  The semiconductor polariton laser promises a new source of coherent light,
which, compared to conventional semiconductor photon lasers, has input-energy
threshold orders of magnitude lower. However, intensity stability, a defining
feature of a coherent state, has remained poor. Intensity noise at many times
of the shot-noise of a coherent state has persisted, which has been attributed
to multiple mechanisms that are difficult to separate in conventional polariton
systems. The large intensity noise in turn limited the phase coherence. These
limit the capability of the polariton laser as a source of coherence light.
Here, we demonstrate a polariton laser with shot-noise limited intensity
stability, as expected of a fully coherent state. This is achieved by using an
optical cavity with high mode selectivity to enforce single-mode lasing,
suppress condensate depletion, and establish gain saturation. The absence of
spurious intensity fluctuations moreover enabled measurement of a transition
from exponential to Gaussian decay of the phase coherence of the polariton
laser. It suggests large self-interaction energies in the polariton condensate,
exceeding the laser bandwidth. Such strong interactions are unique to
matter-wave laser and important for nonlinear polariton devices. The results
will guide future development of polariton lasers and nonlinear polariton
devices.
