  We study the quantum quench in two coupled Tomonaga-Luttinger Liquids (TLLs),
from the off-critical to the critical regime, relying on the conformal field
theory approach and the known solutions for single TLLs. We consider a squeezed
form of the initial state, whose low energy limit is fixed in a way to describe
a massive and a massless mode, and we encode the non-equilibrium dynamics in a
proper rescaling of the time. In this way, we compute several correlation
functions, which at leading order factorize into multipoint functions evaluated
at different times for the two modes. Depending on the observable, the
contribution from the massive or from the massless mode can be the dominant
one, giving rise to exponential or power-law decay in time, respectively. Our
results find a direct application in all the quench problems where, in the
scaling limit, there are two independent massless fields: these include the
Hubbard model, the Gaudin-Yang gas, and tunnel-coupled tubes in cold atoms
experiments.

Statistical Mechanics;Quantum Gases;Strongly Correlated Electrons;High Energy Physics - Theory