Fundamental bound on entanglement generation between interacting Rydberg atoms

Abstract

We analytically derive the fundamental lower bound for the preparation fidelity of a maximally-entangled (Bell) state of two atoms involving Rydberg-state interactions. This bound represents the minimum achievable error $E \geq ( 1 + π/2 ) Γ/B$ due to spontaneous decay $Γ$ of the Rydberg states and their finite interaction strength $B$. Using quantum optimal control methods, we identify laser pulses for preparing a maximally-entangled state of a pair of atomic qubits with an error only $1\%$ above the derived fundamental bound.

Figures (1)

• Figure 1: (a) Level scheme of two two-level atoms with the transition between the stable ground $\ket{g}$ and decaying Rydberg $\ket{r}$ states driven by a laser with Rabi frequency $\Omega$ and detuning $\Delta$. Atoms in state $\ket{r}$ interact dispersively with strength $B$. The system is equivalent to a three-level ladder system with states $\ket{gg}, \, \ket{W}=(\ket{gr} + \ket{rg})/\sqrt{2}, \, \ket{rr}$. (b) Scaling coefficient $\eta$ (with $\eta_{\min} =1+\pi/2$) vs laser pulse duration $T$, as obtained by GRAPE optimization of preparation of target state $\ket{\psi_{\mathrm{f}}}$. Inset shows the preparation fidelity $F$ in the absence of decay. (c) Temporal profiles of the Rabi frequency $\Omega(t)$ (solid cyan line) and detuning $\Delta(t)$ (blue dashed line) for an optimal pulse of duration $BT=6.8$ and area $\Theta\simeq2.31$, and the corresponding dynamics of populations of states $\ket{gg}, \, \ket{W}, \, \ket{rr}$. For this pulse, the corresponding fidelity is $F \simeq 4 \times 10^{-8}$ and scaling coefficient is $\eta \simeq 2.575$. (d) Comparison between the theoretical bounds (solid lines) and the solution identified by GRAPE optimization (open circles). The GRAPE solution $\ket{\psi}$ closely follows the theoretical minimum $P_r(\psi)/\dot{S}(\psi)\simeq G ( S(\psi))$ of Eq. (\ref{['eq:GofS']}), despite the finite pulse duration $T$. In contrast, $1/\dot{S}(\psi)$ diverges for $S \to 0$ more slowly than the theoretical result, which requires $BT\to\infty$ as per Eq. (\ref{['eq:Tint']}).