Methods for studying Tan's contact in the BCS-BEC crossover

In a parallel publication [Phys. Rev. Lett. 132, 263401 (2024)] we demonstrate that photoinduced two-body loss can be used to measure Tan's contact with high precision in a two-component Fermi gas. Here, in the present companion paper, we provide relevant background information on this work and describe in detail the methodology for both the experiments and the data analysis. We first review various theoretical approaches for calculating the contact and identify areas in phase space of the spin-balanced Fermi gas where Tan's contact has not yet been determined. Next, we provide detailed information on our experimental methods, in particular, explaining the measurement and calibration procedures to achieve a high-precision results for the contact. Afterwards, we study the variation of the decay laws of two-body loss in a Fermi gas in the crossover from the Bardeen-Cooper-Schrieffer (BCS) to the Bose-Einstein-condensate (BEC) regime, verifying previous predictions. Finally, we determine the closed-channel fraction of the Fermi gas and compare it to previous measurements and theoretical calculations.