Characterising and Tailoring Spatial Correlations in Multi-Mode Parametric Downconversion.

Photons entangled in their position-momentum degrees of freedom (DoFs) serve as an elegant manifestation of the Einstein-Podolsky-Rosen paradox, while also enhancing quantum technologies for communication, imaging, and computation. The multi-mode nature of photons generated in parametric downconversion has inspired a new generation of experiments on high-dimensional entanglement, ranging from complete quantum state teleportation to exotic multi-partite entanglement. However, precise characterisation 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 formalise a description of the two-photon wavefunction 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 $2D\pi$-measurement. Finally, we discuss how precise knowledge of the collected JTMA enables us to generate tailored high-dimensional entangled states that maximise discrete-variable entanglement measures such as entanglement-of-formation or entanglement dimensionality, and optimise critical experimental parameters such as photon heralding efficiency. By accurately and efficiently characterising 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 multi-mode entanglement.