Mimicking interacting relativistic theories with stationary pulses of light

Photonic quantum simulations of one dimensional many-body systems have attracted renewed interest lately with works on photon crystallization and Luttinger liquids. In this work we show that the quantum Thirring model for interacting fermions in (1$+$1) dimension can be realized using stationary polaritons in hollow waveguides filled with atoms. By controlling optical parameters such as one-photon detunings and external laser intensities, the massless and the massive Thirring models are realizable. Coherently mapping the polaritons into propagating photons allows for the direct probing of the relevant correlation functions and scaling behaviours characteristic of the underlying theories in question.