2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154290We derive the collective low energy excitations of the quantum phase model of interacting lattice bosons within the superfluid state using a dynamical variational approach. We recover the well known sound (or Goldstone) mode and derive a gapped (Higgs type) mode that was overlooked in previous studies of the quantum phase model. This mode is relevant to ultracold atoms in a strong optical lattice potential. We predict the signature of the gapped mode in lattice modulation experiments and show how it evolves with increasing interaction strength.4 pages, 3 figuresOther Condensed Mattertext