  Exciton-polaritons in a microcavity are composite two-dimensional bosonic
quasiparticles, arising from the strong coupling between confined light modes
in a resonant planar optical cavity and excitonic transitions, typically using
excitons in semiconductor quantum wells (QWs) placed at the antinodes of the
same cavity. Quantum phenomena such as Bose-Einstein condensation (BEC),
quantized vortices, and macroscopic quantum states have been reported at
temperatures from tens of Kelvin up to room temperatures, and polaritonic
devices such as spin switches \cite{Amo2010} and optical transistors have also
been reported. Many of these effects of exciton-polaritons depend crucially on
the polariton-polariton interaction strength. Despite the importance of this
parameter, it has been difficult to make an accurate experimental measurement,
mostly because of the difficulty of determining the absolute densities of
polaritons and bare excitons. Here we report the direct measurement of the
polariton-polariton interaction strength in a very high-Q microcavity
structure. By allowing polaritons to propagate over 40 $\mu$m to the center of
a laser-generated annular trap, we are able to separate the polariton-polariton
interactions from polariton-exciton interactions. The interaction strength is
deduced from the energy renormalization of the polariton dispersion as the
polariton density is increased, using the polariton condensation as a benchmark
for the density. We find that the interaction strength is about two orders of
magnitude larger than previous theoretical estimates, putting polaritons
squarely into the strongly-interacting regime. When there is a condensate, we
see a sharp transition to a different dependence of the renormalization on the
density, which is evidence of many-body effects.
