Finite-temperature compressibility of disordered lattice bosons at unit filling

The disordered Bose-Hubbard model is a paradigm for strongly interacting bosons tunneling between adjacent sites of a disordered crystalline material. A complete picture for the disorder-driven, incompressible--to--compressible Mott-insulator--to--Bose-glass transition (which occurs for simultaneously strong interactions and strong disorder) is not fully understood. To probe this problem, we superimpose a cubic disordered optical lattice potential on Bose-Einstein condensates of $^{\mathrm{87}}$Rb atoms and measure the core compressibility by observing how disorder affects double occupancy. Our measurements indicate that a remnant of the zero-temperature phase boundary is visible at finite entropy-per-particle. Furthermore, the physics of this transition can be understood using a single-site disordered model, in contrast to the typical description consisting of rare regions in an infinite system.