  We consider a generalized central spin model, consisting of two central
qubits and an environmental spin chain (with periodic boundary condition) to
which these central qubits are locally and weakly connected either at the same
site or at two different sites separated by a distance $d$. Our purpose is to
study the subsequent temporal generation of entanglement, quantified by
concurrence, when initially the qubits are in an unentangled state. In the
equilibrium situation, we show that the concurrence survives for a larger value
of $d$ when the environmental spin chain is critical. Importantly, a common
feature observed both in the equilibrium and the non-equilibrium situations
while the latter is created by a sudden but global change of the environmental
transverse field, is that the two qubits become maximally entangled for the
critical quenching. Following a non-equilibrium evolution of the spin chain,
our study for $d\neq 0$, indicates that there exists a threshold time above
which concurrence attains a finite value. Additionally, we show that the number
of independent decohering channels (DCs) is determined by $d$ as well as the
local difference of the transverse field of the two underlying Hamiltonians
governing the time evolution. The qualitatively similar behavior displayed by
the concurrence for critical and off-critical quenches, as reported here, is
characterized by analyzing the non-equilibrium evolution of these channels. The
concurrence is maximum when the decoherence factor or the echo associated with
the most rapidly DC decays to zero; on the contrary, the condition when the
concurrence vanishes is determined non-trivially by the associated decay of one
of the intermediate DCs. Analyzing the reduced density of a single qubit, we
also explain the observation that the dephasing rate is always slower than the
unentanglement rate.
