Recently, the full Floquet solution of a Luttinger Liquid with periodically modulated interactions has been derived and resonant wave-vectors have been identified. There, the quantum state can only be stabilized when damping mechanisms of the one-dimensional systems are included. In our work, we investigate the time-periodic Luttinger Liquid under a non-linear perturbation which originates from the Sine-Gordon potential. This term provides interactions among the different modes of the Luttinger Liquid and can potentially confine the parametrically amplified modes. We investigate this model up to fourth order of the phase field using a mean-field approach. The resulting effective model is quadratic in the field operators while the non-linearity remains, due to the explicit dependence of the frequencies on the time- dependent quantum state. Using a self-consistency relation between the number of density wave excitations and the systems’ energy, we discuss the formation of a non-equilibrium steady state and study its stability.

Contact : Alexandre Faribault