\hat{\sigma}^{+}_j e^{-i\omega_d t} \right\}.
\label{ham}\end{aligned}$$

$\hat{a}$ and $\hat{a}^{\dagger}$ are photon annihilation and creation operators for the cavity field with commutation relation $[\hat{a},\hat{a}^{\dagger}] =1$. ${\sigma}^{-}_j$, ${\sigma}^{+}_j$, ${\sigma}^{z}_j$ are usual spin-$\frac{1}{2}$ Pauli operators for the $j^{th}$ atom with commutation relations $[{\sigma}^{+}_j,{\sigma}^{-}_j] ={\sigma}^{z}_j$ and $[{\sigma}^{z}_j,{\sigma}^{\pm}_j] =\pm{\sigma}^{\pm}_j$. $g_j = g_0 f(x_j,y_j)$ is coupling of $j^{th}$ atom with the cavity mode with $g_0=-\mu\sqrt{\omega_c /(2 \hbar \epsilon_0 V)}$ maximum atom-cavity coupling, $f(x,y) = e^{-(x^2+y^2)/w_0^2}$ is mode function of the cavity field, $V = \pi w_0^2L_c/4$ is the mode volume of the cavity, $L_c$ is cavity length, $\mu$ is the transition dipole matrix element between excited and ground state, $2|\Omega|=\frac{\mu}{\hbar}\sqrt{\frac{2I_p}{c\epsilon_0}}$ is Rabi frequency for drive beam, $I_p$ is intensity of the drive beam, $\eta$ is strength of classical cavity probe. $\omega_c$, $\omega_a$, $\omega_d$, $\omega_p$ are frequencies of the cavity (resonance frequency of cavity when it is empty), atom, driving laser and cavity probe respectively and $^{*}$ denotes complex conjugate.

### Rate equations

The evolution equation for the expectation value for an operator $\hat{X}$ can be evaluated using the Heisenberg equation, $\frac{d\langle\hat{X}\rangle}{dt} = \frac{i}{\hbar} \langle [\hat{H},\hat{X}]\rangle$. In addition to this unitary evolution, we introduce non-unitary decay rates $\kappa$ and $\Gamma$ phenomenologically for cavity field and atoms respectively. This 