Non-Markovianity, Loschmidt echo, and criticality: A unified picture

A simple relationship between recently proposed measures of non-Markovianity and the Loschmidt echo is established, holding for a two-level system (qubit) undergoing pure dephasing due to a coupling with a many-body environment. We show that the Loschmidt echo is intimately related to the information flowing out from and occasionally back into the system. This, in turn, determines the non-Markovianity of the reduced dynamics. In particular, we consider a central qubit coupled to a quantum Ising ring in the transverse field. In this context, the information flux between system and environment is strongly affected by the environmental criticality; the qubit dynamics is shown to be Markovian exactly and only at the critical point. Therefore, non-Markovianity is an indicator of criticality in the model considered here.

Figure 1

(a) The Loschmidt echo and (b) the Fisher information flow induced by the coupling of a central spin to the transverse Ising model consisting of $N=4000$ spins for three different values of the transverse field. We choose $J=1$ and $\delta =0.01$. The sensitivity to the critical point ${\lambda }_c^{*}=1$ is dramatic. Note that in (a), the solid black curve and the dashed red curve overlap for initial times.

Figure 2

Non-Markovianity measure $\mathcal{N}$ as a function of the particle number $N$ and the renormalized field ${\lambda }^{*}$. The time integral in $\mathcal{N}$ has been truncated to avoid the finite-size effects. A black line has been added to highlight the area where the measure is exactly zero, corresponding to purely Markovian dynamics.