Single temporal-pulse-modulated parameterized controlled-phase gate for Rydberg atoms

We propose an adiabatic protocol for implementing a controlled-phase gate CZ$_{\theta}$ with continuous $\theta$ of neutral atoms through a symmetrical two-photon excitation process via the second resonance line, $6P$ in $^{87}$Rb, with a single-temporal-modulation-coupling of the ground state and intermediate state. Relying on different adiabatic paths, the phase factor $\theta$ of CZ$_{\theta}$ gate can be accumulated on the logic qubit state $|11\rangle$ alone by calibrating the shape of the temporal pulse where strict zero amplitudes at the start and end of the pulse are not needed. For a wide range of $\theta$, we can obtain the fidelity of CZ$_{\theta}$ gate over $99.7\%$ in less than $1~\mu$s, in the presence of spontaneous emission from intermediate and Rydberg states. And in particular for $\theta=\pi$, we benchmark the performance of the CZ gate by taking into account various experimental imperfections, such as Doppler shifts, fluctuation of Rydberg-Rydberg interaction strength, inhomogeneous Rabi frequency, and noise of driving fields, etc, and show that the predicted fidelity is able to maintain at about $98.4\%$ after correcting the measurement error. This gate protocol provides a robustness against the fluctuation of pulse amplitude and a flexible way for adjusting the entangling phase, which may contribute to the experimental implementation of near-term noisy intermediate-scale quantum (NISQ) computation and algorithm with neutral-atom systems.