High-fidelity Rydberg quantum gate via a two-atom dark state

We propose a two-qubit gate for neutral atoms in which one of the logical state components adiabatically follows a two-atom dark state formed by the laser coupling to a Rydberg state and a strong resonant dipole-dipole exchange interaction between two Rydberg excited atoms. Our gate exhibits optimal scaling of the intrinsic error probability $E\propto {\left(B\tau \right)}^{-1}$ with the interatomic interaction strength $B$ and the Rydberg state lifetime $\tau$. Moreover, the gate is resilient to variations in the interaction strength, and even for finite probability of double Rydberg excitation the gate does not excite atomic motion and experiences no decoherence due to internal-translational entanglement.

Figure 1

Level scheme of two atoms leading to a Rydberg phase gate by three focused laser pulses. States $|0_{\mathrm{c},\mathrm{t}}\rangle$ and $|1_{\mathrm{c},\mathrm{t}}\rangle$ are long-lived qubit basis states of the control ($\mathrm{c}$) and target ($\mathrm{t}$) atoms, and states $|r_{\mathrm{c},\mathrm{t}}\rangle$ and $|a_{\mathrm{c}}\rangle$, $|b_{\mathrm{t}}\rangle$ are Rydberg states with decay rate $\mathrm{\Gamma }$. In steps i and iii the control atom in state $|1_{\mathrm{c}}\rangle$ is resonantly excited and deexcited to state $|r_{\mathrm{c}}\rangle$ by a laser with Rabi frequency ${\mathrm{\Omega }}_{\mathrm{c}}$ and pulse area $\pi$, and in step ii the target atom in state $|1_{\mathrm{t}}\rangle$ is resonantly coupled to state $|r_{\mathrm{t}}\rangle$ by another laser with Rabi frequency ${\mathrm{\Omega }}_{\mathrm{t}}$ and pulse area $2\pi$. Atoms excited to Rydberg states $|r_{\mathrm{c}}\rangle$ and $|r_{\mathrm{t}}\rangle$ strongly interact with each other, $B\gg {\mathrm{\Omega }}_{\mathrm{t}}$, via resonant dipole-dipole exchange process $|r_{\mathrm{c}}{r}_{\mathrm{t}}\rangle \leftrightarrow |a_{\mathrm{c}}{b}_{\mathrm{t}}\rangle$ which leads to suppression of excitation of the target atom in step ii if the control was initially in state $|1_{\mathrm{c}}\rangle$. We assume that atoms in state $|0\rangle$ are decoupled from the laser fields.

Figure 2

Evolution of the two-atom system during step ii of the Rydberg-exchange phase gate. Left: The initial two-qubit state $|0_{\mathrm{c}}{1}_{\mathrm{t}}\rangle$ remains unchanged after step i, and during step ii the target atom undergoes a $2\pi$ Rabi cycle via state $|0_{\mathrm{c}}{r}_{\mathrm{t}}\rangle$. Middle: The initial two-qubit state $|1_{\mathrm{c}}{1}_{\mathrm{t}}\rangle$ is converted to $|r_{\mathrm{c}}{1}_{\mathrm{t}}\rangle$ after step i, and during step ii the resonant dipole-dipole exchange interaction $|r_{\mathrm{c}}{r}_{\mathrm{t}}\rangle \leftrightarrow |a_{\mathrm{c}}{b}_{\mathrm{t}}\rangle$ with strength $B\gg {\mathrm{\Omega }}_{\mathrm{t}}$ leads to the formation of a two-atom dark state $|{\psi }_0\rangle$ and two bright states $|{\psi }_{\pm }\rangle$ shifted by $\pm B$ (right). In our protocol, state $|r_{\mathrm{c}}{1}_{\mathrm{t}}\rangle$ adiabatically follows the dark state $|{\psi }_0\rangle$ as the target pulse ${\mathrm{\Omega }}_{\mathrm{t}}$ is turned on and off.

Figure 3

Error probabilities $E=(1-F)$ of the phase gate, averaged over all the input states, versus $B\tau$. Filled blue circles correspond to smooth $2\pi$ laser pulses ${\mathrm{\Omega }}_{\mathrm{t}}\left(t\right)$ applied to the target qubit, while empty red circles correspond to nonadiabatic (square) pulses of the same area and duration. The results are obtained by numerical simulation of the dynamics of the two-atom system with a non-Hermitian Hamiltonian, as described in Appendix pp2. The solid blue line shows $E=\eta /\left(B\tau \right)$ and the dashed red line shows $E=\eta /\left(B\tau \right)+{\alpha }^2/16$. All Rydberg states decay with the same rate $\mathrm{\Gamma }$, while qubit states $|0\rangle$ and $|1\rangle$ do not decay. We used $\alpha \equiv {\mathrm{\Omega }}_{\mathrm{t}0}/B=0.10472$ ($\eta =37.5$) resulting in the nonadiabatic transition errors $E_{\mathrm{na}}\simeq 2\times {10}^{-6}$ for smooth (shifted Gaussian) pulses.

Figure 4

Energy-level structure of Rydberg states for Stark tuned Förster interaction. Resonant lasers couple the qubit states $|1_{\mathrm{c},\mathrm{t}}\rangle$ of the control and target atoms to the corresponding Rydberg states $|r_{\mathrm{c},\mathrm{t}}\rangle$ (see Fig. 1). A static electric field shifts the Rydberg states making the two-atom states $|r_{\mathrm{c}}{r}_{\mathrm{t}}\rangle$ and $|a_{\mathrm{c}}{b}_{\mathrm{t}}\rangle$ energy degenerate and resonantly coupled via exchange interaction with strength $B$. The forward leakage channel couples $|r_{\mathrm{c}}{r}_{\mathrm{t}}\rangle$ to $|a_{\mathrm{c}}^{\prime }{b}_{\mathrm{t}}^{\prime }\rangle$ with strength $B_{rr}$ and energy defect $\delta {\omega }_{rr}$. The backward leakage channel couples $|a_{\mathrm{c}}{b}_{\mathrm{t}}\rangle$ to $|b_{\mathrm{c}}^{\prime }{a}_{\mathrm{t}}^{\prime }\rangle$ with strength $B_{ab}$ and energy defect $\delta {\omega }_{ab}$.

Figure 5

Schematics of the conventional Rydberg blockade phase gate between the control and target atoms [16]. Atoms in Rydberg states $|r_{\mathrm{c},\mathrm{t}}\rangle$ strongly interact with each other via the static or nonresonant dipole-dipole interaction leading to level shift $B_{\mathrm{sh}}(\gg {\mathrm{\Omega }}_{\mathrm{t}})$ of state $|r_{\mathrm{c}}{r}_{\mathrm{t}}\rangle$.