Robustness of high-fidelity Rydberg gates with single-site addressability

Controlled-phase (cphase) gates can be realized with trapped neutral atoms by making use of the Rydberg blockade. Achieving the ultrahigh fidelities required for quantum computation with such Rydberg gates, however, is compromised by experimental inaccuracies in pulse amplitudes and timings, as well as by stray fields that cause fluctuations of the Rydberg levels. We report here a comparative study of analytic and numerical pulse sequences for the Rydberg cphase gate that specifically examines the robustness of the gate fidelity with respect to such experimental perturbations. Analytical pulse sequences of both simultaneous and stimulated Raman adiabatic passage (STIRAP) are found to be at best moderately robust under these perturbations. In contrast, optimal control theory is seen to allow generation of numerical pulses that are inherently robust within a predefined tolerance window. The resulting numerical pulse shapes display simple modulation patterns and can be rationalized in terms of an interference between distinct two-photon Rydberg excitation pathways. Pulses of such low complexity should be experimentally feasible, allowing gate fidelities of order 99.90–99.99% to be achievable under realistic experimental conditions.

Figure 1

Level scheme for a single atom. The color scheme given here in the online version, blue (darker) for the lower transition and red (lighter) for the upper one, is used throughout the figures of this paper.

Figure 2

Three sequential Blackman pulse pairs implementing a cphase gate. The pulses for the transition $|0\rangle \to |i\rangle$ and $|i\rangle \to |r\rangle$ are acting simultaneously with identical shape and amplitude.

Figure 3

Quantum speed limit for the Rydberg gate using simultaneous Blackman pulse pairs. The time window is only that of the center $2\pi$ pulse in the scheme. As a measure of the breakdown of the Rydberg blockade, the maximum population in the $\left|rr\right.〉$ state during that pulse is shown, as well as the maximum population in the $\left|01\right.〉$ state, as a measure of the breakdown of the adiabatic elimination of the intermediate level. Finally, we show the total gate error obtained when combining the center $2\pi$ pulse of the given duration with two 50-ns $\pi$ pulses on the left atom.

Figure 4

Population and phase dynamics using the simultaneous pulses shown in Fig. 2. Since the population in the intermediary states $\left|1i\right.〉,\left|i1\right.〉,\left|i0\right.〉$ are effectively zero throughout, there are not included in the phase dynamics in panels (d)–(f).

Figure 5

Breakdown of the Rydberg blockade for STIRAP: Only long gate durations allow for amplitudes that are sufficiently large to ensure adiabaticity in STIRAP while being small enough not to break the Rydberg blockade (lower panel).The amplitude of the two central pulse pairs are systematically scanned while the amplitude of the first and last pulse pairs are kept constant.

Figure 6

A sequence of STIRAP pulse pairs to implement the Rydberg cphase gate. The pulse shape for the “blue” transition is given by the solid line, the pulse shape for the “red” transition by the dashed line. While the pulses acting on the left atom can be made very short (limited effectively by the power of the driving laser), the pulses acting on the right atom need to be sufficiently long to avoid breaking the Rydberg blockade.

Figure 7

Mixed scheme: STIRAP pulse pairs for robust population transfer on the left atom (solid and dashed line for “blue” and “red” transition, respectively), and simultaneous pulses for the $2\pi$ rotation of the right atom (identical shape and amplitude for “red” and “blue” transition).

Figure 8

Robustness of the Rydberg gate with respect to Rydberg level fluctuations (top), amplitude fluctuations (middle), and fluctuations in the relative timing between pulses acting on the left and right atoms. All fluctuations are drawn from a Gaussian distribution of width ${\sigma }_{\mathrm{ryd}},{\sigma }_{\Omega }$, and ${\sigma }_{\mathrm{time}}$, respectively. Note that the (%) in the middle panel refers to the percent by which the each pulse was uniformly scaled down. For the Rydberg level, ${\sigma }_{\mathrm{ryd}}=150$ kHz represents a $0.3\%$ variation of $u=57$ MHz.

Figure 9

Amplitudes and spectra of pulses optimized with respect to variations in both two-photon detuning and pulse amplitude, for a gate duration of $T=800$ ns. The pulses and spectra driving the “red” and “blue” transitions are colored as in Fig. 1. The central peaks in the spectra are truncated to emphasize the side peaks. In panel (c), the amplitudes reach a value of 2.0 (red pulse) and 0.8 (blue pulse). In panel (d), the peaks reach 1.4 (red pulse) and 1.0 (blue pulse). Pulses and spectra are shown in the two-color rotating frame. The central frequency of zero corresponds to a laser frequency of the blue pulse that is detuned by ${\Delta }_1$ with respect to the $\left|0\right.〉\to \left|i\right.〉$ transition. For the red pulse, it indicates the frequency for which there is a two-photon resonance with the $\left|0\right.〉\to \left|r\right.〉$ transition. The frequencies matching $\pm {\Delta }_1$ are indicated by vertical dashed gray lines. In panels (c) and (d), the left side-peak is for the red laser, the right side-peak for the blue laser.

Figure 10

Dynamics under the pulses optimized with respect to fluctuations in both the Rydberg level and pulse amplitudes, as shown in Fig. 9. The pulses and spectra driving the “red” and “blue” transitions are colored as in Fig. 1. The intermediate population in the bottom panel (“int”) is integrated over the states $\left|0i\right.〉,\left|i0\right.〉,\left|ii\right.〉,\left|ir\right.〉$, and $\left|ri\right.〉$. The shown dynamics implement the desired cpase gate up to a gate error of $1.04\times {10}^{-4}$. The side-peaks in panels (c) and (d) show the same behavior as those in Fig. 9.

Figure 11

Amplitudes and spectra of pulses optimized with respect to variations in both two-photon detuning and pulse amplitude, for a gate duration of $T=100$ ns. The spectra are drawn on the same scale as in Fig. 9, with the central peaks in panel (c) reaching 2.9 (blue pulse) and 3.0 (red pulse), and 2.5 for both pulses in panel (d).

Figure 12

Dynamics under the optimized pulses shown in Fig. 11. The gate error is $1.92\times {10}^{-4}$.

Figure 13

Expectation value of the gate error in the presence of fluctuations in the $\left|rr\right.〉$ state due to dc electric fields (top), and pulse amplitude fluctuations (bottom). The dashed curve shows the most robust analytical pulse; cf. the dashed curve in Fig. 8. The solid yellow (light gray) and blue (dark gray) lines are for the optimized pulses shown in Figs. 9 and 11, respectively. The dotted line is for a further optimized pulse at $T=800$ ns, without any consideration of limits on the pulse amplitude or complexity. Note that both panels show the robustness for same set of pulses; i.e., the pulses were optimized with respect to both variations in the two-photon detuning and the pulse amplitude.