We derive theoretically an exact relation between Tan's universal contact and the photoexcitation rate of a strongly interacting Fermi gas, in the case of optically transferring fermionic pairs to a more tightly bound molecular state. Our deviation generalizes the relation between Tan's contact and the closed-channel molecular fraction found earlier by Werner, Tarruell, and Castin [Eur. Phys. J. B 68, 401 (2009)]. We use the relation to understand the recent low-temperature photoexcitation measurement in a strongly interacting
6Li Fermi gas [Liu et al., arXiv:1903.12321] and show that there is a reasonable agreement between theory and experiment close to the unitary limit. We propose that our relation can be applied to accurately measure Tan's contact coefficient at finite temperature in future experiments.
A universal ${k}^{\ensuremath{-}4}$ decay of the large-momentum tails of the momentum distribution, fixed by Tan's contact coefficients, constitutes a direct signature of strong correlations in a short-range interacting quantum gas. Here we consider a repulsive multicomponent Fermi gas under harmonic confinement, as in the experiment of G. Pagano et al. [Nat. Phys. 10, 198 (2014)], realizing a gas with tunable $\text{SU}(\ensuremath{\kappa})$ symmetry. We exploit an exact solution at infinite repulsion to show a direct correspondence between the value of the Tan's contact for each of the $\ensuremath{\kappa}$ components of the gas and the Young tableaux for the
SN permutation symmetry group identifying the magnetic structure of the ground state. This opens a route for the experimental determination of magnetic configurations in cold atomic gases, employing only standard (spin-resolved) time-of-flight techniques. Combining the exact result with matrix-product-state simulations, we obtain the Tan's contact at all values of repulsive interactions. We show that a local-density approximation (LDA) on the Bethe-ansatz equation of state for the homogeneous mixture is in excellent agreement with the results for the harmonically confined gas. At strong interactions, the LDA predicts a scaling behavior of the Tan's contact. This provides a useful analytical expression for the dependence on the number of fermions, number of components, and interaction strength. Moreover, using a virial approach, we study the Tan's contact behavior at high temperature and in the limit of infinite interactions, and we show that it increases with the temperature and the number of components. At zero temperature, we predict that the weight of the momentum distribution tails increases with interaction strength and the number of components if the population per component is kept constant. This latter property was experimentally observed in G. Pagano et al. [Nat. Phys. 10, 198 (2014)].