Non-equilibrium steady state solutions of time-periodic driven Luttinger liquids

Controlled time-periodic driving of quantum systems has recently provided a whole range of possibilities for Floquet steady states with quantum engineered properties. In order to study the interplay of many-body correlations with time-periodic driving we now consider a Tomonaga-Luttinger liquid with periodically changing interactions in the steady state, which is proposed to be realized by an interacting Bose gas confined to a one-dimensional Lieb-Liniger model. Without assuming any high-frequency approximation, we are able to analyse the time-dependent Schrödinger equation by developing a Floquet-Bogliubov ansatz, which is solved in terms of Mathieu functions. Remarkably, the characteristic Mathieu exponent can also become complex valued in regions of frequency and momentum space, which does not correspond to a steady state solution. For the experimental systems this implies an instability when multiples of driving frequency approximately matches twice the dispersion energy, which is observable by the creation of a large number of collective density wave excitations at the corresponding wave numbers.