scattered by a super-radiant pair (decay rate $2\Gamma_{0}$, saturation intensity of the dimer transition $2I_{\rm sat}$, with $I_{\rm sat}$ the atomic saturation intensity) irradiated by a light with intensity $I\ll I_{\rm sat}$ is $P_{\rm dimer}=\hbar\omega_{0}(2\Gamma_{0})I/(4I_{\rm sat})$. Finally, the power scattered by a single atom irradiated by the same light detuned by $|\Delta|$ with respect to the atomic transition is $P_{0}\approx\hbar\omega_{0}\Gamma_{0}I/(2I_{\rm sat})(\Gamma_{0}/2\Delta)^{2}$. Combining the expressions above, we find the ratio of the total incoherent power scattered by the cloud to the power scattered by a single atom: $P_{\rm incoh}/P_{0}\sim N^{2}[16\pi/(3k^{3}V)]$. The critical number of atoms $N_{\rm c}$ where the transition between scattering by individual atoms to scattering by dimers occurs is thus $N_{c}\sim 3k^{3}V/(16\pi)$. The transition point is thus predicted to be independent of the detuning, in agreement with numerical calculations performed for various detunings. Also, for our parameters, $N_{c}\approx 25$, in good agreement with the value observed in Fig. 2, considering the simplicity of the model.

We now calculate the coherent power $P_{\rm coh}^{\rm dim}$ scattered by the dimers and compare it to the coherent scattering due to individual atoms $P_{\rm coh}^{\rm at}$. This latter one is given by $P_{\rm coh}^{\rm at}=N^{2}P_{0}\,\Omega/(4\pi)$, as for the intermediate detunings the cloud is still in the single scattering regime. Here, $\Omega/(4\pi)$ is the solid angle of the diffraction pattern. The power coherently scattered by the dimers is $P_{\rm coh}^{\rm dim}=N_{\rm p}^{2}P_{\rm dimer}\,\Omega/(4\pi)$ as the dimers are spread in a dilute way in the same volume than the atoms. Finally, we get $P_{\rm coh}^{\rm dim}/P_{\rm coh}^{\rm at}=[8\pi/(3k^{3}V)]^{2}N^{2}(\Gamma_{0}/\Delta)^{2}$. For $N\lesssim 500$ and $|\Delta|/\Gamma_{0}=100-500$, $P_{\rm coh}^{\rm dim}/P_{\rm coh}^{\rm at}\ll 1$, thus making the contribution of the dimers to coherent scattering negligible.

Appendix C Derivation of the detailed balance used in Sec.VI

Here we give an explicit derivation of Eq. 6. We consider the ensemble (volume $V$) of discrete atoms and a spherical surface $\Sigma$ with a radius $r\gg 1/k$. The conservation of energy in steady-state state tells us that the flux of the time-averaged Poyting vector through $\Sigma$ compensates for the time-averaged power dissipated by the current in the volume $V$:

[TABLE]

with ${\bf H}={\bf B}/\mu_{0}$ (non magnetic medium) and $\bf J$ the current density in the ensemble. This expression is valid in a microscopic model. We now decompose the fields into the incoming component, the coherent scattered component and the incoherent part: ${\bf E}={\bf E}_{\rm i}+\langle{\bf E}_{\rm sc}\rangle+\delta{\bf E}_{\rm sc}$ and ${\bf H}={\bf H}_{\rm i}+\langle{\bf H}_{\rm sc}\rangle+\delta{\bf H}_{\rm sc}$, with $\langle.\rangle$ denoting an ensemble average. We also decompose the current density into an ensemble average and a fluctuating part: ${\bf J}=\langle{\bf J}\rangle+\delta{\bf J}$. As $\oint_{\Sigma}{\bf E}_{\rm i}\times{\bf H}_{\rm i}^{*}\cdot d{\bf S}=0$, the extinction power $P_{\rm ext}$ taken from the incident field is, after expansion and ensemble average:

[TABLE]

with $\langle{\bf E}\rangle={\bf E}_{\rm i}+\langle{\bf E}_{\rm sc}\rangle$. For a cloud of cold atoms, no energy is dissipated in the medium as all incident power is re-scattered. Therefore:

[TABLE]

In this case, $P_{\rm ext}=P_{\rm coh}+P_{\rm incoh}$ using the definitions of the coherent (Eq. 3) and incoherent (Eq. 4) scattered powers.

Now, for the homogeneous medium with an effective permittivity equivalent to the ensemble of atoms, $\delta{\bf J}={\bf 0}$, $\delta{\bf E}=\delta{\bf E}_{\rm sc}={\bf 0}$ and $\delta{\bf H}_{\rm sc}={\bf 0}$, and Eq. (14) yields

[TABLE]

As P_{\rm coh}={1\over 2}{\rm Re}\Big{[}\oint_{\Sigma}\langle{\bf E}_{\rm sc}\rangle\times\langle{\bf H}_{\rm sc}^{*}\rangle\cdot d{\bf S}\Big{]}, we get by identification:

[TABLE]

or equivalently, $P_{\rm incoh}=P_{\rm abs}^{\rm macro}$ .