  We study the Loschmidt echo (LE) in a central spin model in which a central
spin is globally coupled to an environment (E) which is subjected to a small
and sudden quench at $t=0$ so that its state at $t=0^+$, remains the same as
the ground state of the initial environmental Hamiltonian before the quench;
this leads to a non-equilibrium situation. This state now evolves with two
Hamiltonians, the final Hamiltonian following the quench and its modified
version which incorporates an additional term arising due to the coupling of
the central spin to the environment. Using a generic short-time scaling of the
decay rate, we establish that in the early time limit, the rate of decay of the
LE (or the overlap between two states generated from the initial state evolving
through two channels) close to the quantum critical point (QCP) of E is
independent of the quenching. We do also study the temporal evolution of the LE
and establish the presence of a crossover to a situation where the quenching
becomes irrelevant. In the limit of large quench amplitude the non-equilibrium
initial condition is found to result in a drastic increase in decoherence at
large times, even far away from a QCP. These generic results are verified
analytically as well as numerically, choosing E to be a transverse Ising chain
where the transverse field is suddenly quenched.
