the number of excitations of the system, and $\Delta_n$ is a phase shift interaction term. The latter can be obtained, e.g., in atomic systems, by adding an off-resonant driving to another excited state ${|e'\rangle}$, as depicted in figure \[fig1\](b), which results in an Stark shift $\Delta_n=|\Omega'|^2/(\omega_{{\rm a}}-\omega_L')$. In general, the way of implementing $\Omega_n$ and $\Delta_n$ will depend on the particular system.

For completeness, it is worth mentioning that $\Lambda$ systems can also be mapped to effective TLS by using an off-resonant Raman transition as depicted in figure \[fig1\](c). By adiabatically eliminating the excited state ${|e'\rangle}$, one can formally project the dynamics to the two metastable states, $\{{|g\rangle},{|e\rangle}\}$, and find a similar light-matter hamiltonian as the one of equation \[Hint\], with the advantage that the effective TLS defined by $\{{|g\rangle},{|e\rangle}\}$ will be long-lived as they are encoded in metastable states. For example, by switching both $\Omega_g$ and $\Omega_e$ at the same time with detuning $\delta(\gg |\Omega_g|,|\Omega_e|)$ as depicted in figure \[fig1\](c), we can implement a coherent driving term with effective $\Omega = \frac{\Omega_g \Omega_e^*}{4 \delta}$. By switching $\delta$ in this case big enough one can neglect spontaneous emission processes as they will be proportional to $\Gamma^* \left(\frac{|\Omega_e|^2+|\Omega_g|^2}{4\delta^2}\right)$. Moreover, if we 