Enhancement of entanglement transfer in a spin chain by phase-shift control

We study the effect of a phase shift on the amount of transferable two-spin entanglement in a spin chain. We consider a ferromagnetic Heisenberg or $XY$ spin chain, both numerically and analytically, and two mechanisms to generate a phase shift, the Aharonov-Casher effect, and the Dzyaloshinskii-Moriya interaction. In both cases, the maximum attainable entanglement is shown to be significantly enhanced, suggesting its potential usefulness in quantum information processing.

Figure 1

(Color online) Examples of configurations for the Aharonov-Casher effect in a ring-shaped spin chain. In (a), the $z$ axis is taken to be parallel to the direction perpendicular to the plane that contains the spin chain. With an electric field $\vec{E}$, which is directed to the radial direction, the term $\vec{\mu }\times \vec{E}\left(\vec{x}\right)\bullet d\vec{x}$ in Eq. (5) takes its largest value. Alternatively, as in (b), the directions for the spin and the electric field can be swapped to have the same $\vec{\mu }\times \vec{E}\left(\vec{x}\right)$. The magnetic moment of spin eigenstate, $\mid \uparrow ⟩$ or $\mid \downarrow ⟩$, is parallel to the radial direction.

Figure 2

Energy spectrum $E_k=-\cos (k+\theta )$ (in the units of $J$). (a) A nonzero $\theta$ changes the dispersion relation represented with open circles (no phase shift) to the one with filled circles $(\theta \ne 0)$. The number of sites $N$ is taken to be 8. (b) The energy spectrum when $N$ is large.

Figure 3

(Color online) Schematic picture of entanglement transfer in a spin chain. (a) The configuration of the Heisenberg ring and the entanglement it has at $t=0$, with a spatially separated system that is here represented as the 0th spin. (b) The ideal goal of the entanglement transfer operation: we wish to transfer as much entanglement with the 0th spin as possible to a specific (target) spin in the chain.

Figure 4

(Color online) An example of plots of the concurrence $C$ transferred in a five-spin chain (ring) in the presence of a phase shift. The initial entanglement [of the form $(\mid 01⟩+\mid 10⟩)∕\sqrt{2}$] is in the first spin of the ring and an isolated (0th) spin. The phase shift $\theta$ is an extra phase a magnon acquires when traveling to a neighboring site. The concurrence $C$ is evaluated for the pair of the 0th and third spins. Units for $t$ and $\theta$ are $\hslash$ seconds and radian, respectively, where $\hslash$ is the Planck constant.

Figure 5

(Color online) Comparison of the maximum concurrence attainable $C_{\max }$ between the cases with and/or without phase shift. The blue plots (diamonds) show the concurrence between the 0th and $l$th spins when the 0th and first spins have the entanglement $(\mid 01⟩+\mid 10⟩)∕\sqrt{2}$ at $t=0$. The filled and open diamonds correspond to nonzero and zero phase shift, respectively. The red plots (triangles) are for the concurrence between the $l$th and $(l+1)$th spins when the first and the second spins are initially entangled in the same form. The horizontal axis represents the total number $N$ of sites in the chain and the location $l$ $(2\le l\le N)$ at which the concurrence is evaluated.