  We study the time dependence of the decoherence factor (DF) of a qubit
globally coupled to an environmental spin system (ESS) which is driven across
the quantum critical point (QCP) by varying a parameter of its Hamiltonian in
time $t$ as $1 -t/\tau$ or $-t/\tau$, to which the qubit is coupled starting at
the time $t \to -\infty$; here, $\tau$ denotes the inverse quenching rate. In
the limit of weak coupling, we analyze the time evolution of the DF in the
vicinity of the QCP (chosen to be at $t=0$) and define three quantities,
namely, the generalized fidelity susceptibility $\chi_F(\tau)$ (defined right
at the QCP), and the decay constants $\alpha_1 (\tau)$ and $\alpha_2 (\tau)$
which dictate the decay of the DF at a small but finite $t$($>0$). Using a
dimensional analysis argument based on the Kibble-Zurek healing length, we show
that $\chi_F(\tau)$ as well as $\alpha_1 (\tau)$ and $\alpha_2(\tau)$ indeed
satisfy universal power-law scaling relations with $\tau$ and the exponents are
solely determined by the spatial dimensionality of the ESS and the exponents
associated with its QCP. Remarkably, using the numerical t-DMRG method, these
scaling relations are shown to be valid in both the situations when the ESS is
integrable and non-integrable and also for both linear and non-linear variation
of the parameter. Furthermore, when an integrable ESS is quenched far away from
the QCP, there is a predominant Gaussian decay of the DF with a decay constant
which also satisfies a universal scaling relation.
