  We explore non-equilibrium processes in two-dimensional conformal field
theories (2d CFTs) due to the growth of operators induced by inhomogeneous and
homogeneous Hamiltonians by investigating the time dependence of the partition
function, energy density, and entanglement entropy. The non-equilibrium
processes considered in this paper are constructed out of the Lorentzian and
Euclidean time evolution governed by different Hamiltonians. We explore the
effect of the time ordering on entanglement dynamics so that we find that in a
free boson CFT and RCFTs, this time ordering does not affect the entanglement
entropy, while in the holographic CFTs, it does. Our main finding is that in
the holographic CFTs, the non-unitary time evolution induced by the
inhomogeneous Hamiltonian can retain the initial state information longer than
in the unitary time evolution.
