Quench-induced growth of distant entanglement from product and locally entangled states in spin chains

We study the problem of entangling two spins at the distant ends of a spin chain by exploiting the nonequilibrium dynamics of the system after a sudden global quench. As initial states we consider a canted or spiral order product state of the spins and singlets of neighboring pairs of spins. We find that within the class of canted order initial states, no entanglement is generated at any time except for the special case of the Néel state. While an earlier work has shown that the Néel state is indeed an excellent starting resource for the dynamical generation of long-distance entanglement, the curious fact that this is the sole point within a large class of initial product states of the spins was not noted. On the other hand, we find that an initial state which is a series of nearest-neighbor Bell states, and well motivated by some physical realizations, is also a good starting resource for end to end entanglement in a way similar to that of the Néel state. The scheme is shown to be robust to random single spin flip in the initial Néel state as well as to randomness of the couplings.

Figure 1

This figure shows the canted order state considered as an initial state [Eq. (2)].

Figure 2

Dynamics of the concurrence and the fully entangled fraction of spins 1 and $N$ of the chain for $N=24$ as a function of the canting angle $\alpha$. The scaled time $t$ is in units of $1/J$.

Figure 3

Dynamics of the concurrence and the fully entangled fraction of spins 1 and $N$ of the chain for $N=50$ as a function of the canting angle $\alpha$. The scaled time $t$ is in units of $1/J$.

Figure 4

The full width at half maximum $\Delta \alpha$ of the peak in entanglement around $\alpha =\pi$ at the optimal time versus the chain length.

Figure 5

Dynamics of the concurrence and the fully entangled fraction of spins 1 and $N$ of the chain for $N=24$ spins with a series of Bell pairs as an initial state. The scaled time $t$ is in units of $1/J$.

Figure 6

Dynamics of the concurrence and the fully entangled fraction of spins 1 and $N$ of the chain for $N=50$ spins with a series of Bell pairs as an initial state. The scaled time $t$ is in units of $1/J$.

Figure 7

The fully entangled fraction for an initial Néel state with a random single spin flip with probability $N\epsilon =0$, 0.05, 0.1, and 0.15 (top to bottom) for (a) $N=24$ and (b) $N=50$.

Figure 8

The concurrence and fully entangled fraction for a series of Bell pairs as an initial state with random offsets in the couplings through the chain $\delta =0$, 0.1, and 0.2 (top to bottom) for $N=10$ (average taken over 100 realizations).

Figure 9

(a) Maximum fully entangled fraction of ${\rho }_{1,N}$ versus the chain length. (b) The time at which the first peak of entanglement occurs versus the chain length.

Figure 10

This figure shows the quantum walk of excitations (fermions) starting from an initial state [Eq. (2)]. The $k=7$ fermion is shown to generate an equal distribution of probabilities at either end at time $t\sim \frac{N}{4J}$, whereas the $k=3$ fermion generates a state with unequal probabilities at the ends.