Mean-Field Description of Cooperative Scattering by Atomic Clouds

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.


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 r j , whose atomic transition has frequency ω a , linewidth Γ and dipole d (polarization effects are neglected). The system is driven by a monochromatic plane wave with electric field E 0 , frequency ω 0 and wave vector k 0 , detuned from the atomic transition by ∆ 0 = ω 0 − ω 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 β j [25]: where Ω 0 = dE 0 /ℏ is the Rabi frequency and 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, nλ 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 λ 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 σ 2 (where σ = 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/σ 2 ≫ 1. Under this hypothesis, the particles can be described by a smooth density n(r) and their probability to be excited by a field β(r, t). By approximating the sum over j by an integral over the smooth density, i.e. j → drn(r), Eq.(1) turns into Using where Y n,m (θ, ϕ) 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 ′ . Taking θ as the polar angle with respect the direction of the wave vector k 0 , we can expand By substituting it in Eq.(3) and assuming a radial distribution, n(r), we obtain where and 2π 0 dϕ π 0 dθ sin θ Y * n,m (θ, ϕ)e ik0r cos θ = 2δ m,0 π(2n + 1)i n j n (k 0 r), multiplying Eq.(6) by Y * n,m (θ, ϕ) and integrating over the angles, we obtain: If α n,m (0) = 0, the only components different from zero are those for m = 0. So, defining α n = (2n + 1)/4π α n,0 and since Y n,0 (θ, ϕ) = (2n + 1)/4π P n (cos θ) where P n (x) are the Legendre polynomial, we write where α n (t) is the solution of the following differential equation where We observe that F n (r) has a real part and an imaginary part. The real part is Re{F n (r)} = λ n j n (k 0 r) where 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 Im{F n (r)} = 4π y n (k 0 r) and contributes to the cooperative Lamb shift, arising from the cosine term of the kernel of Eq.(3). When the detuning ∆ 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: we obtain The far-field amplitude of the radiation scattered by N atoms along the direction of the wave-vector k = k 0 (sin θ cos ϕ, sin θ sin ϕ, cos is where where J 0 (x) is the zero-order Bessel function and we used the integral The angular distribution of the power scattered by N atoms is where P 1 = ck 4 0 d 2 /(32π 2 ϵ 0 ). The total scattered power is obtained by integrating over the solid angle, giving For a continuous distribution, By integrating over the solid angle 4π, the total scattered power is

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.
A. Uniform sphere [10,11] For an uniform sphere of radius R and density n(r) = N/V where V = (4π/3)R 3 and 0 < r < R, Taking r = R and defining k 0 R = σ, we obtain Since we obtain where is the collective decay rate of the mode n. By inserting these expressions in Eq.(11) with r = R we obtain, for where δ = ∆ 0 /Γ and ω n = [y n (σ)/j n (σ)]λ 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 If the pump is switched off after the steady-state is reached (taken as the time t = 0), Then and 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 + λ n )Γ > Γ 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 antisymmetric 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 σ ≪ 1, only the term n = 0, with λ 0 ≈ N , decays fast (Dicke superradiance [1]), while all the other terms with n ≥ 1 are suppressed by a factor σ 2n . The collective shift is ω 0 ∼ −N/2σ. The case of a large cloud is illustrated by Fig.1 and 2, showing λ n /N and ω n /N for σ = 20, as obtained from Eq. (29). We observe that for σ ≫ 1 and n < σ, λ n ≈ 3N/2σ 2 ≡ λ N (dashed blue line in Fig.1) is almost independent on n and drops to zero for n > σ, approximately as The collective Lamb shift ω n in the limit σ ≫ 1 and n < σ is approximately ω n ∼ −(λ N /2) cot(σ − nπ/2) ∼ (3N/4σ 2 ){tan σ, − cot σ}, where the first value is for n odd and the second for n even (dashed blue line and dashdotted red line in Fig.2, respectively). We observe that ω n changes sign with n and, with the exception for the values of σ where tan σ or cot σ are large, it averages to zero and gives a negligible contribution. For large detuning, δ ≫ 1, it can be neglected. Fig.3 shows the average excitation probability ⟨|β(t)| 2 ⟩ vs Γt for δ = 10, σ = 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 λ n ≈ λ N , giving 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 λ n ≈ λ N and i n (2n + 1)j n (k 0 r)P n (cos θ) . We observe that the MF solution initially follows the fast superradiant decay as exp(−λ N Γt) and later the single-atom decay exp(−Γt). 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.
where σ = k 0 σ R and I n (x) is the nth-order modified Bessel function. Taking the limit r → ∞ in Eq.(12), we obtain the same equation (30) for α n (t) and the same expression (33) as for the uniform sphere, where the collective shift ω n = (λ n /2) lim r→∞ {y n (k 0 r)/j n (k 0 r)} may be neglected. For σ large, all the modes up to n ∼ σ are significant and The spectrum can be treated as a continuum, with λ n ≈ λ(η) = (N/2σ 2 ) exp(−η 2 /2σ 2 ) (where η = n + 1/2). Fig.6 shows the discrete values λ n /N vs n for σ = 20 from Eq.(40) (columns) and its continuous approximation (41) (red continuous line). Then, the sum in Eq.(33) can be approximated by an integral, where we have set x = λ(η). In the limit δ ≫ 1, where Γ sr = (N/2σ 2 )Γ is the superradiant decay rate. Instead, for δ = 0 where γ(a, x) = x 0 e −u u a−1 du is the lower incomplete gamma function. For large times, it can be approximated by Hence, the decay of the excitation is not exponential, neither in the superradiant regime: at short times the decay rate is Γ sr and at later times the excitation decays as exp(−Γt)/(Γ sr t), before the slower subradiant decay takes place at time larger than 1/Γ. Fig.7 shows ⟨|β(t)| 2 ⟩ vs Γt for δ = 10 and a Gaussian sphere with σ = 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 Γ sr , but at later times the exact discrete model shows that the decay is subradiant, with a rate less than the single-atom value Γ (shown by the dotted black line in Fig.7). Figure 8 shows ⟨|β(t)| 2 ⟩ vs time for the same case of Fig.7, except that now δ = 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.

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 + 4δ 2 ) ≪ 1, but still cooperative when b 0 ≫ 1 and δ ≫ 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 Γ sr ∼ N Γ/(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 Γ will dominate, whereas for an extended cloud many modes are present, up to a number n ∼ 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 Γ. 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.