  Taking inspiration from the state-of-the art knowledge of the Bose-Hubbard
(BH) model and recent methodological developments in its fermionic counterpart,
this work deals with the study of the collective dynamics of a lattice Bose gas
beyond the mean-field picture through a quantum description of its elementary
excitations. The Hamiltonian quantization, performed via a Bogoliubov
quadratization of the BH action within the Gutzwiller approach, allows to
expand the effective action of the theory up to second order in the
fluctuations around the mean-field solution, as well as to prefigure the
possibility of identifying the main decay vertices of the collective modes and
other effects that are not evident at the second-order level. This quantum
description extends the standard Bogoliubov approach to the study of superfluid
Bose systems for comprising higher excitation branches, including the Higgs
mode in the superfluid phase, and identifies their physical meaning together
with appropriate observables which could be taken into consideration for their
experimental characterization. The ultimate aim of the quantization procedure
is the determination of fundamental quantities as the depletion of the
condensate and the effective superfluid fraction, which are not accessible by a
mean-field description or not completely characterized in the regime of strong
interactions.
