We study single-particle excitations of a Bose gas in an optical lattice in the Mott-insulator phase. The characteristic feature of the single-particle spectrum in the Mott-insulator phase is the existence of an energy gap between the particle and hole excitations. We show that the single-particle excitation energies and associated energy gap in the Mott-insulator phase can be directly probed by an output-coupling experiment. We apply the general expression for the output current derived by Luxat and Griffin, which is given in terms of the single-particle Green's functions of a trapped Bose gas, to the Mott-insulator phase using the Bose-Hubbard model. The energy spectrum of the momentum-resolved output current exhibits two characteristic peaks corresponding to the particle and hole excitations, and thus it can be used to detect the transition point from the Mott insulator to superfluid phase where the energy gap disappears.
We derive the equation of state of bosons in an optical lattice in the framework of the Bose-Hubbard model. Near the density-driven Mott transition, the expression of the pressure $P(\ensuremath{\mu},T)$ versus chemical potential and temperature is similar to that of a dilute Bose gas but with renormalized mass
m∗ and scattering length
a∗. Here
m∗ is the mass of the elementary excitations at the quantum critical point governing the transition from the superfluid phase to the Mott-insulating phase, while
a∗ is related to their effective interaction at low energy. We use a nonperturbative renormalization-group approach to compute these parameters as a function of the ratio
t/U between hopping amplitude and on-site repulsion.