photon number $$\langle \hat{a}^\dagger\hat{a}(t)\rangle=\langle U^\dagger(t)\hat{a}^\dagger\hat{a} U(t)\rangle=\langle {\cal T}e^{i\int_0^t d\tau \{H^{in}(\tau)+H_{nl}(\tau)\}}U^\dagger_0(t)\hat{a}^\dagger\hat{a}U_0(t){\cal T}e^{-i\int_0^t d\tau \{H^{in}(\tau)+H_{nl}(\tau)\}}\rangle$$ is given as the sum of the following three parts: (1) the system operator contribution $$\begin{aligned}
\langle \hat{a}^\dagger\hat{a}(t)\rangle_s=d_{21}(\kappa t,0)d_{12}(\kappa t,0)+d_{23}(\kappa t,0)d_{14}(\kappa t,0)(n_{th}+1)+d_{24}(\kappa t,0)d_{13}(\kappa t,0)n_{th};
\label{c1}\end{aligned}$$ (2) the noise drive contribution $$\begin{aligned}
\langle \hat{a}^\dagger\hat{a}(t)\rangle_n&=&2\int_0^{\kappa t} d\tau d_{21}(\kappa t,\tau)d_{12}(\kappa t,\tau)+\frac{2s_m}{Q}\int_0^{\kappa t} d\tau d_{23}(\kappa t,\tau)d_{14}(\kappa t,\tau)(n_{th}+1)\nonumber\\
&+&\frac{2s_m}{Q}\int_0^{\kappa t} d\tau d_{24}(\kappa t,\tau)d_{13}(\kappa t,\tau)n_{th};
\label{c2}\end{aligned}$$ (3) the contribution $$\begin{aligned}
\langle \hat{a}^\dagger\hat{a}(t)\rangle_c &=&\big |{\cal E}\int_0^{\kappa t} d\tau e^{i\delta s_m  \tau}+i \frac{{\cal E}}{s_m \delta}\int_0^{\kappa t} d\tau d_{11}(\kappa t,\tau)F(\tau)-i\frac{{\cal E}}{s_m \delta}\int_0^{\kappa t} d\tau d_{12}(\kappa t,\tau)F^\ast(\tau)\nonumber\\
&+&iG_m|\frac{{\cal E}}{s_m\delta}|^2 \int_0^{\kappa t} d\tau d_{13}(\kappa t,\tau)|F(\tau)|^2e^{is_m\tau}-iG_m|\frac{{\cal E}}{s_m\delta}|^2 \int_0^{\kappa t} d\tau d_{14}(\kappa t,\tau)|F(\tau)|^2 e^{-is_m\tau}\big|^2
\label{c3}\end{aligned}$$ purely from the coherent drive term, with $F(\tau)=e^{i\delta s_m \tau}-1$. In Eq. (\[c3\]) we have also included the contribution from the pure drive action $U_0(t)={\cal T}\exp\{-i\int_0^t d\tau H_e(\tau)\}$. The notation $d_{ij}(t,\tau)=[{\cal T}e^{\int_\tau^t dt' \hat{M}(t')}]_{ij}$ represents a matrix element for $i,j=1,2,3,4$. For a sufficiently high drive intensity $E$, the third contribution from the coherent drive term dominates over the two others.

We here study the cavity photon number evolutions in the weak coupling regime of small effective coupling intensity $J=(g/\omega_m)(E/\kappa)$, as a complement of the discussion on the relatively large $J$ in Figs. 2(a) and 2(c) 