Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits

We present experimental results on two-qubit Rydberg-blockade quantum gates and entanglement in a two-dimensional qubit array. Without postselection against atom loss we achieve a Bell state fidelity of $0.73\pm 0.05$. The experiments are performed in an array of single Cs atom qubits with a site to site spacing of $3.8\mu \mathrm{m}$. Using the standard protocol for a Rydberg-blockade $C_Z$ gate together with single qubit operations we create Bell states and measure their fidelity using parity oscillations. We analyze the role of ac Stark shifts that occur when using two-photon Rydberg excitation and show how to tune experimental conditions for optimal gate fidelity.

Figure 1

Level structure (left) and pulse sequence (right) for two-qubit Rydberg-blockade $C_Z$ gate. The qubits are encoded in ground hyperfine states $|0\rangle ,|1\rangle$ while $|r\rangle$ is a high lying Rydberg state.

Figure 2

Level diagram (a) and ac Stark shifts (b) for two-photon Rydberg excitation.

Figure 3

Experimental data used to determine optical beam powers. The figure shows measurements on the control site only and fitted frequencies are indicated in the plots. (a) Ground-state Ramsey oscillations with 459-nm laser used to extract ${\mathrm{\Omega }}_1$. (b) Ground-Rydberg Rabi oscillations used to extract ${\mathrm{\Omega }}_2$ once ${\mathrm{\Omega }}_1$ and ${\mathrm{\Delta }}_1$ are known. (c) Ground-Rydberg Ramsey experiment used to measure ${\mathrm{\Delta }}_{\mathrm{diff},\mathrm{gR}}$. The straight line is an aid to the eye and is not a fit, but suggests a frequency $<0.1\mathrm{MHz}$. The inset shows a faster ground-Rydberg Ramsey oscillation using other parameters. Beam powers extracted from these measurements are given in Table 2.

Figure 4

Experimental $C_X$ eye diagram on next-nearest-neighbor sites separated by $7.6\mu \mathrm{m}$. The blue curve is from loading an atom in both control and target sites and represents a blockaded data set. The red curve is from data with no atom in the control site. The two curves have a $\pi$ phase shift with respect to each other as expected. The phase at the minimum of the no blockade curve is ${\varphi }_{01}-{\varphi }_{00}$ which gives the $C_{X,\mathrm{tc}}$ gate.

Figure 5

State preparation (left) and ${\overline{C}}_X$ gate population matrix (right) on next-nearest-neighbor sites, 7.6 $\mu \mathrm{m}$ apart, using parameters measured in Sec. 3b. The average statistical uncertainty of the data is $\pm 0.008$ for the state preparation and $\pm 0.02$ for the ${\overline{C}}_X$ gate.

Figure 6

Bell state population measurement and parity oscillation measurement with no loss correction using next-nearest-neighbor sites. The populations measured for the Bell state are $|00\rangle :0.54\pm 0.06, |01\rangle :0.03\pm 0.02, |10\rangle :0.05\pm 0.03, |11\rangle :0.38\pm 0.06$. The fit to the parity curve gives $|C_1|=0.27\pm 0.02$ and $|C_2|=0.006$. This results in an entanglement fidelity $F=0.73\pm 0.05$ and $F=0.79\pm 0.05$ when corrected for atom loss.

Figure 7

The entanglement fidelity as a function of the fractional change of the optical powers $P_{459},P_{1038}$ and the detuning ${\mathrm{\Delta }}_1$ from the optimal values. The center point of each plot uses the parameters found experimentally in Sec. 3b and the color scale is the fidelity normalized by the value at the center of each panel.

Figure 8

Ground-state population after a $2\pi$ Rydberg pulse (a) and after a $\pi -$ gap $-\pi$ Rydberg pulse sequence with variable gap time (b).

Figure 9

Calculated ground-Rabi frequency ${\mathrm{\Omega }}_{\mathrm{R}}$ (left), resonant Stark shift of $|6s_{1/2},f=4\rangle$ due to the 459-nm light ${\mathrm{\Delta }}_{11}^{\mathrm{r}}$ (center), and total ground-Rydberg differential Stark shift ${\mathrm{\Delta }}_{\mathrm{gR}}$ including resonant and nonresonant contributions (right), for two-photon excitation of Cs $|82s,m_j=-1/2\rangle$ starting in $|4,0\rangle$ using parameters from Table 2 for the control site. Solid blue curves account for the $7p_{1/2}$ hyperfine structure, while the dashed yellow curves do not include hyperfine corrections. The spontaneous emission plot only accounts for scattering from $7p_{1/2}$ as the Rydberg state decay is negligible for the parameters used. The gate experiments used ${\nu }_1=0.83\mathrm{GHz}$.