  Phase fluctuations determine the low-energy properties of quantum
condensates. However, at the condensation threshold, both density and phase
fluctuations are relevant. While strong emphasis has been given to the
investigation of phase fluctuations, which dominate the physics of the quantum
system away from the critical point -- number fluctuations have been much less
explored, even in thermal equilibrium. In this work, we report experimental
observation and theoretical description of fluctuations in a
circularly-confined non-equilibrium Bose-Einstein condensate of polaritons near
the condensation threshold. We observe critical fluctuations, which combine the
number fluctuations of a single-mode condensate state and competition between
different states. The latter are analogous to mode hopping in photon lasers.
Our theoretical analysis indicates that this phenomenon is of a quantum
character, while classical noise of the pump is not sufficient to explain the
experiments. The manifestation of a critical quantum state competition unlocks
new possibilities for the study of condensate formation while linking to
practical realizations in photonic lasers.
