much faster that the one of the system (Markov approximation), the evolution of $\rho$, the reduced density matrix for the atom-like system, can be described by a Markovian master equation of the form $d \rho /d t = {\cal L}\left[\rho\right]$ [@lehmberg70a; @lehmberg70b; @chang12a; @gardiner_book00a], with the superoperator $$\label{mequation1}
    {\cal L}\left[\rho\right] = 
    \sum_{n,m} \Gamma_{n,m} \left( \sigma_{ge}^n \rho \sigma_{eg}^m - \rho \sigma_{eg}^m \sigma_{ge}^n \right)
    + \mathrm{H.c.} \,,$$ where $$\Gamma_{n,m} = \frac{\Gamma_{\oned}}{2} e^{i q(\omega_{{\rm a}}) |z_n - z_m|}\,,$$ where $\Gamma_\oned$, the decay into waveguide modes, that we will assume to be larger than the rate of spontaneous emission into all other modes, $\Gamma^* \ll \Gamma_\oned$ as this is the regime we are interested in. Moreover, as the propagation lengths of the waveguide modes for many implementations are long ($L_\mathrm{p} \gg \lambda_\mathrm{a}$), the atoms can be separated several wavelengths apart and can therefore be individually addressed as depicted in figure \[fig1\](a). In particular, we assume to control the TLS state through the Hamiltonian (in the interaction picture with respect to $H_\mathrm{qb}$) $$H_\mathrm{las}= \sum_{n} \frac{1}{2} \left( \Omega_n \sigma_{ge}^n  + \mathrm{H.c.} \right) + \Delta_n \sigma_{ee}^n,
    \label{eq:Hlas}$$ where $\Omega_n$ is the amplitude of the coherent driving (that we consider to be resonant, i.e., $\omega_L=\omega_{{\rm a}}$) which controls 