single-site resolved control fields. In green, we depict the pairing that we will use to engineer the computational qubits.(b) Level structure of a TLS with a coherent driving with amplitude (detuning) $\Omega_n$ ($\Delta_n$) with an additional level to which transitions can be driven off-resonantly to engineer the $\Delta_n \sigma_{ee}^n$ term of the hamiltonian. (c) A three-level system in which the excited state is driven far off-resonantly can be made approximately equivalent to a TLS with modified parameters as shown in the legend.[]{data-label="fig1"}](figure1v3.pdf){width="98.00000%"}

The composite system is described by the Hamiltonian $H = H_0 + H_\II$, where $H_0$ is the free term given by $H_0 = H_\mathrm{qb} + H_\mathrm{field}$, (using $\hbar = 1$) $$H_\mathrm{qb} = \omega_{{\rm a}}\sum_{n=1}^N \sigma^{n}_{ee}, \
    H_\mathrm{field} = \sum_q \omega_q a^\dagger_q a_q,
    \label{H0}$$ where $\omega_{{\rm a}}$ is the TLS energy, $\sigma_{ij}^n={|i\rangle}_n{\langle j|}_n$ are atomic operators, and $\omega_q$ is the energy dispersion relation of the waveguide modes. We consider a dipolar coupling of the form $$H_\II  = \sum_{n} \left( \sigma_{ge}^n E(z_n) + \mathrm{H.c.} \right),
    \label{Hint}$$ with $E(z) = \sum_q g_q (a_q e^{i q z}+{{a_q^{\dagger}}} e^{-i q z})$, and $g_q$ the single photon coupling constant. When the system-reservoir coupling is weak (Born approximation) and the evolution of the 1d-reservoir is 