Pair size versus coherence length and quantum geometry in the spin-orbit coupled BCS-BEC crossover

We investigate the interplay between pairing correlations and quantum geometry in the spin-orbit-coupled BCS-BEC crossover. Working in the helicity basis, we derive the exact pair-size tensor for the two-body bound states and show that it separates into an intraband contribution and a quantum-geometric interband contribution governed by the quantum metric. We extend this decomposition to two distinct many-body length scales at zero temperature: the Cooper-pair size from mean-field BCS theory and the coherence length from Gaussian fluctuation theory. Although the two almost coincide in the BCS regime, the former measures the internal extent of a pair while the latter measures the long-wavelength response of the order parameter. Evaluating both for three-dimensional Rashba, three-dimensional Weyl, and two-dimensional Rashba spin-orbit couplings, we find that the pair size decreases monotonically with spin-orbit coupling, whereas the coherence length develops a pronounced minimum, associated with the collapse of the noninteracting Fermi surface onto the ring or sphere of helicity-band minima, before growing again as the composite bosons become weakly interacting. The quantum-geometric contributions reach up to about 30\% of the squared pair size and squared coherence length, tracking the same changes in the helicity Fermi-surface topology.