Dynamics of decoherence: universal scaling of the decoherence factor

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.