Adiabatic flux insertion and growing of Laughlin states of cavity Rydberg polaritons

Recently, the creation of photonic Landau levels in a twisted cavity has been demonstrated in Nature \textbf{534}, 671 (2016). Here we propose a scheme to adiabatically transfer flux quanta in multiples of $3\hbar$ simultaneously to all cavity photons by coupling the photons through flux-threaded cones present in such cavity setup. The flux transfer is achieved using external light fields with orbital angular momentum and a near-resonant dense atomic medium as mediator. Furthermore, coupling the cavity fields to a Rydberg state in a configuration supporting electromagnetically induced transparency, fractional quantum Hall states can be prepared. To this end a growing protocol is used consisting of a sequence of flux insertion and subsequent single-photon insertion steps. We discuss specifically the growing of the $\nu=1/2$ bosonic Laughlin state, where we first repeat the flux insertion twice creating a double quasi-hole excitation. Then, the hole is refilled using a coherent pump and the Rydberg blockade.