# Mean-Field Description of Cooperative Scattering by Atomic Clouds

Nicola Piovella  Dipartimento di Fisica ”Aldo Pontremoli”, Università degli
Studi di Milano, Via Celoria 16, I-20133 Milano, Italy

###### Abstract

We present analytic expressions for the scattering of light by an extended
atomic cloud. We obtain the solution for the mean-field excitation of
different atomic spherical distributions driven by an uniform laser, including
the initial build-up, the steady-state and the decay after the laser is
switched off. We show that the mean-field model does not describe subradiant
scattering, due to negative interference of the photons scattered by $N$
discrete atoms.

##  I Introduction

The cooperative emission from a system of $N$ two-level excited atoms has been
the object of an intense investigation is the past, starting from the
pioneering studies on Dicke superradiance [1]. On the other hand, the
diffusive regime of cooperative scattering in dense medium has been study
extensively in the past by a diagramatic approach [2], where light travels
over a distance much larger than the mean free path. More recently, optical
properties of atomic clouds beyond the single-atom level have been studied
theoretically [3, 4] and experimentally [5], where the connections between the
perturbative diffusive theory and the coupled-dipole description have been
investigated. Also, cooperative scattering from dense cold atomic clouds has
been the object of intense investigation [6]. These studies are complementary
to a different regime where the light scattering induces a dipole-dipole
interaction between the atom pairs, leading to the cooperative processes of
superradiance and subradiance. This regime is characterized to be dominated by
single-scattering of photons by many atoms, whereas to the diffusive regime is
dominated by multiple scattering. The transition between single and multiple
scattering is controlled by the optical thickness parameter
$b(\Delta)=b_{0}/(1+4\Delta^{2}/\Gamma^{2})$[7, 8], where $b_{0}$ is the
resonant optical thickness, $\Delta$ is the detuning of the laser frequency
from the atomic resonance frequency and $\Gamma$ is the transition linewidth.
In this contest, a new kind of single-photon superradiance has been proposed
by Scully and coworkers [9, 10, 11], from an extended ensemble of $N$ atoms
prepared by absorption of a single photon and exhibiting superradiant decay. A
bridge between this single-photon superradiance and the more classical process
of cooperative scattering of an incident laser by $N$ atoms [12] has been
proposed by a series of theoretical and experimental papers [13, 14, 15, 16,
17]. A more intriguing effect in such systems is subradiance, also initially
proposed by Dicke in 1954 [1], i.e. destructive interference effect leading to
the partial trapping of light in the system. This effect has been predicted
[18] and then observed [19] in a system of driven cold atoms, after that the
laser is abruptly switched off and the emitted photons detected in a given
direction. From a theoretical point of view, subradiance has been investigated
mostly studying the eigenvalues of the system and identified in the most long-
lived modes, surviving after the more fast superradiant modes have been
extinguished [20, 21, 22]. The analysis was based on the numerical solution of
the coupled-dipole model of $N$ two-level atoms driven by an uniform laser
field. A continuous-distribution version of this model allows for an
analytical treatment of the problem. This has been done extensively in a
series of papers by Svidzinsky et al. [10, 11], considering the temporal decay
of the system initially prepared in some given excited state. The stationary
problem of the system driven by an uniform laser has been studied in terms of
the collective modes in ref.[23, 24]. However, the cooperative decay after
that the laser has been switched off has not yet been studied by using the
continuous-distribution model (or the so-called mean-field (MF) model), except
in ref.[22] where however the solution has been obtained numerically.

The aim of this paper is to provide analytical expressions for the excitation
of the driven system and for the scattered light intensity. This will
encompass both the analytical work by Svidzinsky et al. [10, 11], who did not
consider a driven system, and the numerical results of ref.[22].

We outline that the MF model assumes a coherent interaction between the
scatterers, neglecting granularity and fluctuations in the atomic
distribution. These ingredients are necessary in order to describe the random
walk of the scattered photons, leading to the diffusive regime for
sufficiently dense samples [7, 8]. Hence, the MF model is not able to describe
the diffusive regime, where the atom scatters a photon many times within a
mean-free path. As already mentioned, multiple-scattering regime is
characterized by a large optical thickness $b=L/\ell_{sc}$, where $L$ is the
size of the medium, $\ell_{sc}=1/n\sigma_{sc}$ is the mean-free pass, $n$ is
the atomic density, $\sigma_{sc}=(6\pi/k^{2})/(1+4\Delta^{2}/\Gamma^{2})$ is
the scattering cross-section and $k$ is the laser wavenumber. Hence, the MF
model is valid for small optical thickness
$b(\Delta)=b_{0}/(1+4\Delta^{2}/\Gamma)$, i.e. for large resonant optical
thickness $b_{0}$ and large detuning, such that $b(\Delta)\ll 1$.

The paper is organized as follows. In sec. II we present the general MF
equations for a continuous atomic distribution. In sec. III we derive the
expression for the average quantities and the scattered light intensity and
power. The particular cases of uniform, parabolic and Gaussian radial
distribution are discussed in sec. IV and compared with the numerical solution
of the discrete model. Conclusions are summarized in sec. V.

##  II General equations

From a microscopic point of view and using a dipole approximation, our medium
is composed of an ensemble of $N$ two-level atoms with position
$\mathbf{r}_{j}$, whose atomic transition has frequency $\omega_{a}$,
linewidth $\Gamma$ and dipole $d$ (polarization effects are neglected). The
system is driven by a monochromatic plane wave with electric field $E_{0}$,
frequency $\omega_{0}$ and wave vector $\mathbf{k}_{0}$, detuned from the
atomic transition by $\Delta_{0}=\omega_{0}-\omega_{a}$. In the linear regime
and in the Markov approximation (valid if the decay time is larger than the
photon time-of-flight through the atomic cloud), the problem reduces to the
following differential equation for the atomic dipole amplitudes $\beta_{j}$
[25]:

$\displaystyle\frac{d\beta_{j}}{dt}$ $\displaystyle=$
$\displaystyle\left(i\Delta_{0}-\frac{\Gamma}{2}\right)\beta_{j}-\frac{i\Omega_{0}}{2}e^{i\mathbf{k}_{0}\cdot\mathbf{r}_{j}}-\frac{\Gamma}{2}\sum_{m\neq
j}G_{jm}\beta_{m}(t).$ (1)

where $\Omega_{0}=dE_{0}/\hbar$ is the Rabi frequency and

$G_{jm}=\frac{\exp(ik_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|)}{ik_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}=\frac{\sin(k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|)}{k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}-i\frac{\cos(k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|)}{k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}$
(2)

The kernel $G_{jm}$ describes the coupling between the dipoles, mediated by
the photons exchanged between the dipoles. It has a real component (sine
term), describing the cooperative atomic decay, and an imaginary component
(cosine term) describing the cooperative Lamb shift [26]. The latter becomes
significant when the number of atoms in a cubic optical wavelength,
$n\lambda^{3}$, is larger than unity, such that the contribution from the
virtual photons becomes relevant.

In light scattering experiments, disorder plays a role when the number of
atoms projected onto a cross section perpendicular to the incident beam is
small enough so that a light mode focused down to the diffraction limit (that
is $\lambda^{2}$) would be able to resolve and count the atoms. In other
words, the stochastic fluctuations induced by the random positions of the
atoms can be neglected when the total number of atoms $N$ is larger than the
number of modes $\sigma^{2}$ (where $\sigma=k_{0}R$ and $R$ is the transverse
size of the system) that fit into the cloud’s cross section, i.e. when the
optical density is $b_{0}=3N/\sigma^{2}\gg 1$. Under this hypothesis, the
particles can be described by a smooth density $n(\mathbf{r})$ and their
probability to be excited by a field $\beta(\mathbf{r},t)$. By approximating
the sum over $j$ by an integral over the smooth density, i.e.
$\sum_{j}\rightarrow\int d\mathbf{r}n(\mathbf{r})$, Eq.(1) turns into

$\displaystyle\frac{\partial\beta(\mathbf{r},t)}{\partial
t}=\left(i\Delta_{0}-\frac{\Gamma}{2}\right)\beta(\mathbf{r},t)-\frac{i}{2}\Omega_{0}e^{i\mathbf{k}_{0}\cdot\mathbf{r}}-\frac{\Gamma}{2}\int
d\mathbf{r}^{\prime}n(\mathbf{r}^{\prime})\frac{\exp(ik_{0}|\mathbf{r}-\mathbf{r}^{\prime}|)}{ik_{0}|\mathbf{r}-\mathbf{r}^{\prime}|}\beta(\mathbf{r}^{\prime},t).$
(3)

Using

$\frac{\exp(ik_{0}|\mathbf{r}-\mathbf{r}^{\prime}|)}{ik_{0}|\mathbf{r}-\mathbf{r}^{\prime}|}=4\pi\sum_{n=0}^{\infty}\sum_{m=-n}^{n}j_{n}(k_{0}r_{<})Y_{n,m}(\theta,\phi)Y^{*}_{n,m}(\theta^{\prime},\phi^{\prime})h^{(1)}_{n}(k_{0}r_{>})$
(4)

where $Y_{n,m}(\theta,\phi)$ are the spherical harmonics, $j_{n}(r)$ and
$h^{(1)}_{n}(r)=j_{n}(r)+iy_{n}(r)$ are the spherical Bessel and Hankel
functions of first kind, respectively, and $r_{<}$ ($r_{>}$) is the smaller
(larger) between $r$ and $r^{\prime}$. Taking $\theta$ as the polar angle with
respect the direction of the wave vector $\mathbf{k}_{0}$, we can expand

$\beta(\mathbf{r},t)=\sum_{n,m}\alpha_{n,m}(t)\beta_{n}(r)Y_{n,m}(\theta,\phi).$
(5)

By substituting it in Eq.(3) and assuming a radial distribution, $n(r)$, we
obtain

$\displaystyle\sum_{n^{\prime},m^{\prime}}\left\\{\dot{\alpha}_{n^{\prime},m^{\prime}}-\left(i\Delta_{0}-\frac{\Gamma}{2}\right)\alpha_{n^{\prime},m^{\prime}}\right\\}\beta_{n^{\prime}}(r)Y_{n^{\prime},m^{\prime}}(\theta,\phi)=-\frac{i}{2}\Omega_{0}e^{ik_{0}r\cos\theta}$
(6) $\displaystyle-$
$\displaystyle\frac{\Gamma}{2}(4\pi)\int_{0}^{\infty}dr^{\prime}r^{\prime
2}n(r^{\prime})\int
d\Omega^{\prime}\sum_{n^{\prime\prime},m^{\prime\prime}}j_{n^{\prime\prime}}(k_{0}r_{<})Y_{n^{\prime\prime},m^{\prime\prime}}(\theta,\phi)Y^{*}_{n^{\prime\prime},m^{\prime\prime}}(\theta^{\prime},\phi^{\prime})h^{(1)}_{n^{\prime\prime}}(k_{0}r_{>})$
$\displaystyle\times$
$\displaystyle\sum_{n^{\prime},m^{\prime}}\alpha_{n^{\prime},m^{\prime}}\beta_{n^{\prime}}(r^{\prime})Y_{n^{\prime},m^{\prime}}(\theta^{\prime},\phi^{\prime}),$

where $d\Omega^{\prime}=d\phi^{\prime}\sin\theta^{\prime}d\theta^{\prime}$.
Since

$\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\,Y^{*}_{n,m}(\theta,\phi)Y_{n^{\prime},m^{\prime}}(\theta,\phi)=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}$
(7)

and

$\displaystyle\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\,Y^{*}_{n,m}(\theta,\phi)e^{ik_{0}r\cos\theta}=2\delta_{m,0}\sqrt{\pi(2n+1)}i^{n}j_{n}(k_{0}r),$
(8)

multiplying Eq.(6) by $Y_{n,m}^{*}(\theta,\phi)$ and integrating over the
angles, we obtain:

$\displaystyle\left\\{\dot{\alpha}_{n,m}-\left(i\Delta_{0}-\frac{\Gamma}{2}\right)\alpha_{n,m}\right\\}\beta_{n}(r)$
$\displaystyle=$
$\displaystyle-i\Omega_{0}\delta_{m,0}\sqrt{\pi(2n+1)}i^{n}j_{n}(k_{0}r)$
$\displaystyle-$
$\displaystyle\frac{\Gamma}{2}(4\pi)\alpha_{n,m}\int_{0}^{\infty}dr^{\prime}r^{\prime
2}n(r^{\prime})j_{n}(k_{0}r_{<})h^{(1)}_{n}(k_{0}r_{>})\beta_{n}(r^{\prime}).$

If $\alpha_{n,m}(0)=0$, the only components different from zero are those for
$m=0$. So, defining $\alpha_{n}=\sqrt{(2n+1)/4\pi}\,\alpha_{n,0}$ and since
$Y_{n,0}(\theta,\phi)=\sqrt{(2n+1)/4\pi}\,P_{n}(\cos\theta)$ where $P_{n}(x)$
are the Legendre polynomial, we write

$\beta(r,\theta,t)=\sum_{n=0}^{\infty}\alpha_{n}(t)j_{n}(k_{0}r)P_{n}(\cos\theta)$
(10)

where $\alpha_{n}(t)$ is the solution of the following differential equation

$\displaystyle\left\\{\dot{\alpha}_{n}-\left(i\Delta_{0}-\frac{\Gamma}{2}\right)\alpha_{n}\right\\}j_{n}(k_{0}r)$
$\displaystyle=$
$\displaystyle-i\frac{\Omega_{0}}{2}(2n+1)i^{n}j_{n}(k_{0}r)-\frac{\Gamma}{2}F_{n}(r)\alpha_{n}.$
(11)

where

$\displaystyle F_{n}(r)$ $\displaystyle=$ $\displaystyle
4\pi\left\\{h^{(1)}_{n}(k_{0}r)\int_{0}^{r}dr^{\prime}r^{\prime
2}n(r^{\prime})j^{2}_{n}(k_{0}r^{\prime})+j_{n}(k_{0}r)\int_{r}^{\infty}dr^{\prime}r^{\prime
2}n(r^{\prime})j_{n}(k_{0}r^{\prime})h^{(1)}_{n}(k_{0}r^{\prime})\right\\}$

We observe that $F_{n}(r)$ has a real part and an imaginary part. The real
part is $\mathrm{Re}\\{F_{n}(r)\\}=\lambda_{n}j_{n}(k_{0}r)$ where

$\lambda_{n}=4\pi\int_{0}^{\infty}drr^{2}n(r)j^{2}_{n}(k_{0}r)$ (13)

is the collective decay rate of the mode $n$ and it corresponds to the
contribution of the sine term of the kernel of Eq.(3). The imaginary part is

$\mathrm{Im}\\{F_{n}(r)\\}=4\pi\left\\{y_{n}(k_{0}r)\int_{0}^{r}dr^{\prime}r^{\prime
2}n(r^{\prime})j^{2}_{n}(k_{0}r^{\prime})+j_{n}(k_{0}r)\int_{r}^{\infty}dr^{\prime}r^{\prime
2}n(r^{\prime})j_{n}(k_{0}r^{\prime})y_{n}(k_{0}r^{\prime})\right\\}$ (14)

and contributes to the cooperative Lamb shift, arising from the cosine term of
the kernel of Eq.(3). When the detuning $\Delta_{0}$ is much larger than the
collective Lamb shift, the sine-kernel provides a good approximation to the
solution.

##  III Average quantities

Using the expansion (10), we can calculate the average:

$\displaystyle\langle|\beta(t)|^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{2\pi}{N}\int_{0}^{\pi}d\theta\sin\theta\int_{0}^{\infty}r^{2}n(r)|\beta(r,\theta,t)|^{2}dr$
$\displaystyle=$
$\displaystyle\frac{2\pi}{N}\sum_{n,m=0}^{\infty}\alpha_{m}^{*}(t)\alpha_{n}(t)\int_{0}^{\infty}r^{2}n(r)j_{m}(k_{0}r)j_{n}(k_{0}r)dr\int_{-1}^{1}P_{m}(x)P_{n}(x)dx.$

Using

$\int_{-1}^{1}dx\,P_{m}(x)P_{n}(x)=\frac{2}{2n+1}\delta_{m,n},$ (16)

we obtain

$\displaystyle\langle|\beta|^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{n=0}^{\infty}\frac{|\alpha_{n}(t)|^{2}\lambda_{n}}{2n+1}.$
(17)

The far-field amplitude of the radiation scattered by $N$ atoms along the
direction of the wave-vector
$\mathbf{k}=k_{0}(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ is

$\displaystyle E_{s}(\mathbf{k})$ $\displaystyle=$ $\displaystyle
E_{1}\sum_{j=1}^{N}\beta_{j}e^{-i\mathbf{k}\cdot\mathbf{r}_{j}}$ (18)

where $E_{1}=(dk_{0}^{2}/4\pi\epsilon_{0}r)\exp(ik_{0}r)$. For a continuous
distribution,

$\displaystyle E_{s}(\mathbf{k})$ $\displaystyle=$ $\displaystyle
E_{1}\int_{0}^{2\pi}d\phi^{\prime}\int_{0}^{\pi}\sin\theta^{\prime}d\theta^{\prime}\int_{0}^{\infty}r^{\prime
2}n(r^{\prime})\beta(r^{\prime},\theta^{\prime})e^{-ik_{0}r^{\prime}[\sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime})+\cos\theta\cos\theta^{\prime}]}dr^{\prime}$
(19) $\displaystyle=$ $\displaystyle 2\pi
E_{1}\int_{0}^{\pi}\sin\theta^{\prime}d\theta^{\prime}\int_{0}^{\infty}r^{\prime
2}n(r^{\prime})\beta(r^{\prime},\theta^{\prime})J_{0}(k_{0}r^{\prime}\sin\theta\sin\theta^{\prime})e^{-ik_{0}r^{\prime}\cos\theta\cos\theta^{\prime}}dr^{\prime}$
$\displaystyle=$ $\displaystyle
E_{1}\sum_{n=0}^{\infty}\alpha_{n}i^{-n}\lambda_{n}P_{n}(\cos\theta)$

where $J_{0}(x)$ is the zero-order Bessel function and we used the integral

$\displaystyle\int_{0}^{\pi}\sin\theta^{\prime}P_{n}(\cos\theta^{\prime})J_{0}(k_{0}r^{\prime}\sin\theta\sin\theta^{\prime})e^{-ik_{0}r^{\prime}\cos\theta\cos\theta^{\prime}}d\theta^{\prime}=2i^{-n}j_{n}(k_{0}r^{\prime})P_{n}(\cos\theta).$
(20)

The angular distribution of the power scattered by $N$ atoms is

$\displaystyle\frac{dP}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{c\epsilon_{0}}{2}|E_{s}(\mathbf{k})|r^{2}=\frac{P_{1}}{4\pi}\left|\sum_{j=1}^{N}\beta_{j}e^{-i\mathbf{k}\cdot\mathbf{r}_{j}}\right|^{2}$
(21) $\displaystyle=$
$\displaystyle\frac{P_{1}}{4\pi}\left\\{\sum_{j}|\beta_{j}|^{2}+\sum_{j}\sum_{m\neq
j}\beta_{j}\beta_{m}^{*}e^{-i\mathbf{k}\cdot(\mathbf{r}_{j}-\mathbf{r}_{m})}\right\\}$

where $P_{1}=ck_{0}^{4}d^{2}/(32\pi^{2}\epsilon_{0})$. The total scattered
power is obtained by integrating over the solid angle, giving

$P=P_{1}\sum_{j=1}^{N}\sum_{m=1}^{N}\beta_{j}\beta_{m}^{*}\frac{\sin(k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|)}{k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}.$
(22)

For a continuous distribution,

$\frac{dP}{d\Omega}=\frac{P_{1}}{4\pi}\left\\{N\langle|\beta|^{2}\rangle+\left|\sum_{n=0}^{\infty}\alpha_{n}i^{-n}\lambda_{n}P_{n}(\cos\theta)\right|^{2}\right\\}$
(23)

By integrating over the solid angle $4\pi$, the total scattered power is

$P=P_{1}\sum_{n=0}^{\infty}\frac{|\alpha_{n}|^{2}\lambda_{n}(1+\lambda_{n})}{2n+1}.$
(24)

##  IV Specific radial distribution

We consider three different spherical distributions for which exact analytic
expressions can be obtained. These include a sphere with uniform, parabolic
and Gaussian profile.

###  IV.1 Uniform sphere [10, 11]

For an uniform sphere of radius $R$ and density $n(r)=N/V$ where
$V=(4\pi/3)R^{3}$ and $0<r<R$,

$F_{n}(r)=\frac{3N}{R^{3}}\left\\{h^{(1)}_{n}(k_{0}r)\int_{0}^{r}dr^{\prime}r^{\prime
2}j^{2}_{n}(k_{0}r^{\prime})+j_{n}(k_{0}r)\int_{r}^{R}dr^{\prime}r^{\prime
2}j_{n}(k_{0}r^{\prime})h^{(1)}_{n}(k_{0}r^{\prime})\right\\}$ (25)

Taking $r=R$ and defining $k_{0}R=\sigma$, we obtain

$F_{n}(\sigma)=\frac{3N}{\sigma^{3}}h^{(1)}_{n}(\sigma)\int_{0}^{\sigma}dxx^{2}j^{2}_{n}(x).$
(26)

Since

$\displaystyle\int x^{2}j^{2}_{n}(x)dx$ $\displaystyle=$
$\displaystyle\frac{x^{3}}{2}\left\\{j_{n}^{2}(x)-j_{n-1}(x)j_{n+1}(x)\right\\}$
(27)

we obtain

$F_{n}(\sigma)=h^{(1)}_{n}(\sigma)\lambda_{n}$ (28)

where

$\displaystyle\lambda_{n}$ $\displaystyle=$
$\displaystyle\frac{3N}{2}\left\\{j^{2}_{n}(\sigma)-j_{n-1}(\sigma)j_{n+1}(\sigma)\right\\}$
(29)

is the collective decay rate of the mode $n$. By inserting these expressions
in Eq.(11) with $r=R$ we obtain, for $j_{n}(\sigma)\neq 0$,

$\displaystyle\dot{\alpha}_{n}-\Gamma\left\\{i(\delta-\omega_{n})-\frac{1}{2}(1+\lambda_{n})\right\\}\alpha_{n}=-i\frac{\Omega_{0}}{2}(2n+1)i^{n}$
(30)

where $\delta=\Delta_{0}/\Gamma$ and
$\omega_{n}=[y_{n}(\sigma)/j_{n}(\sigma)]\lambda_{n}/2$ is the collective Lamb
shift of the mode $n$. Equation (30) can be straightforwardly integrated and,
once inserted in Eq. (10), leads to the following expression for the
excitation amplitude

$\beta(r,\theta,t)=\frac{\Omega_{0}}{\Gamma}\sum_{n=0}^{\infty}\frac{i^{n}(2n+1)j_{n}(k_{0}r)P_{n}(\cos\theta)}{2(\delta-\omega_{n})+i(1+\lambda_{n})}\left[1-e^{i(\delta-\omega_{n})\Gamma
t-(1+\lambda_{n})\Gamma t/2}\right]$ (31)

If the pump is switched off after the steady-state is reached (taken as the
time $t=0$),

$\displaystyle\beta^{\mathrm{(free)}}(r,\theta,t)=\frac{\Omega_{0}}{\Gamma}\sum_{n=0}^{\infty}\frac{i^{n}(2n+1)j_{n}(k_{0}r)P_{n}(\cos\theta)}{2(\delta-\omega_{n})+i(1+\lambda_{n})}e^{i(\delta-\omega_{n})\Gamma
t-(1+\lambda_{n})\Gamma t/2}.$ (32)

Then

$\langle|\beta^{\mathrm{(free)}}|^{2}\rangle=\frac{\Omega_{0}^{2}}{N\Gamma^{2}}\sum_{n=0}^{\infty}\frac{(2n+1)\lambda_{n}}{4(\delta-\omega_{n})^{2}+(1+\lambda_{n})^{2}}e^{-(1+\lambda_{n})\Gamma
t},$ (33)
$\frac{dP}{d\Omega}=\frac{P_{1}}{4\pi}\left\\{\sum_{n=0}^{\infty}\frac{(2n+1)\lambda_{n}e^{-(1+\lambda_{n})\Gamma
t}}{4(\delta-\omega_{n})^{2}+(1+\lambda_{n})^{2}}+\left|\sum_{n=0}^{\infty}\frac{(2n+1)\lambda_{n}P_{n}(\cos\theta)}{2(\delta-\omega_{n})^{2}+i(1+\lambda_{n})}e^{-i\omega_{n}\Gamma
t-(1+\lambda_{n})\Gamma t/2}\right|^{2}\right\\}$ (34)

and

$P(t)=P_{1}\sum_{n=0}^{\infty}\frac{(2n+1)\lambda_{n}(1+\lambda_{n})e^{-(1+\lambda_{n})\Gamma
t}}{4(\delta-\omega_{n})^{2}+(1+\lambda_{n})^{2}}.$ (35)

We observe that this solution does not describe the subradiant decay after the
laser is cut off, since every mode has a decay rate
$(1+\lambda_{n})\Gamma>\Gamma$ i.e. larger than the single-atom decay. The MF
model is unable to describe subradiance, experimentally observed in [27] and
theoretically discussed in [18]: single photon subradiance arises from the
anti-symmetric states of $N$ atoms, in which only a single excitation among
$N$ is present [1, 28]. Hence, it can be described only by the discrete model
of Eq.(1). Conversely, single-photon superradiance can be well described by
the MF model, as it will discussed in the following.

For a small cloud, with $\sigma\ll 1$, only the term $n=0$, with
$\lambda_{0}\approx N$, decays fast (Dicke superradiance [1]), while all the
other terms with $n\geq 1$ are suppressed by a factor $\sigma^{2n}$. The
collective shift is $\omega_{0}\sim-N/2\sigma$. The case of a large cloud is
illustrated by Fig.1 and 2, showing $\lambda_{n}/N$ and $\omega_{n}/N$ for
$\sigma=20$, as obtained from Eq.(29). We observe that for $\sigma\gg 1$ and
$n<\sigma$, $\lambda_{n}\approx 3N/2\sigma^{2}\equiv\lambda_{N}$ (dashed blue
line in Fig.1) is almost independent on $n$ and drops to zero for $n>\sigma$,
approximately as

$\lambda_{n}\approx\frac{3\pi
N}{4n^{2}n!^{2}}\left(\frac{\sigma^{2}}{4}\right)^{n}.$ (36)

The collective Lamb shift $\omega_{n}$ in the limit $\sigma\gg 1$ and
$n<\sigma$ is approximately
$\omega_{n}\sim-(\lambda_{N}/2)\cot(\sigma-n\pi/2)\sim(3N/4\sigma^{2})\\{\tan\sigma,-\cot\sigma\\}$,
where the first value is for $n$ odd and the second for $n$ even (dashed blue
line and dash-dotted red line in Fig.2, respectively). We observe that
$\omega_{n}$ changes sign with $n$ and, with the exception for the values of
$\sigma$ where $\tan\sigma$ or $\cot\sigma$ are large, it averages to zero and
gives a negligible contribution. For large detuning, $\delta\gg 1$, it can be
neglected.

Fig.3 shows the average excitation probability $\langle|\beta(t)|^{2}\rangle$
vs $\Gamma t$ for $\delta=10$, $\sigma=20$ and $N=10^{3}$: the continuous red
line is the MF solution, obtained from Eq.(31), whereas the dash black line is
the numerical solution of Eqs.(1). The Timed-Dicke approximated solution [13,
14, 29, 30] can be obtained by assuming $\lambda_{n}\approx\lambda_{N}$,
giving

$\langle|\beta(t)|^{2}\rangle=\frac{1}{N\Gamma^{2}}\frac{\Omega_{0}^{2}}{4\delta^{2}+(1+\lambda_{N})^{2}}\left|1-e^{i\delta\Gamma
t-(1+\lambda_{N})\Gamma t/2}\right|^{2}.$ (37)

This solution, reported in Fig.3 by the dashed blue line, is in good agreement
with the exact solution, confirming that the driving laser brings the atoms
into a state well described by the Timed-Dicke approximation, where the
remaining subradiant part is only a small fraction of it.

When the laser is cut off, at short times the decay is superradiant, with
$\lambda_{n}\approx\lambda_{N}$ and

$\displaystyle\beta^{\mathrm{(free)}}(r,\theta,t)$ $\displaystyle\approx$
$\displaystyle\frac{(\Omega_{0}/\Gamma)}{2\delta+i(1+\lambda_{N})}e^{i\delta\Gamma
t-(1+\lambda_{N})\Gamma
t/2}\sum_{n=0}^{\infty}i^{n}(2n+1)j_{n}(k_{0}r)P_{n}(\cos\theta)$ (38)
$\displaystyle=$
$\displaystyle\frac{(\Omega_{0}/\Gamma)}{2\delta+i(1+\lambda_{N})}e^{ik_{0}r\cos\theta+i\delta\Gamma
t-(1+\lambda_{N})\Gamma t/2}.$

Fig.4 shows
$\langle|\beta^{\mathrm{(free)}}(t)|^{2}\rangle/\langle|\beta^{\mathrm{(free)}}(0)|^{2}\rangle$
vs $\Gamma t$ in semi-log scale for the same parameters of Fig.3, after the
laser is cut off. The continuous blue line is the MF solution, Eq. (33), the
dashed black line is the numerical solution of Eqs.(1), the dashed-dotted red
line is the Timed-Dicke superradiant decay $\exp(-\lambda_{N}\Gamma t)$ and
the dotted black line is the single-atom decay $\exp(-\Gamma t)$. We observe
that the MF solution initially follows the fast superradiant decay as
$\exp(-\lambda_{N}\Gamma t)$ and later the single-atom decay $\exp(-\Gamma
t)$. Instead, the discrete solution shows a subradiant decay, slower than the
single-atom decay. This behavior is peculiar of the discrete system and can
not be caught by the MF model.

Figure 1: $\lambda_{n}/N$ for an uniform sphere with $\sigma=20$. The dashed
blue line is the value $\lambda_{N}/N=3/2\sigma^{2}$.

Figure 2: $\omega_{n}/N$ for an uniform sphere with $\sigma=20$. The dashed
blue line is $\omega_{N}/N=3/4\sigma^{2}\tan\sigma$, the dash-dotted red line
is the value $\omega_{N}/N=-3/4\sigma^{2}\cot\sigma$.

Figure 3: $\langle|\beta(t)|^{2}\rangle$ (in units of
$(\Omega_{0}/\Gamma)^{2}$) vs $\Gamma t$ for $\delta=10$ and an uniform sphere
with $\sigma=20$ and $N=10^{3}$, from the analytical MF solution (continuous
red line), from the numerical solution of the discrete equations (1) (dash-dot
black line) and from the Timed-Dicke approximated solution, (37) (dash blue
line).

Figure 4:
$\langle|\beta^{\mathrm{(free)}}(t)|^{2}\rangle/\langle|\beta^{\mathrm{(free)}}(0)|^{2}\rangle$
vs $\Gamma t$ for $\delta=10$ and an uniform sphere with $\sigma=20$ and
$N=10^{3}$, from the analytical MF solution (continuous blue line) and from
the numerical solution of the discrete equations (1) (dashed black line). The
dashed-dotted red line is the Timed-Dicke approximation,
$\exp(-\lambda_{N}\Gamma t)$ and the dotted black line is the single atom
decay $\exp(-\Gamma t)$.

###  IV.2 Parabolic profile

An other case which can be solved analytically is a sphere with a parabolic
profile, with radial density $n(r)=(15N/8\pi R^{3})(1-r^{2}/R^{2})$ and
$0<r<R$. In this case we obtain

$\displaystyle\lambda_{n}$ $\displaystyle=$
$\displaystyle\frac{15N}{2}\left\\{\frac{1}{3}j_{n}^{2}(\sigma)-\frac{1}{2}j_{n+1}(\sigma)j_{n-1}(\sigma)-\frac{1}{6}j_{n-1}^{2}(\sigma)\right.$
(39) $\displaystyle+$
$\displaystyle\left.\frac{1}{3\sigma}\left(n+\frac{3}{2}\right)j_{n-1}(\sigma)j_{n}(\sigma)\right.$
$\displaystyle-$
$\displaystyle\left.\frac{1}{3\sigma^{2}}\left(n+\frac{3}{2}\right)\left[\left(n+\frac{1}{2}\right)j_{n}^{2}(\sigma)-\left(n-\frac{1}{2}\right)j_{n+1}(\sigma)j_{n-1}(\sigma)\right]\right\\}.$

where $\sigma=k_{0}R$. The other expressions, obtained from the uniform sphere
in Sec.IV.1, remain valid. Fig.5 shows $\lambda_{n}/N$ for $\sigma=20$, as
obtained from Eq.(39). For $\sigma\gg 1$ and $n\ll\sigma$,
$\lambda_{n}\approx(5N/2\sigma^{2})$.

Figure 5: $\lambda_{n}/N$ for a sphere with parabolic profile, with
$\sigma=20$. The dashed blue line is the value $\lambda_{N}/N=5/2\sigma^{2}$.

###  IV.3 Gaussian profile

For a Gaussian profile, with density
$n(r)=[N/(2\pi)^{3/2}\sigma_{R}^{3}]\exp(-r^{2}/2\sigma_{R}^{2})$, we obtain
[23]

$\lambda_{n}=N\sqrt{\frac{\pi}{2\sigma}}e^{-\sigma^{2}}I_{n+1/2}(\sigma^{2})$
(40)

where $\sigma=k_{0}\sigma_{R}$ and $I_{n}(x)$ is the $n$th-order modified
Bessel function. Taking the limit $r\rightarrow\infty$ in Eq.(LABEL:Fn), we
obtain the same equation (30) for $\alpha_{n}(t)$ and the same expression (33)
as for the uniform sphere, where the collective shift
$\omega_{n}=(\lambda_{n}/2)\lim_{r\rightarrow\infty}\\{y_{n}(k_{0}r)/j_{n}(k_{0}r)\\}$
may be neglected. For $\sigma$ large, all the modes up to $n\sim\sigma$ are
significant and

$\lambda_{n}\approx\frac{N}{2\sigma^{2}}e^{-(n+1/2)^{2}/2\sigma^{2}}.$ (41)

The spectrum can be treated as a continuum, with
$\lambda_{n}\approx\lambda(\eta)=(N/2\sigma^{2})\exp(-\eta^{2}/2\sigma^{2})$
(where $\eta=n+1/2$). Fig.6 shows the discrete values $\lambda_{n}/N$ vs $n$
for $\sigma=20$ from Eq.(40) (columns) and its continuous approximation (41)
(red continuous line).

Figure 6: $\lambda_{n}/N$ vs $n$ for a sphere with Gaussian profile, with
$\sigma=20$, for the exact discrete expression (40) and its continuous
approximation (41) (red continuous line).

Then, the sum in Eq.(33) can be approximated by an integral,
$\sum_{n=0}^{\infty}(2n+1)\rightarrow 2\int_{0}^{\infty}\eta d\eta$ to get

$\displaystyle\langle|\beta^{\mathrm{(free)}}|^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{2\Omega_{0}^{2}}{N\Gamma^{2}}\int_{0}^{\infty}\frac{\eta\lambda(\eta)}{4\delta^{2}+(1+\lambda(\eta))^{2}}e^{-(1+\lambda(\eta))\Gamma
t}d\eta$ (42) $\displaystyle=$
$\displaystyle\left(\frac{\Omega_{0}}{\Gamma}\right)^{2}\frac{2\sigma^{2}}{N}\int_{0}^{N/2\sigma^{2}}\frac{dx}{4\delta^{2}+(1+x)^{2}}e^{-(1+x)\Gamma
t},$

where we have set $x=\lambda(\eta)$. In the limit $\delta\gg 1$,

$\displaystyle\langle|\beta^{\mathrm{(free)}}|^{2}\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{\Omega_{0}}{2\delta\Gamma}\right)^{2}\frac{e^{-\Gamma
t}}{\Gamma_{sr}t}\left(1-e^{-\Gamma_{sr}t}\right),$ (43)

where $\Gamma_{sr}=(N/2\sigma^{2})\Gamma$ is the superradiant decay rate.
Instead, for $\delta=0$

$\displaystyle\langle|\beta^{\mathrm{(free)}}|^{2}\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{\Omega_{0}}{\Gamma}\right)^{2}\frac{2\sigma^{2}}{N}\Gamma
t\\{\gamma(-1,(\Gamma+\Gamma_{sr})t)-\gamma(-1,\Gamma t)\\},$ (44)

where $\gamma(a,x)=\int_{0}^{x}e^{-u}u^{a-1}du$ is the lower incomplete gamma
function. For large times, it can be approximated by

$\displaystyle\langle|\beta^{\mathrm{(free)}}|^{2}\rangle$
$\displaystyle\approx$
$\displaystyle\left(\frac{\Omega_{0}}{\Gamma}\right)^{2}\frac{e^{-\Gamma
t}}{\Gamma_{sr}t}\left[1-\left(\frac{\Gamma_{sr}}{\Gamma+\Gamma_{sr}}\right)^{2}e^{-\Gamma_{sr}t}\right],$
(45)

Hence, the decay of the excitation is not exponential, neither in the
superradiant regime: at short times the decay rate is $\Gamma_{sr}$ and at
later times the excitation decays as $\exp(-\Gamma t)/(\Gamma_{sr}t)$, before
the slower subradiant decay takes place at time larger than $1/\Gamma$. Fig.7
shows $\langle|\beta(t)|^{2}\rangle$ vs $\Gamma t$ for $\delta=10$ and a
Gaussian sphere with $\sigma=20$ and $N=10^{3}$, from the analytical MF
solution (continuous red line) and from the numerical solution of the discrete
equations (1) (dashed blue line). We observe a good agreement between the MF
and the discrete models as long as the laser is on. Just after the laser is
cut, the two solutions show that the excitation decays superradiantly, with a
rate $\Gamma_{sr}$, but at later times the exact discrete model shows that the
decay is subradiant, with a rate less than the single-atom value $\Gamma$
(shown by the dotted black line in Fig.7).

Figure 7: $\langle|\beta(t)|^{2}\rangle$ (in units of
$(\Omega_{0}/\Gamma)^{2}$) vs $\Gamma t$ for $\delta=10$ and a Gaussian sphere
with $\sigma=20$ and $N=10^{3}$, from the analytical MF solution (continuous
red line) and from the numerical solution of the discrete equations (1)
(dashed blue line). The dotted black line is the single-atom decay, as
$\exp(-\Gamma t)$.

Figure 8 shows $\langle|\beta(t)|^{2}\rangle$ vs time for the same case of
Fig.7, except that now $\delta=0$. In this case the MF solution (red
continuous line) does not reproduce well the exact discrete solution (dashed
blue line), neither when the laser is on. This confirms that the MF solution
does not describe the multiple-scattering regime (and hence the diffusion
regime), characterized by a large optical thickness $b=b_{0}/(1+4\delta^{2})$
(where $b_{0}=3N/\sigma^{2}$ is the resonant optical thickness). In the case
of Fig.8, $\delta=0$ and $b=b_{0}\sim 7.5$, whereas in the case of Fig.7
$\delta=10$ and $b\ll 1$. In the MF model the interaction is coherent and
dominated by collective modes: in order to describe the diffusive dynamics,
where the particles scatter many photons in a mean-free path, the model must
include granularity and fluctuations, which are missed assuming a smooth,
continuous density distribution.

Figure 8: $\langle|\beta(t)|^{2}\rangle$ (in units of
$(\Omega_{0}/\Gamma)^{2}$) vs $\Gamma t$ for $\delta=0$ and a Gaussian sphere
with $\sigma=20$ and $N=10^{3}$, from the analytical MF solution (continuous
red line) and from the numerical solution of the discrete equations (1)
(dashed blue line). The dotted black line is the single-atom decay, as
$\exp(-\Gamma t)$.

Finally Fig. 9 shows the total scattered power vs time $P_{s}(t)$ (in units of
the single-atom value $P_{1}$), calculated from the MF model, Eq.(35),
(continuous red line) and for the exact discrete model, Eq.(22), (dashed blue
line). The parameters are those of Fig.7. The MF solution describes rather
well the exact behavior, also if the transient oscillations are more strongly
damped in the exact solution. Just after the laser is cut, the decay rate is
superradiant, with a rate $\Gamma_{sr}=N\Gamma/2\sigma^{2}$ proportional to
the resonant optical thickness. Subradiant decay occurs at later times, after
the power has decreased by several orders of magnitude.

Figure 9: $P/P_{1}$ vs $\Gamma t$ for $\delta=10$ and a Gaussian sphere with
$\sigma=20$ and $N=10^{3}$, from the analytical MF solution (continuous red
line) and from the numerical solution of the discrete equations (22) (dashed
blue line).

##  V Conclusions

The aim of this paper has been to provide an analytical description of the
cooperative light scattering by an ensemble of atoms driven by an uniform
laser beam. We have compared the mean-field (MF) model, where a continuous
atomic distribution is assumed, to the numerical results from the discrete
coupled dipoles model. The MF model describes a coherent interaction between
the atoms, neglecting multiple scattering and diffusion effects due to the
random walk of the photon within a mean-free pass distance. For these reasons,
the validity of the MF model is limited to a regime with small optical
thickness $b=b_{0}/(1+4\delta^{2})\ll 1$, but still cooperative when $b_{0}\gg
1$ and $\delta\gg 1$. In this regime the MF model gives a rather accurate
description of the atomic excitation and of the scattered light intensity when
the laser is on, but is unable to describe the subradiant decay after the
laser is cut off. This suggests that subradiance is intrinsically related to
the discreetness of the system and to the anti-symmetric properties of the
single-excitation $N$-atomic states. Contrarily to previous works, we do not
assume an initial preparation of the atoms in a superposition of states with a
single excitation (the so-called Dicke states), but the excitation is provided
by a classical uniform laser. The atomic system reaches a stationary state
which is dominated by the Timed-Dicke symmetric state. When the laser is cut,
the early decay is superradiant, with a rate $\Gamma_{sr}\sim
N\Gamma/(k_{0}R)^{2}$, where $R$ is the size of the atomic cloud. The MF
solution can be expressed in terms of collective modes whose features depend
on the atomic distribution. We discussed the cases of uniform, parabolic and
Gaussian spherical distribution. When the cloud’s size is smaller than an
optical wavelength, a single mode with decay rate $N\Gamma$ will dominate,
whereas for an extended cloud many modes are present, up to a number $n\sim
k_{0}R$: the fastest modes are those with a decay rate proportional to the
resonant optical thickness $b_{0}$, down to the slower ones with decay rate
$\Gamma$. So, the last surviving modes when the laser is off are those with a
single-atom decay rate. In this sense, the subradiant component of the excited
state is lost in a MF description.

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