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arXiv:1808.03473 · Physical Review A

Fast three-qubit Toffoli quantum gate based on three-body Förster resonances in Rydberg atoms

Ilya I. Beterov, I. N. Ashkarin, Elena A. Yakshina, Denis B. Tretyakov, Vasily M. Entin, I. I. Ryabtsev, Patrick Cheinet, P. Pillet, Mark E. Saffman

Published Oct 17, 2018• 56 citations
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Abstract

We propose a scheme of fast three-qubit Toffoli quantum gate for ultracold neutral-atom qubits. The scheme is based on Stark-tuned three-body F\"orster resonances, which we have observed in our recent experiment [D. B. Tretyakov et al., Phys. Rev. Lett. 119, 173402 (2017)]. The three-body resonance corresponds to a transition when the three interacting atoms change their states simultaneously, and it occurs at a different dc electric field with respect to the two-body F\"orster resonance. A combined effect of three-body and two-body F\"orster interactions in external electric and magnetic fields near the three-body resonance results in complex coherent behavior of the populations and phases of collective states of a three-atom system. We have found that it is possible to obtain experimental conditions suitable to implement three-qubit Toffoli gate with 98.3% fidelity and less than 3 $\ensuremath{\mu}\mathrm{s}$ duration.

DOI
10.1103/physreva.98.042704

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We propose a scheme of fast three-qubit Toffoli quantum gate for ultracold neutral-atom qubits. The scheme is based on Stark-tuned three-body F"orster resonances, which we have observed in our recent experiment [D. B. Tretyakov et al., Phys. Rev. Lett. 119, 173402 (2017)]. The three-body resonance corresponds to a transition when the three interacting atoms change their states simultaneously, and it occurs at a different dc electric field with respect to the two-body F"orster resonance. A combined effect of three-body and two-body F"orster interactions in external electric and magnetic fields near the three-body resonance results in complex coherent behavior of the populations and phases of collective states of a three-atom system. We have found that it is possible to obtain experimental conditions suitable to implement three-qubit Toffoli gate with 98.3% fidelity and less than 3 \ensuremathμs\ensuremath{\mu}\mathrm{s}\ensuremathμs duration.
We have developed an improved scheme of a three-qubit Toffoli gate based on a fine-structure-state-changing three-body Stark-tuned Rydberg interaction. This scheme is a substantial improvement of our previous proposal [I. I. Beterov et al., Phys. Rev. A 98, 042704 (2018)]. Due to the use of a different type of three-body F\"orster resonance we substantially simplified the scheme of laser excitation and phase dynamics of collective three-body states. This type of F\"orster resonance exists only in systems with more than two atoms, while the two-body resonance is absent. We reduced the sensitivity of the gate fidelity to fluctuations of external electric field and eliminated the necessity to use external magnetic field for fine-tuning of the resonant electric-field value, compared to the previous scheme of Toffoli gate based on Rydberg atoms. A gate fidelity of >99% was demonstrated in the calculations.
We have observed recently the Stark-tuned three-body F\"orster resonances $3\ifmmode\times\else\texttimes\fi{}n{P}_{3/2}(|M|)\ensuremath{\rightarrow}n{S}_{1/2}+(n+1){S}_{1/2}+n{P}_{3/2}(|{M}^{*}|)$ at long-range interactions of a few cold Rb Rydberg atoms [D. B. Tretyakov et al., Phys. Rev. Lett. 119, 173402 (2017)]. The three-body resonance appears at a different dc electric field with respect to the ordinary two-body resonance $2\ifmmode\times\else\texttimes\fi{}n{P}_{3/2}(|M|)\ensuremath{\rightarrow}n{S}_{1/2}+(n+1){S}_{1/2}$ and corresponds to a transition when the three interacting atoms change their states simultaneously (two atoms go to the SSS states, and the third atom remains in the PPP state but changes its moment projection), with the negligible contribution of the two-body resonance to the population transfer. It thus has a Borromean character and represents an effective three-body operator, which can be used to directly control the three-body interactions in quantum simulations and quantum gates implemented with Rydberg atoms. In this paper we theoretically investigate the coherence of such three-body resonances and we show that high-contrast Rabi-like population oscillations are possible for the localized Rydberg atoms in a certain spatial configuration. This paves the way to implementing three-qubit quantum gates and quantum simulations based on three-body Rydberg interactions.
Application of few-body Förster resonances for implementation of multiqubit quantum gate circuits has been investigated. New types of three-atom Borromean transitions based on the relay atom have been proposed and numerically studied. In particular, a Stark-controlled non-isolated three-atom Förster resonance between high-lying n = 80, 81, 82 S − P states of Rb atoms isolated in individual optical traps has been modeled. Isolated three-atom Förster resonance has also been demonstrated for n = 70, 71 states of Rb atoms. The resonances were investigated in a fixed spatial configuration, allowing us to demonstrate the coherent population and phase dynamics of the collective states involved. Three-qubit Toffoli gates schemes have been developed and numerically modeled based on the demonstrated resonances. Also, a generalized doubly controlled phase CCPHASE gate scheme has been developed based on the radiofrequency-induced three-body Förster resonance. Additionally, a similar quantum gate scheme has been proposed based on two-atom RF-induced Förster resonance with controlled displacement. The fast performance and high fidelity of the proposed schemes, as well as their potential robustness to errors, allow us to expect a successful experimental implementation in the near future.
Three-body Förster resonances controlled by a dc electric field are of interest for the implementation of three-qubit quantum gates with single atoms captured in optical traps and laser-excited into strongly interacting Rydberg states. In Ref. [P. Cheinet et al., Quantum Electronics 50(3), 213 (2020)], we proposed and analyzed a new type of three-body Förster resonance 3×nP3/2→nS1/2+(n+1)S1/2+nP1/2{\rm 3}\times nP_{3/2} \to nS_{1/2} +(n+1)S_{1/2} +nP_{1/2}3×nP3/2​→nS1/2​+(n+1)S1/2​+nP1/2​ that can be realized with Rb Rydberg atoms for an arbitrary principal quantum number nnn. Its peculiarity is that the third atom goes into a state with a total angular moment J=1/2J=1/2J=1/2, which has no Stark structure, so two-body Förster resonances are completely absent. In the present work, an extended theoretical study of this three-body Förster resonance is performed for various spatial configurations of three interacting Rb Rydberg atoms and conditions for their experimental implementation are determined. It was found that one of the resonances has a weak dependence of the resonant electric field on the distance between atoms and is therefore most suitable for performing experiments to observe coherent oscillations of populations of collective three-body states and implement three-qubit quantum gates based on them.
Abstract The three-body Förster resonances 3 × nP3/2(|M|) →nS1/2 + (n + 1)S1/2 + nP3/2(|M*|), controlled by a constant electric field, were realised earlier by the authors in an ensemble of several cold Rydberg Rb atoms. One of the drawbacks of such resonances for potential application in three-qubit quantum gates is the proximity of the two-body Förster resonance 2 × nP3/2 → nS1/2 + (n + 1)S1/2, as well as the possibility of their implementation only for states with values of the principal quantum numbers n ⩽38. A three-body resonance of a new type, 3 × nP3/2 → nS1/2 + (n + 1)S1/2 + nP1/2, which can be realised for arbitrary n, is proposed and analysed. Its specific feature is also that the third atom transits into a state with a different total angular momentum J = 1/2, which has no Stark structure, so that the two-body resonance is completely absent. Numerical calculations showed that for not too strong interaction, it is possible to observe coherent three-body oscillations of the populations of collective states, which is of interest for developing new schemes of three-qubit quantum gates controlled by an electric field.
Three-body F\"orster resonances controlled by a dc electric field are of interest for the implementation of three-qubit quantum gates with single atoms in optical traps using their laser excitation into strongly interacting Rydberg states. In our recent theoretical paper [Zh. Eksp. Teor. Fiz. 168, 14 (2025)], it was found that the proposed earlier three-body F\"orster resonance $3\ifmmode\times\else\texttimes\fi{}n{P}_{3/2}\ensuremath{\rightarrow}n{S}_{1/2}+(n+1){S}_{1/2}+n{P}_{1/2}$ in Rb Rydberg atoms presents a splitting, with one of the split components having weaker dependence of the resonant electric field (and the corresponding dynamic shift) on the distance RRR between the atoms. Here we study this effect in more detail, since this resonance appears as the most suitable for performing experiments on observing coherent oscillations of populations of collective three-body states and implementing three-qubit quantum gates based on them. For a linear spatial configuration of three interacting Rydberg atoms, the physical mechanism of this phenomenon is revealed and analytical formulas are obtained that describe the behavior of split structure of the F\"orster resonance depending on RRR. It is found that the splitting is a measure of the energy of the resonant dipole-dipole exchange interaction with an excitation hopping between neighboring Rydberg states SSS and PPP.