Non-Markovian qubit dynamics induced by Coulomb crystals

We investigate the backflow of information in a system with a second-order structural phase transition, namely, a quasi-one-dimensional Coulomb crystal. Using standard Ramsey interferometry which couples a target ion (the system) to the rest of the chain (a phononic environment), we study the non-Markovian character of the resulting open system dynamics. We study two different time scales and show that the backflow of information pinpoints both the phase transition and different dynamical features of the chain as it approaches criticality. We also establish an exact link between the backflow of information and the Ramsey fringe visibility.

Figure 1

Short-time-scale dynamics. (a) Time evolution of the trace distance for the maximizing pair $\left\{\right|+\rangle ,|-\rangle \}$ for $N=200$, far from the critical point (black line, $\Delta =0.1$) and very close to it (red line, $\Delta ={10}^{-5}$). (b) Non-Markovianity measure $\mathcal{N}$ for short-time-scale truncation as a function of $\Delta$ for $N=100$ (blue solid line) and $N=1000$ (green dashed line).

Figure 2

Long-time-scale dynamics. (a) Time evolution of the trace distance for the maximizing pair $\left\{\right|+\rangle ,|-\rangle \}$, for $N=300$, far from the critical point (black, $\Delta =0.1$) and very close to it (red, $\Delta ={10}^{-4}$). (b) Non-Markovianity measure $\mathcal{N}$ for long time-scale truncation as a function of $\Delta$ for $N=300$.

Figure 3

Short-time-scale non-Markovianity measure ${\mathcal{N}}_{\text{cr}}$ at criticality as a function of the number of ions $N$.