We 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.Comment: 4 pages, 3 figure