Qubit-Photon Bound States: Crossover from Waveguide to Cavity Regime

When the energy of a quantum emitter aligns with the photonic bandgap of the waveguide, variety of distinct quantum optical phenomena emerges. The paradigmatic example of such nontrivial interaction results in the formation of a qubit-photon bound state (QPBS) where a single photon is exponentially localized around the qubit position. Superconducting quantum circuits has been extensively used for studying QPBS where the photonic band gaps are generated in two-dimensional periodic structures such as photonic crystals and metamaterials. Nonetheless, in all the experiments reported so far, the exponential nature of the bound state wave function is justified by making two assumptions (1) that the waveguide is infinite in length and (2) the coupling to the external ports are absent. These assumptions are difficult to achieve for a physical device and the transition from an experimentally realizable finite-size system to the theoretically assumed infinite device size has never been thoroughly understood. Here, we have realized an alternative platform based on a three-dimensional (3D) rectangular waveguide that has a naturally occurring low-frequency cutoff for generating QPBS as proposed in Shahmoon and Kurizki [Phys. Rev. A 87, 033831 (2013)]. In its theoretical description, we explicitly take into consideration the finite nature of the 3D waveguide arising due to its metallic boundaries and coupling to microwave ports and demonstrate how the infinite waveguide limit arises in such a finite system.