Photons are elementary particles of light, and their interactions in vacuum are extremely weak. The seclusion of photons makes them perfect carriers of classical and quantum information, but also poses difficulties for employing them in quantum information technologies. Recent years have seen tremendous experimental progress in the development of synthetic quantum systems where strong and controllable coupling between single photons is achieved. In a variety of solid-state and optical platforms, propagating photons are coupled with local emitters such as atoms, quantum dots, NV centers, or superconducting qubits. Despite the different nature of the platforms, many of these systems can be described using the same theoretical framework called waveguide quantum electrodynamics (WQED).

Dissipation is an inevitable ingredient of many synthetic quantum systems and is a source of error in quantum information applications. Despite its important role in experimental systems, the implications of dissipation in scattering theory havenot been fully explored. Chapter 2 discusses our discovery of the dissipation-induced bound states in WQED systems. The appearance of these bound states is in a one-to-one correspondence with zeros in the single-photon transmission. We also formulate a dissipative version of Levinson's theorem by looking at the relation between the number of bound states and the winding number of the transmission phases.

In Chapter 3, we study three-body loss in Rydberg polaritons. Despite past theoretical and experimental studies of the regime with dispersive interaction, the dissipative regime is still mostly unexplored. Using a renormalization group technique to solve the quantum three-body problem, we show how the shape and strength of dissipative three-body forces can be universally enhanced for Rydberg polaritons. We demonstrate how these interactions relate to the transmission through a single-mode cavity, which can be used as a probe of the three-body physics in current experiments.

The high level of control of the synthetic quantum systems behind WQED offers many inspirations for theoretical studies. In Chapter 4 of this dissertation, we explore a new direction of scattering theory motivated by the controllability of dispersion relations in synthetic quantum systems. We study single-particle scattering in one dimension when the dispersion relation is E(k)=k^m, where m>= 2 is an integer. For a large class of interactions, we discover that the S-matrix evaluated at an energy E->0 converges to a universal limit that is only dependent on m. We also give a generalization of Levinson's theorem for these more general dispersion relations in WQED systems.