Loschmidt echo, emerging dual unitarity and scaling of generalized temporal entropies after quenches to the critical point

We discuss how all relevant objects involved in the calculation of the Loschmidt echo of a product state after a quench to a conformal invariant critical point can be predicted by using conformal field theories. We check such prediction with tensor networks finding excellent agreement. In particular we are able to predict and confirm that in such an out-of-equilibrium scenario, we observe an emerging dual-unitarity of the evolution. We also show how to extract the universal information of the underlying CFT from such out-of-equilibrium protocol, including, the central charge, the operator content, and the generalized temporal entropies. In particular we show that using state-of-the art tensor networks algorithms such calculations only require resources that increase polynomially with the duration of the quench, thus providing an example of numerically efficiently solvable out-of-equilibrium scenario.