Investigation of three-body F\"orster resonance for various spatial configurations of the three interacting Rubidium 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 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 F\"orster resonance ${\rm 3}\times nP_{3/2} \to nS_{1/2} +(n+1)S_{1/2} +nP_{1/2}$ that can be realized with Rb Rydberg atoms for an arbitrary principal quantum number $n$. Its peculiarity is that the third atom goes into a state with a total angular moment $J=1/2$, which has no Stark structure, so two-body F\"orster resonances are completely absent. In the present work, an extended theoretical study of this three-body F\"orster 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.