Characterizing and Tailoring Spatial Correlations in Multimode Parametric Down-Conversion

Photons entangled in their position-momentum degrees of freedom serve as an elegant manifestation of the Einstein-Podolsky-Rosen paradox, while also enhancing quantum technologies for communication, imaging, and computation. The multimode nature of photons generated in parametric down-conversion has inspired a generation of experiments on high-dimensional entanglement, ranging from complete quantum state teleportation to exotic multipartite entanglement. However, precise characterization of the underlying position-momentum state is notoriously difficult due to limitations in detector technology, resulting in a slow and inaccurate reconstruction riddled with noise. Furthermore, theoretical models for the generated two-photon state often forgo the importance of the measurement system, resulting in a discrepancy between theory and experiment. Here we formalize a description of the two-photon wave function in the spatial domain, referred to as the collected joint-transverse momentum amplitude (JTMA), which incorporates both the generation and measurement system involved. We go on to propose and demonstrate a practical and efficient method to accurately reconstruct the collected JTMA using a simple phase-step scan known as the 2Dπ measurement. Finally, we discuss how precise knowledge of the collected JTMA enables us to generate tailored high-dimensional entangled states that maximize discrete-variable entanglement measures such as entanglement of formation or entanglement dimensionality, and optimize critical experimental parameters such as photon heralding efficiency. By accurately and efficiently characterizing photonic position-momentum entanglement, our results unlock its full potential for discrete-variable quantum information science and lay the groundwork for future quantum technologies based on multimode entanglement.