Quantum photonics with active feedback loops

We develop a unified theoretical framework for the efficient description of multi-photon states generated and propagating in loop-based optical networks which contain nonlinear elements. These active optical components are modeled as nonlinear media, resembling a two-mode squeezer. First, such nonlinear components can be seeded to manipulate quantum states of light, as such enabling photon addition protocols. And, second, they can function as an amplifying medium. To prove the practical importance of our approach, the impact of multiple round trips is analyzed for states propagating in experimentally relevant loop configurations of networks, such as time-multiplexed driven quantum walks and iterative photon-number state generation protocols. Our method not only enables us to model such complex systems but also allows us to propose alternative setups that overcome previous limitations. To characterize the systems under study, we provide exact expressions for fidelities with target states, success probabilities of heralding-type measurements, and correlations between optical modes, including many realistic imperfections. Moreover, we provide an easily implementable numerical approach by devising a vector-type representation of photonic states, measurement operators, and passive and active processes.