Figure 1

Central spin 1/2 (qubit) interacts with the Ising spin environment whose quantum phase transition is driven by an external magnetic field. For large initial and final fields—right and left panels, respectively—the environment is in the paramagnetic phase: its spins align with the field. For small fields ($|g(t)|<1$, center panel), the Ising chain enters a ferromagnetic phase, where its spins try to align with $x$ or $-x$, breaking the symmetry of the Ising Hamiltonian.Reuse & Permissions

Figure 2

Decoherence during a quench: a quasiperiodic regime between the critical points. Black solid line shows the result obtained from numerical integration of Eq. (10). Our analytical approximations are superimposed on it as a blue dashed line: Eq. (16) for $g(t)\in (-1,1)$ and Eq. (19) for $g(t)<-1$. The green dashed-dotted line is the adiabatic result equivalent to ground-state fidelity (${\tau }_Q\to \infty$ limit): Eq. (A1) combined with Eq. (8). Numerical solution almost perfectly overlaps with analytical results away from the critical points: $|g(t)-g_c|>\mathrm{ĝ},\delta$. The main plot shows that for moderate environment sizes—$N=1000$ here—almost perfect revivals of coherence take place between the critical points, but coherence is virtually lost after driving the environment through the second critical point. The inset shows the same, but for $N=30\hspace{0.5em}000$. It illustrates that for very large environments, there is little coherence left after passing the first critical point, and the qubit is completely decohered after crossing the second critical point. Note that ground-state fidelity provides the envelope for the revivals of coherence between the critical points. We used here $\delta ={10}^{-2}$ and ${\tau }_Q=250$.Reuse & Permissions

Figure 3

Same as in Fig. 2, except we consider here $N$th root of the decoherence factor $D$ illustrating details of dynamics of decoherence in the large $N$ limit. Numerical result (black line) is obtained by calculating $D^{1/N}$ for $N=2000$ environmental spins. $N$th root of ground-state fidelity (green dashed-dotted line) is calculated for the same $N$. The blue dashed line shows our analytical approximations: Eqs. (16) and (19).Reuse & Permissions

Figure 4

Decoherence during a quench: a Gaussian regime between the critical points. Solid black line shows the result coming from numerical integration of Eq. (10). The dashed blue line shows our analytical approximations: Eq. (17) between the critical points [$g(t)\in (-1,1)$] and Eq. (19) past the second critical point [$g(t)<-1$]. The inset shows $N$th root of the data from the main plot: it illustrates details of the revivals of coherence past the second critical point. The parameters for this plot are: $N=1000$, $\delta =4\times {10}^{-4}$, and ${\tau }_Q=250$. Except for $\delta$, they are the same as in Fig. 2. The switch from coherence revivals (Fig. 2) to monotonic decay of coherence between the critical points is a result of lowering the qubit-environment coupling $\delta$ below the $\pi /16{\tau }_Q$ threshold.Reuse & Permissions

Figure 5

Decoherence factor $F_k$ during the quench. Solid black lines show numerics based on Eq. (10). The blue dashed lines illustrate our analytical approximations. Before reaching the critical point at $g_c=1$, modes evolve adiabatically: (a) made for $g(t)=2$. The adiabatic result [Eq. (13)] fits numerics for all momenta. Between the critical points—(b) and (c) made for $g(t)=0$—low $k$ modes are excited due to driving across $g_c=1$, while the large $k$ modes evolve adiabatically. The former is illustrated in (b), where Eq. (15) fits numerics, while the latter is depicted in (c) where Eq. (13) matches numerics. (d)–(f), prepared for $g(t)=-2$, show what happens after passing the second critical point at $g_c=-1$: low $k$ modes are still excited due to driving across $g_c=1$, intermediate modes evolve adiabatically, while large $k$ modes are excited by crossing the $g_c=-1$ critical point. This is theoretically described by Eqs. (15), (13), and (18) depicted in (d), (e), and (f), respectively. Note that despite the tiny range of variation of the adiabatically evolving modes— (a), (c), and (e)—the striking envelope of the quasiperiodic revivals of coherence between the critical points comes from these modes. Solid black lines (numerics) have been obtained for $N=2000$, ${\tau }_Q=250$, and $\delta ={10}^{-2}$. The same ${\tau }_Q$ and $\delta$ have been used to get the dashed lines (analytical approximations).Reuse & Permissions