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We investigate the finite temperature momentum distribution of bosonicnoncondensate particles inside a 3D optical lattice near the superfluid to Mottinsulator transition point, treating the quantum fluctuation and thermalfluctuation effects on equal footing. We explicitly address the differentmomentum ($q$) dependence of quasi-particles excitations resulted from thermaland quantum origin: the former scales as $|\bfq|^{-2}$ and hence is dominant inthe small momentum region, while the later scales as $|\bfq|^{-1}$ and hencedominant in the large momentum limit. Analytic and semi-analytic results arederived, providing a unique method to determine the temperature, condensatedensity, coherent length and/or single particle gap etc. inside the opticallattice. Our results also agree with the scaling theory of a quantum $XY$ modelnear the transition point. Experimental implication of the TOF measurement isalso discussed.

Thermal and quantum noncondensate particles near the superfluid to Mott insulator transition