  We present and analyze a protocol in which polaritons in a noncoplanar
optical cavity form fractional quantum Hall states. We model the formation of
these states and present techniques for subsequently creating anyons and
measuring their fractional exchange statistics. In this protocol, we use a
rapid adiabatic passage scheme to sequentially add polaritons to the system,
such that the system is coherently driven from $n$- to $(n+1)$-particle
Laughlin states. Quasiholes are created by slowly moving local pinning
potentials in from outside the cloud. They are braided by dragging the pinning
centers around one another, and the resulting phases are measured
interferometrically. The most technically challenging issue with implementing
our procedure is that maintaining adiabaticity and coherence requires that the
two-particle interaction energy $V_0$ be sufficiently large compared to the
single-polariton decay rate $\gamma$, $V_0 /\gamma \gg 10 N^2 \ln N$, where $N$
is the number of particles in the target state. While this condition is very
demanding for present-day experiments where $V_0 /\gamma\sim 50$, our protocol
presents a significant advance over the existing protocols in the literature.
