Dynamical structure factor and a new method to measure the pairing gap in two-dimensional attractive Fermi-Hubbard model

The measurement of the pairing gap is crucial for investigating the physical properties of superconductors or superfluids. We propose a strategy to measure the pairing gap through the dynamical excitations. With the random phase approximation (RPA), we study the dynamical excitations of a two-dimensional attractive Fermi-Hubbard model by calculating its dynamical structure factor. Two distinct collective modes emerge: a Goldstone phonon mode at transferred momentum q=[0,0] and a roton mode at q=[π,π]. The roton mode exhibits a sharp molecular peak in the low-energy regime. Notably, the area under the roton molecular peak scales with the square of the pairing gap, which holds even in three-dimensional and spin-orbit coupled (SOC) optical lattices. This finding suggests an experimental approach to measure the pairing gap in lattice systems by analyzing the dynamical structure factor at q=[π,π]