Colloquium: Strongly interacting photons in one-dimensional continuum

Photon-photon scattering in vacuum is extremely weak. However, strong effective interactions between single photons can be realized by employing strong light-matter coupling. These interactions are a fundamental building block for quantum optics, bringing many-body physics to the photonic world and providing important resources for quantum photonic devices and for optical metrology. This Colloquium reviews the physics of strongly interacting photons in one-dimensional systems with no optical confinement along the propagation direction. It focuses on two recently demonstrated experimental realizations: superconducting qubits coupled to open transmission lines and interacting Rydberg atoms in a cold gas. Advancements in the theoretical understanding of these systems are presented in complementary formalisms and compared to experimental results. The experimental achievements are summarized alongside a description of the quantum optical effects and quantum devices emerging from them.