Engineering Correlation and Entanglement Dynamics in Spin Systems

We show that correlation and entanglement dynamics of spin systems can be precisely controlled and engineered using only a small number of external physical control parameters. We first point out that the correlation dynamics of such systems can be understood in terms of spin-wave propagation, giving a simple physical explanation of the behavior seen in a number of recent works. We then extend this picture to more realistic translationally invariant systems prepared in product states. Since spin waves propagate according to a system’s dispersion relation which typically depends on external physical parameters, this insight provides a convenient way to understand how dynamics can be controlled. We demonstrate these ideas in a simple example system, showing that correlations can be made to propagate in well-defined packets whose speed can be engineered in advance, controlled during the evolution, or even stopped altogether.

Figure 1

For $\gamma =1.1$, $\lambda =2.0$, all the wave-packet envelopes ($2S$, $2CS$, and $2S^2$ from Eq. (2), solid curves, inset) are similar in form, peaked around a nearly linear region of the dispersion relation $\epsilon (\varphi )$ (dashed curve, inset) with gradient 2.1. Thus the correlations $C_{zz}(x,t)$ (main plot) propagate in well-defined packets at a speed given by the gradient.

Figure 2

For $\gamma =10.0$, $\lambda =0.9$, the wave-packet envelopes (solid curves, inset; cf. Fig. 1) are spread over the entire frequency range. The maximum propagation speed of the correlations $C_{zz}(x,t)$ (main plot) is given by the maximum gradient 19.8 of the dispersion relation $\epsilon \varphi$ (dashed curve, inset), faster than in Fig. 1 but with higher dispersion.

Figure 3

Starting from $\gamma =1.1$, $\lambda =2.0$ as in Fig. 1, $\gamma$ and $\lambda$ are smoothly changed to move from the dashed to the dotted dispersion relation (inset), increasing the correlation speed.

Figure 4

The system is initially allowed to evolve with ${\gamma }_0=0.9$ ${\lambda }_0=0.5$, then “quenched” at time $t_1=20.0$ to ${\gamma }_1=0.1$, ${\lambda }_1=10.0$. Some of the correlations $C_{zz}(x,t)$ are frozen at the separation ($x\approx 20$) they reached at $t_1$. Others propagate according to the new dispersion relation, or are “reflected”.