  We study the non-equilibrium evolution of concurrence of a Bell pair
constituted of two qubits, through the measurement of Loschmidt echo (LE) under
the scope of generalized central spin model. Having detected the Griffiths
phase via derivative of LE in equilibrium, we show that in the non-equilibrium
situation, the spin chain requires a temporal window to realize the effect of
disorder. We show that within this timescale, LE falls off exponentially and
this decay is maximally controlled by the initial states and coupling strength.
Our detail investigation suggests that there actually exist three types of
exponential decay, a Gaussian decay in ultra short time scale followed by two
exponential decay in the intermediate time with two different decay exponents.
The effect of the disorder starts appearing in the late time power law fall of
LE where the power law exponent is strongly dependent on disorder strength and
the final state but almost independent of initial states and coupling strength.
This feature allows us to indicate the presence of Griffiths phase. To be
precise, continuously varying critical exponent and wide distribution of
relaxation time imprint their effect in LE in the late time limit where the
power law fall is absent for quenching to a Griffiths phase. Here, LE vanishes
following the fast exponential fall. Interestingly, for off-critical quenching
LE attains a higher saturation value for increasing disorder strength,
otherwise vanishes for a clean spin chain, referring to the fact that disorder
prohibits the rapid decay of entanglement in long time limit. Moreover, we show
that disorder is also able to destroy the light cone like nature of post quench
quasi-particles as LE does not sense the singular time scales appearing for
clean spin chain with qubits coupled at symmetric positions.
