Figure 1

(a) Level scheme of atoms interacting with the probe field \(\boldsymbol{E}_{p}\) detuned by \(\Delta _{p}\) from the atomic transition resonance, while the excited atomic state \(| e \rangle \) decays to the ground state \(| g \rangle \) with rate \(\Gamma _{e} = 2\pi \times 6\text{ MHz}\). (b) Two-dimensional array of atoms (black filled circles) with the lattice spacing \(s=532\text{ nm}\) smaller than the wavelength \(\lambda _{e} = 780\text{ nm}\) of the atomic transition \(| e \rangle \to | g \rangle \). The atomic positions can deviate from the equilibrium lattice positions (open gray circles). (c) Transmission (T, green), reflection (R, red) and scattering (S, brown) spectra for an incoming coherent probe pulse (\(\bar{n}_{p} = 1\)) of duration \(\tau = 2~\mu \text{s}\) (Gaussian envelope) in a Gaussian spatial mode with waist \(w_{0} = 3 \lambda _{e} = 2.34~\mu \text{m}\) focused at, and normal to, the atomic layer in the xy plane. Inset shows the transmission, reflection and scattering of the probe field at the collective resonance frequency \(\Delta _{p} = \Delta \) (\(\Delta = 0.172 \Gamma _{e} \simeq 2\pi \times 1\text{ MHz}\)) vs the (Gaussian) atomic position uncertainly (standard deviations) \(\sigma _{x} = \sigma _{y}\) while \(\sigma _{z} = 0\) (dashed lines), and \(\sigma _{z}\) while \(\sigma _{x} = \sigma _{y}=0.01~\mu \text{m}\) (dotted lines). The graph are obtained from Monte Carlo simulations of Eqs. (14a)–(14c) for the dissipative dynamics of the stochastic atomic wavefunction averaged over 100-200 independent trajectories, in conjunction with Eqs. (15a), (15b) and (16)