Studying the low-entropy Mott transition of bosons in a three-dimensional optical lattice by measuring the full momentum-space density

We report on a combined experimental and theoretical study of the low-entropy Mott transition for interacting bosons trapped in a three-dimensional (3D) cubic lattice – namely, the interaction-induced superfluid-to-normal phase transition in the vicinity of the zero-temperature Mott transition. Our analysis relies on the measurement of the 3D momentum distribution, which allows us to extract the momentum-space density $\rho (\mathbf{k}=\mathbf{0})$ at the center of the Brillouin zone. Upon varying the ratio between the interaction $U$ and the tunnelling energy $J$ across the superfluid transition, we observe that $\rho (\mathbf{k}=\mathbf{0})$ exhibits a sharp transition at a value of $U\mathrm{/}J$ consistent with the bulk prediction from quantum Monte Carlo. In addition, the variation of $\rho (\mathbf{k}=\mathbf{0})$ with $U\mathrm{/}J$ exhibits a critical behavior consistent with the expected 3D XY universality class. Our results show that the tomographic reconstruction of the momentum distribution of ultracold bosons can reveal traits of the critical behavior of the superfluid transition even in an inhomogeneous trapped system.