Entanglement resonance in driven spin chains

We consider a spin-1/2 anisotropic $XY$ model with time-dependent spin-spin coupling as means of creating long-distance entanglement. We predict the emergence of significant entanglement between the first spin and the last spin whenever the ac part of the coupling has a frequency matching the Zeeman splitting. In particular, we find that the concurrence assumes its maximum with a vanishing dc part. Mapping the time-dependent Hamiltonian within a rotating-wave approximation to an effective static model provides qualitative and quantitative understanding of this entanglement resonance. Numerical results for the duration of the entanglement creation and its length dependence substantiate the effective static picture.

Figure 1

(Color online) Maximum concurrence obtained in a time window up to $4N/\text{max}\left(J_0,J_1\right)$, between spins 1 and $N$ for different frequencies and chain lengths $N$ with $J_1=0.1B$, $\gamma =1$, and $J_0=0$ (solid), $0.01B$ (dashed), $0.1B$ (dashed-dotted). The effective anisotropy thus has the values $\tilde{\gamma }=\gamma J_1/2J_0=\infty$, 5, 0.5.

Figure 2

(Color online) Maximum obtained concurrence between the end spins for ${\omega }_d=2B$ as a function of the dc interaction $J_0$ for various chain lengths, $J_1=0.1B$ and $\gamma =1$. The solid lines are obtained with the full time-dependent Hamiltonian (1), while the dashed lines mark the RWA solution.

Figure 3

(Color online) Entanglement dynamics for a chain of eight spins with driving frequency ${\omega }_d=2B$ for two different effective anisotropies $\tilde{\gamma }=\gamma J_1/2J_0$. As in Fig. 1, $J_1=0.1B$, $\gamma =1$. Solid lines mark the exact numerical solution, while the dashed lines are computed within RWA. The symbols mark the time evolution for switching off the driving (squares) and for changing the frequency to $\omega =3B$ (circles) after the concurrence maximum is reached at time $t_{\text{arrival}}$. Both results cannot be distinguished for the chosen resolution.

Figure 4

(Color online) Length dependence of the first maximum of the concurrence between ends, $C_{1,N}\left(t\right)$, and the corresponding fully entangled fraction $f$ for RWA Hamiltonian (5) with $J_0=0$. The inset shows the length dependence of the arrival time at which the entanglement assumes its maximum.