Splitting of the three-body Förster resonance in Rb Rydberg atoms as a measure of dipole-dipole interaction strength

TL;DR

The paper addresses measuring three-body Rydberg interactions via the fine-structure-state-changing (FSSC) three-body Förster resonance in a linear chain of three Rb atoms. It develops a compact analytical framework by reducing a five-level model to an effective three-level system, deriving explicit, distance-dependent formulas for the resonance detunings $oldsymbol{δ}_ ext{±}$, the splitting $oldsymbol{δ}_0$, and the coupling $oldsymbol{Ω}_0$, with key scalings $oldsymbol{Ω_{12}=2Ω/9}$, $oldsymbol{Ω_{25}=rac{√2}{9}Ω}$, $oldsymbol{Ω_{45}=Ω/9}$ and $oldsymbol{Ω=4.9×10^{4}R^{-3}}$ (MHz). The two split resonances follow $oldsymbol{δ}_+(R)=7698/R^3-1.52×10^6/R^6$ and $oldsymbol{δ}_-(R)=-7698/R^3-1.52×10^6/R^6$, with $oldsymbol{Ω}_0(R)=1.224×10^6/R^6$, enabling a direct probe of dipole–dipole and van der Waals interactions. Comparison with full numerical simulations including Zeeman sublevels shows good agreement for resonance positions and heights, validating the analytical model and its use for extracting interaction strengths; the work also identifies the $oldsymbol{δ_+}$ branch as particularly robust to atomic-position fluctuations, suggesting practical routes to observe coherent three-body population oscillations and implement three-qubit gates. Overall, the study provides a theoretically grounded, experimentally relevant method to quantify three-body Rydberg interactions via resonance splitting and shifts.

Abstract

Three-body Förster 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. Eksper. Teor. Fiz. 168(1), 14 (2025)] it was found that the proposed earlier three-body Förster resonance $3\times nP_{3/2} \to nS_{1/2} +(n+1)S_{1/2} +nP_{1/2} $ in Rb Rydberg atoms has a splitting, with one of the split components having weaker dependence of the resonant electric field (and the corresponding dynamic shift) on the distance $R$ between the atoms. Here we study this effect in more detail, since such a resonance is 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 Förster resonance depending on $R$. 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 $S$ and $P$.

Figures (4)

• Figure 1: (a) Numerically calculated Stark structure of the FSSC three-body Förster resonance $3\times 70P_{3/2} \to 70S_{1/2} +71S_{1/2} +70P_{1/2}$ for three Rb Rydberg atoms. The energies W of various three-body collective states are shown versus the controlling dc electric field. Intersections between collective states (labeled by numbers) correspond to the Förster resonances. (b) Simplified scheme of the FSSC three-body Förster resonance $3\times 70P_{3/2} \to 70S_{1/2} +71S_{1/2} +70P_{1/2}$ for three Rb Rydberg atoms. The initially populated collective state is state 1. The intermediate collective state is state 2, with one atom remaining in the initial $70P_{3/2}$ state. The final collective state is state 3, with the changed fine-structure component of the $P$ state. The energy defects $\Delta _{1}$ and $\Delta _{2}$ are controlled by the dc electric field. The three-body Förster resonance occurs at $\Delta _{1} \approx \Delta _{2}$. (c) The linear spatial configuration of the three Rydberg atoms considered in this paper.
• Figure 2: (a) Rydberg states (labeled $a-d$) in a single Rb atom related to the FSSC three-body Förster resonance $3\times 70P_{3/2} \to 70S_{1/2} +71S_{1/2} +70P_{1/2}$. (b) Collective states of the three interacting Rb Rydberg atoms. Their labels $ijk$ indicate the related states of Fig. \ref{['Fig2']}(a) and take into account all possible atom permutations. Red arrows indicate interaction-induced transitions from the initial state $bbb$ ($3\times 70P_{3/2}$) to the intermediate states of the kind $70S_{1/2} +71S_{1/2} +70P_{3/2}$. Blue arrows indicate interaction-induced transitions from the intermediate states to the final states of the kind $70S_{1/2} +71S_{1/2} +70P_{1/2}$. Green horizontal arrows indicate always-resonant exchange transitions corresponding to the excitation hopping between $S$ and $P$ Rydberg atoms. (c) Reduced scheme of the three-body Förster resonance that takes into account the symmetries and identities of some transitions in Fig. \ref{['Fig2']}(b). State 1 is the same as state $bbb$. State 2 represents identical states $acb, bac, bca, cab$. State 3 represents identical states $abc, cba$. State 4 represents identical states $acd, dac, dca, cad$. State 5 represents identical states $adc, cda$. States 2-3 and 4-5 still experience always-resonant exchange transitions, which should result in their mixing and dynamic splitting. (d) Final reduced scheme of the three-body Förster resonance. States 2 and 3 of Fig. \ref{['Fig1']}(c) are replaced by two split states labeled as 2. States 4 and 5 of Fig. \ref{['Fig1']}(c) are also replaced by two split states labeled as 3. As the splittings due to always-resonant dipole-dipole interaction are significant, in the analytical calculations we can take into account only one of the two states in each mixed state 2 or 3.
• Figure 3: Analytically (a)-(d) and numerically (e)-(h) calculated spectra of the FSSC three-body Förster resonance $3\times 70P_{3/2} \to 70S_{1/2} +71S_{1/2} +70P_{1/2}$ in Rb Rydberg atoms for the interaction time of 1 $\mu$s and various interatomic distances $R=$15, 10, 9, and 8 $\mu$m. Analytical calculations have been done with Eqs. (\ref{['Eq6']}) and (\ref{['Eq8']}). Numerical calculations have been done with the full theoretical model developed by us earlier in Refs. 1427. A good agreement in positions and heights of the split resonances is observed, thus justifying the validity of Eqs. (\ref{['Eq6']}) and (\ref{['Eq8']}) to be used in measuring Rydberg interaction strength.
• Figure 4: Dependences of the shifts of the centers of the two split FSSC three-body resonances $3\times 70P_{3/2} \to 70S_{1/2} +71S_{1/2} +70P_{1/2}$ in Fig. \ref{['Fig3']} on the interatomic distance $R$ for the analytical (blue curves) and numerical (green circles) theoretical models. The shifts are recalculated from V/cm to the MHz scale using Eqs. (\ref{['Eq7']}).