Compressibility and the Equation of State of an Optical Quantum Gas in a Box

The compressibility of a medium, quantifying its response to mechanical perturbations, is a fundamental quantity determined by the equation of state. For gases of material particles, studies of the mechanical response are well established, in fields from classical thermodynamics to cold atomic quantum gases. In the quantum degenerate ideal Bose gas in two dimensions, however, a peculiar prediction so far has remained unseen: the compressibility of the gas is expected to become infinitely large. Here we demonstrate a measurement of the equation of state as well as the compressibility of a homogeneously trapped two-dimensional quantum gas of light inside a nanostructured dye-filled optical microcavity. Upon reaching quantum degeneracy we observe signatures of Bose-Einstein condensation in the finite-size system, causing a sharp increase of the density response to an external force, hiniting at the infinite compressibility of the uniform two-dimensional Bose gas.