  In this paper, we study theoretically atomic quantum simulations of a U(1)
gauge-Higgs model on a three-dimensional (3D) spatial lattice by using an
extended Bose-Hubbard model with intersite repulsions on a 3D optical lattice.
Here, the phase and density fluctuations of the boson variable on each site of
the optical lattice describe the vector potential and the electric field on
each link of the gauge-model lattice, respectively. The target gauge model is
different from the standard Wilson-type U(1) gauge-Higgs model because it has
plaquette and Higgs interactions with asymmetric couplings in the space-time
directions. Nevertheless, the corresponding quantum simulation is still
important as it provides us with a platform to study unexplored time-dependent
phenomena characteristic of each phase in the general gauge-Higgs models. To
determine the phase diagram of the gauge-Higgs model at zero temperature, we
perform Monte-Carlo simulations of the corresponding 3+1-dimensional U(1)
gauge-Higgs model, and obtain the confinement and Higgs phases. To investigate
the dynamical properties of the gauge-Higgs model, we apply the
Gross-Pitaevskii equations to the extended Bose-Hubbard model. We simulate the
time-evolution of an electric flux that initially is put on a straight line
connecting two external point charges. We also calculate the potential energy
between this pair of charges and obtain the string tension in the confinement
phase. Finally, we propose a feasible experimental setup for the atomic
simulations of this quantum gauge-Higgs model on the 3D optical lattice. These
results may serve as theoretical guides for future experiments.
