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arXiv:1807.09125 · Physical Review A

Coherence of three-body Förster resonances in Rydberg atoms

I. I. Ryabtsev, Ilya I. Beterov, Denis B. Tretyakov, Elena A. Yakshina, Vasily M. Entin, Patrick Cheinet, P. Pillet

Published Nov 13, 2018• 15 citations
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Abstract

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 $S$ states, and the third atom remains in the $P$ 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.

DOI
10.1103/physreva.98.052703

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We have observed recently the Stark-tuned three-body F"orster resonances 3\ifmmode×\else\texttimes\finP3/2(∣M∣)\ensuremath→nS1/2+(n+1)S1/2+nP3/2(∣M∗∣)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}^{*}|)3\ifmmode×\else\texttimes\finP3/2​(∣M∣)\ensuremath→nS1/2​+(n+1)S1/2​+nP3/2​(∣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×\else\texttimes\finP3/2(∣M∣)\ensuremath→nS1/2+(n+1)S1/22\ifmmode\times\else\texttimes\fi{}n{P}_{3/2}(|M|)\ensuremath{\rightarrow}n{S}_{1/2}+(n+1){S}_{1/2}2\ifmmode×\else\texttimes\finP3/2​(∣M∣)\ensuremath→nS1/2​+(n+1)S1/2​ 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.
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.
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.
Three-body Förster resonances at long-range interactions of Rydberg atoms were first predicted and observed in Cs Rydberg atoms by Faoro et al. [Nat. Commun. 6, 8173 (2015)NCAOBW2041-172310.1038/ncomms9173]. In these resonances, one of the atoms carries away an energy excess preventing the two-body resonance, leading thus to a Borromean type of Förster energy transfer. But they were in fact observed as the average signal for the large number of atoms N≫1. In this Letter, we report on the first experimental observation of the three-body Förster resonances 3×nP_{3/2}(|M|)→nS_{1/2}+(n+1)S_{1/2}+nP_{3/2}(|M^{*}|) in a few Rb Rydberg atoms with n=36, 37. We have found here clear evidence that there is no signature of the three-body Förster resonance for exactly two interacting Rydberg atoms, while it is present for N=3-5 atoms. This demonstrates the assumption that three-body resonances can generalize to any Rydberg atom. As such resonance represents an effective three-body operator, it can be used to directly control the three-body interactions in quantum simulations and quantum information processing with Rydberg atoms.
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.
Rubidium Rydberg atoms in either |m_{j}| sublevel of the 36p_{3/2} state can exchange energy via Stark-tuned Förster resonances, including two-, three-, and four-body dipole-dipole interactions. Three-body interactions of this type were first reported and categorized by Faoro et al. [Nat. Commun. 6, 8173 (2015)NCAOBW2041-172310.1038/ncomms9173] and their Borromean nature was confirmed by Tretyakov et al. [Phys. Rev. Lett. 119, 173402 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.173402]. We report the time dependence of the N-body Förster resonance N×36p_{3/2,|m_{j}|=1/2}→36s_{1/2}+37s_{1/2}+(N-2)×36p_{3/2,|m_{j}|=3/2}, for N=2, 3, and 4, by measuring the fraction of initially excited atoms that end up in the 37s_{1/2} state as a function of time. The essential features of these interactions are captured in an analytical model that includes only the many-body matrix elements and neighboring atom distribution. A more sophisticated simulation reveals the importance of beyond-nearest-neighbor interactions and of always-resonant interactions.