Counterdiabatic driving at Rydberg excitation for symmetric $C_Z$ gates with ultracold neutral atoms

TL;DR

This work develops and analyzes counterdiabatic driving for symmetric $C_Z$ gates with Rydberg blockade in ultracold neutral atoms. By embedding a time-dependent CD term into analytically shaped ARP pulses, it achieves substantial gate-speedups—down to $2T\approx 0.1$ µs—while maintaining high entanglement fidelity. The study covers single-photon, two-photon, and three-photon excitation schemes, showing near-unit Bell fidelities (up to $\mathcal{F}\simeq 0.9999$ for single-photon, $\mathcal{F}\approx 0.998$–$0.996$ for two-/three-photon) under realistic lifetimes and blockade strengths, with the three-photon scheme offering Doppler- and addressing-related advantages. Overall, the analytic pulse profiles and the demonstrated robustness position CD driving as a practical route to fast, high-fidelity symmetric $C_Z$ gates in neutral-atom quantum processors.

Abstract

We extend the scheme of neutral atom Rydberg $C_Z$ gate based on double sequence of adiabatic pulses applied symmetrically to both atoms using counterdiabatic driving in the regime of Rydberg blockade. This provides substantial reducing of quantum gate operation times (at least five times) compared to previously proposed adiabatic schemes, which is important for high-fidelity entanglement due to finite Rydberg lifetimes. We analyzed schemes of adiabatic rapid passage with counterdiabatic driving for single-photon, two-photon and three-photon schemes of Rydberg excitation for rubidium and cesium atoms. We designed laser pulse profiles with fully analytical shapes and calculated the Bell fidelity taking into account atomic lifetimes and finite blockade strengths. We show that the upper limit of the Bell fidelity reaches ${\mathcal F}\simeq0.9999$ in a room-temperature environment.

Figures (7)

• Figure 1: (color online) (a) Energy level structure of atom qubit with logical states represented by ground state sublevels $|0\rangle,|1\rangle$ and Rydberg state $|r\rangle$ coupled to state $|1\rangle$ by resonant laser radiation with Rabi frequency $\Omega_0\left(t\right)$ and detuning $\delta\left(t\right)$. (b) Rydberg atoms in state $|r\rangle$ interact with strength $\sf{B}$ which results in Rydberg blockade. When two nearby atoms in ground state $|1\rangle$ are illuminated by resonant laser radiation, only one atom is excited to Rydberg state. The two-atom system is effectively a two-level system with states $|11\rangle$ and $\frac{1}{\sqrt{2}}\left(|r1\rangle+|1r\rangle\right)$ with enhanced coupling $\sqrt{2}\Omega_0\left(t\right)$ between them. (c) Double adiabatic sequence results in accumulation of $\pi$ phase shift after excitation and de-excitation of two-atom system prepared initially in each of the states $|01\rangle$, $|10\rangle$, $|11\rangle$. The state $|00\rangle$ remains unaffected. The phase-shifted pulse $\Omega_{\rm{CD}}\left(t\right)$ represents a counterdiabatic term which is necessary to speed-up the gate.
• Figure 2: (color online) (a) Rabi frequency $\Omega_0\left(t\right)$ (blue) and counterdiabatic drive $\Omega_{CD}\left(t\right)$ (red). (b) Detuning $\delta\left(t\right)$. (c) Population $P_{01}$ of the ground state $|01\rangle$. (d) Population $P_{11}$ of the ground state $|11\rangle$ with pulse parameters from Eq. (\ref{['eq10']}) (solid line) and from Eq. (\ref{['eq14']}) (dashed line). (e) Phase $\varphi_{01}$ of the ground state $|01\rangle$. (f) Phase $\varphi_{11}$ of the ground state $|11\rangle$ with pulse parameters from Eq. (\ref{['eq10']}) (solid line) and from Eq. (\ref{['eq14']}) (dashed line). The gate time $2T$=0.1$~\mu$s defines Rabi frequency $\Omega_{0\rm max}=2 \pi/T$ and detuning $\delta_0= \pi/T$ of the pulses from Eq. (\ref{['eq15']}), which are centered at $\pm T/2$ and have $w=T/4$.
• Figure 3: (color online) (a) Density plot of Bell fidelity as function of Rabi frequency $\Omega_{0 \rm max}$ and detuning $\delta_0$. (b) Dependence of Bell state infidelity on the ratio between the driving Rabi frequency $\Omega_0$ and optimal value of the Rabi frequency $\Omega_0^{\rm opt}$, separately calculated for each $C_Z$ scheme for the gate time $2T=0.1\,\mu\rm s$: our gate scheme (squares), Levine-Pichler gate (triangles) and time-optimal gate (circles). (c) Dependence of Bell state infidelity using $C_Z$ gate with ARP pulses on blockade strength $\sf B$.
• Figure 4: (color online) (a) Scheme of the atomic energy levels used in numeric simulation. (b) Populations of the $|10\rangle$, $|r0\rangle$ and $|d\rangle$ states for the initial state $|10\rangle$. The population in $|d\rangle$ is defined as $1-{\rm Tr}_{0,1,r}[\rho]$. (c) Populations of the $|11\rangle$, $|1r\rangle+|r1\rangle$, $|rr\rangle$ and $|d\rangle$ states for the initial state $|11\rangle$. (d) Dependence of infidelity of Bell state on blockade strength $\sf{B}$.
• Figure 5: (color online) Scheme of two-photon Rydberg excitation in Cs and Rb atoms with Rabi frequencies $\Omega_1(t), \Omega_2(t)$ via intermediate state $|p\rangle$ at detuning $\Delta$.