Quench spectroscopy for Lieb-Liniger bosons in the presence of harmonic trap

Quench spectroscopy has emerged as a powerful technique for probing the energy spectrum of various quantum phases for quantum systems from out-of-equilibrium dynamics. While its efficacy has been demonstrated in the homogeneous systems theoretically, most experimental setups feature a confining potential, such as a harmonic trap, which complicates the practical implementations. In this work, we experimentally probe the quench spectroscopy for one-dimensional bosons in optical lattices with the presence of a harmonic trap, and compare our results with the density matrix renormalization group simulation. For the Mott insulator phase, although a gap is still observed, the band signal is broadened along the frequency space and cut at the half Brillouin zone, which can be explained by the nearest-neighbor tunneling excitations under harmonic confinement. Comparing with the superfluid spectrum, we can see a clear distinction between the two phases and find that the inverse quench with larger amplitude yields the clearest spectrum. Our work offers pivotal insights into conducting quench spectroscopy effectively in practical systems.