We have described a protocol for preparing anyons in an optical cavity built by carefully aligning a set of high-quality mirrors. The three stages of the protocol are illustrated in the animation below. First, one drives the cavity with lasers to sequentially inject photons, building up the N-polariton Laughlin state |Φ_{N}⟩. This is achieved by pumping photons with angular momentum at the outer rim of the cloud and appropriately sweeping the drive frequency. Second, one produces anyonic quasihole excitations by moving additional lasers in from the edge of the cloud. Finally, one uses the same lasers to drag the quasiholes around one another and measures the resulting phase using an interferometric technique described in the main text.

The cavity is coherently driven by lasers to sequentially add polaritons to the system. The driving freuqency is swept through resonance to adiabatically transfer the system from the n- to the n+1-particle Laughlin states: |Φ_{n}⟩ → |Φ_{n+1}⟩. The sweep rate must be slow compared to the energy splittings between the Laughlin states and other many-body states, which are set by the two-particle interaction energy V_{0} (zeroth Haldane pseudopotential).

The following simulations correspond to the optical drives given by Eqs. (6)--(8) in the main article, with c_{Ω} = 4, c_{δ} = 5.33. Note that ρ denotes the polariton density.

**(a)** Adiabatic drive: the drive frequency is swept adiabatically, adding one polariton over a duration 4τ = 60/V_{0}. The system is driven through Laughlin states |Φ_{n}⟩ of uniform density and increasing particle number n, as in Fig. 5(a) of the main article. Thus the area of the cloud grows linearly with n. **(b)** Nonadiabatic drive: the frequency sweeps are faster, each occurring over a duration 4τ = 32/V_{0}. The evolution is no longer adiabatic, giving rise to vortices and ripples in the density.

Quasiholes are produced by moving additional lasers in from the edge of the cloud. These local pinning potentials are described by the Hamiltonian in Eqs. (11) and (12) in the main text. To suppress unwanted edge excitations during this process, the potentials must be swept slowly compared to the excitation energy ε set by the harmonic confinement of the polaritons.

The following simulations correspond to a trap-induced splitting ε = 0.06V_{0} and potential strength U_{0} = 50V_{0}. As before, ρ denotes the polariton density.

**(a)** Adiabatic sweep: two local potentials from opposite sides are swept inward at a rate |∂_{t}(r_{0}/l)| = 0.04V_{0} < ε. The system adiabatically follows the instantaenous ground state, forming a quasihole on either side. **(b)** Nonadiabatic sweep: the potentials are swpet too fast, |∂_{t}(r_{0}/l)| = 0.08V_{0} > ε, exciting surface modes and large deformations.

Quasiholes are braided by stirring the pinning lasers around one another. The quasiholes adiabatically follow the laser potentials provided the rotation rate is sufficiently small. The adiabaticity condition is different for clockwise and countercloskwise rotations, as the (effective) magnetic field breaks time-reversal symmetry. Braiding too close to the boundary causes edge excitations.

The following simulations correspond to uniform rotations with a potential strength U_{0} = 50V_{0} and trap-induced splitting ε = 0.06V_{0} [except (d) where ε = 0].

**(a)** Adiabatic braiding: two quasiholes at r_{0}/l = ±1.6 are adiabatically exchanged by slowly stirring the pinning lasers through 180° over an interval T_{b} = 80/V_{0}. **(b)** Nonadiabatic braiding - counterclockwise: two quasiholes at r_{0}/l = ±1.5 are dragged counterclockwise through 180° over an interval T_{b} = 6/V_{0}, exciting vortices and ripples in the density. **(c)** Nonadiabatic braiding - clockwise: quasiholes at r_{0}/l = ±1.5 are exchanged clockwise over an interval T_{b} = 40/V_{0}, exciting surface modes and density ripples. **(d)** Nonadiabatic - edge excitations: quasiholes near the edge at r_{0}/l = ±2.4 are exchanged clockwise over an interval T_{b} = 20/V_{0}, exciting edge modes.