Measurement of Tan's contact in SU($N$) fermions

Contact interactions play a fundamental role for understanding correlated quantum systems from microscopic models. For example, the 1/$k^4$ momentum tail, the weight of which is known as Tan's contact, reveals a set of universal relations in interacting atomic systems. Nevertheless, the experimental measurement of contact has still remained unexplored for multi-component fermions with high spins. Here, we report on the measurement of $s$-wave contact parameter in a cold spin-balanced gas of $^{173}$Yb atoms with SU($N$) symmetry. The high-momentum tail is recorded after time-of-flight expansion, and Tan's contact is directly quantified with a tunable number of spin component $N$ without changing the number of atoms per component. We experimentally verify the linear increase in the contact with $N$ providing experimental confirmation of the SU($N$) interaction. Furthermore, we investigate the momentum distribution of SU($N$) fermions at the low momentum regime showing that multi-component fermions with large $N$ exhibit a more bosonic behavior. Our measurement directly reveals the interaction effect in a multi-component Fermi gas with SU($N$) symmetry, and paves the way for study of many-body physics with high spins.