A Concept of Two-Point Propagation Field of a Single Photon: A Way to Picometer X-ray Displacement Sensing and Nanometer Resolution 3D X-ray Micro-Tomography

We introduce the two-point propagation field (TPPF), a real-valued, phase-sensitive quantity defined as the functional derivative of the single-photon detection probability with respect to an infinitesimal opaque perturbation placed between source and detection slits. The TPPF is analytically derived and shown to exhibit a stable, high-frequency sinusoidal structure (6.7 nm period) near the detection slit. This structure enables shot-noise-limited displacement sensing at 15 pm precision using routinely available synchrotron fluxes and practical nanofabricated slit/comb geometries, requiring mechanical stability only over the final 0.5 mm. The same principle provides a foundation for future nanometer-resolution 3D X-ray microtomography of bulk samples, potentially resulting in a reduced radiation dose. Two conceptual strategies, a central blocker and off-axis multi-slit arrays, are estimated to lower the required incident fluence by more than one order of magnitude each, yielding combined reductions of two to three orders of magnitude with near-term detector development. The TPPF concept, originally developed in a perturbative study of single-particle propagation, thus bridges fundamental quantum measurement questions with practical high-resolution X-ray metrology and imaging.