  Exceptional points (EPs) with their intriguing spectral topology have
attracted considerable attention in a broad range of physical systems, with
potential sensing applications driving much of the present research in this
field. Here we theoretically demonstrate the realization of EPs in a system
with significant nonlinearity, a non-equilibrium exciton-polariton condensate.
With the possibility to control loss and gain and nonlinearity by optical
means, this system allows for a comprehensive analysis of the interplay of
nonlinearities (Kerr-type and saturable gain) and non-Hermiticity. Not only do
we find that EPs can be intentionally shifted in parameter space by the
saturable gain, we also observe intriguing rotations and intersections of
Riemann surfaces, and find nonlinearity-enhanced sensing capabilities. Our
results are quite general in nature and illustrate the potential of tailoring
spectral topology and related phenomena in non-Hermitian systems by
nonlinearity.
