Figure 3

Transmission (T, green), reflection (R, red) and scattering (S, brown) spectra of the two-dimensional array of four-level atoms for an incoming coherent probe pulse (\(\bar{n}_{p} = 1\)) of duration \(\tau = 2~\mu \text{s}\) focused at the atomic layer in the xy plane. The driving field has the Rabi frequency \(\Omega _{d}= 2\pi \times 2.0\text{ MHz}\) and detuning \(\Delta _{d} = -0.172\Gamma _{e}\) to provide a two-photon resonance for the probe field at the collective resonance frequency \(\Delta _{p} = \Delta =0.172\Gamma _{e}\). The cavity mode is assumed resonant, \(\Delta _{c}=0\), and couples to the Rydberg transition with strength \(\eta = 2\pi \times 2.0\text{ MHz}\). The decay rates of the Rydberg states are \(\Gamma _{s,r} = 10^{-3} \Gamma _{e}\) with the other parameters as in Fig. 1. The cavity mode is empty, \(n_{c}=0\) and \(\Omega _{c} =0\), (long dashed lines); or contains one photon, \(n_{c}=1\) and \(\Omega _{c} = \eta \), (solid lines). For reference, we also show the response of two level atoms as in Fig. 1(c) (thin dashed lines). Inset shows the transmission, reflection and scattering of the probe field at the collective resonance frequency \(\Delta _{p} = \Delta \) vs the atomic position uncertainly \(\sigma _{x} = \sigma _{y}\) while \(\sigma _{z} = 0.01~\mu \text{m}\). The graph are obtained from Monte Carlo simulations of Eqs. (19a)–(19d) in conjunction with Eqs. (15a), (15b) and (16)