Entanglement of neutral-atom chains by spin-exchange Rydberg interaction

Conditions to achieve an unusually strong Rydberg spin-exchange interaction are investigated and proposed as a means to generate pairwise entanglement and realize a swap-like quantum gate for neutral atoms. Ground-state entanglement is created by mapping entangled Rydberg states to ground states using optical techniques. A protocol involving swap-gate and pairwise entanglement operations is predicted to create global entanglement of a chain of $N$ atoms in a time that is independent of $N$.

Figure 1

Schematic illustration for generating entanglement between two ${}^{87}\mathrm{Rb}$ atoms. (a) Geometry considered with atoms (black dots) in tightly focused optical traps (shaded regions) along the quantization axis $\mathbf{z}$. (b) Laser excitation of atom $\alpha =A,B$ by two-photon transitions. The two-photon transition via the $5P$ ($6P$) state of atom $\alpha$ has an effective Rabi frequency ${\Omega }_{\alpha }$ (${\Omega }^{\alpha }$). (c) Relevant levels in the excitation of Rydberg Bell state $|r_{+}\rangle$.

Figure 2

Schematic of the entanglement protocol via pulses 1, 2, and 3. During pulse 1 (2), only the two-photon transition via $5P$ ($6P$) state of atom $A$ ($B$) is excited with Rabi frequency ${\Omega }_A$ (${\Omega }^B$). During pulse 3, the four two-photon transitions via $5P$ and $6P$ states of atoms $A$ and $B$ are turned on and the magnitudes of these two-photon Rabi frequencies are set equal to $\Omega$. In the left column we show the bare two-atom basis, while on the right we highlight the superposition states involved.

Figure 3

Evolution of the two-atom state during pulses 2 and 3 of the entanglement protocol, starting from $|\Uparrow \uparrow \rangle$. We plot the populations of the relevant states from the basis in Eq. (7): The red (gray) solid line is $|{\mathcal{G}}_{+}{|}^2$; the blue (dark gray) dashed line is $|{\mathcal{R}}_{+}{|}^2$; the green (light gray) dotted line is ${\sum }_{\eta }\left[\right|{\mathcal{U}}_{\eta }{|}^2+|{\mathcal{D}}_{\eta }{|}^2]$.

Figure 4

Schematic of the swap-like gate protocol. During the $2\pi$ pulse, $|r_{-}\rangle$, $|\Downarrow \Downarrow \rangle$ and $|\Uparrow \Uparrow \rangle$ are hardly excited due to Rydberg blockade, while $|r_{+}\rangle$ will be excited, forming a $\Lambda$ system with $|\downarrow \Downarrow \rangle$ and $|\uparrow \Uparrow \rangle$.

Figure 5

Schematic illustration of a protocol to generate entanglement between 12 ${}^{87}\mathrm{Rb}$ atoms loaded in a one-dimensional lattice. The pairwise entanglement protocol creates entanglement between atoms $A_j$ and $B_j$ ($C_j$ and $D_j$) during step 1 (2), while the swap-like gate protocol entangles atoms $B_j$ and $C_j$ ($D_j$ and $A_{j+1}$) during step 3 (4).

Figure 6

Schematic illustration for generating an effective right-hand or left-hand polarized light field from two linearly polarized light fields related by a $\pi /2$ or $3\pi /2$ phase difference.

Figure 7

Probability distribution of fidelity for the ground Bell state when each of the four two-photon Rabi frequencies in pulse 3 obeys a uniform distribution in $[1-\epsilon ,1+\epsilon ]\Omega$, where $\epsilon =0.1\left(0.2\right)$ for the open circles and dots.

Figure 8

State evolution starting from $|\uparrow \Uparrow \rangle$. The (red) solid, (blue) dashed, (green) dotted, and (magenta) dash-dotted curves represent population on state ${|\uparrow \Uparrow \rangle }_{B_1,C_1}$, ${|\downarrow \Downarrow \rangle }_{B_1,C_1}$, $|r_{+}{\rangle }_{B_1,C_1}$, and $|r_{-}{\rangle }_{B_1,C_1}$, respectively. Here $\Omega =89$ kHz. At the end of pulse duration $t_{\text{pulse}8}=11.12\mu \text{s}$ (this duration is $0.12\mu \text{s}$ shorter than $2\pi /\Omega$), ${|\langle \Psi |\downarrow \Downarrow \rangle }_{B_1,C_1}{|}^2=96.649\%$.