Quantum Quenches in Coupled Luttinger Liquids: Truncated Spectrum Approach vs Semi-Classics

We consider a quantum quench of multiple Luttinger liquids. Here we initialize the system by supposing the Luttinger liquids are in their ground state. At t=0 we turn on a tunnel coupling between the liquids. We then study the subsequent time evolution of quantities such as the mode occupation numbers. We show that there are discrepancies between the results as determined by the truncated spectrum approach and a semi-classical treatment. We argue that this difference is real through unitary perturbation theory which is exact at short times. At late times the evolution of the system can be understood through the spectrum of sine-Gordon. We provide evidence that the non-classical part of this spectrum plays a role. We comment on the relevance of these results for experimental quantum quenches on two coupled Luttinger liquids.