Searching optimal conditions for quantum gates application with the new 3-body F\"orster resonances in Rb and Cs Rydberg atoms

Three body resonant interactions between Rydberg atoms are considered in order to perform few-body quantum gates. So far, the resonances found in cesium or rubidium atoms relied on an adjacent two-body resonance which ceases to exist for principal quantum numbers above $n\simeq 40$. We have proposed recently a new class of 3-body interaction resonances in alkali-metal Rydberg atoms [P. Cheinet $\mathit{\text{et}}\mathit{\text{al.}}$, Quant. Elect. $\mathbf{\text{50}}$, 213 (2020)], which circumvienes this limit. We investigate here the relative strength between this new class of 3-body interaction resonance and quasi-forbidden 2-body interaction resonances in rubidium and cesium Rydberg atoms. We then identify the best case scenario for detecting and using this 3-body interaction.