Physical Review A

Role of multilevel Rydberg interactions in electric-field-tuned Förster resonances

Jorge Douglas Massayuki Kondo, Donald W. Booth, Luís F. Gonçalves, James P. Shaffer, Luís Gustavo Marcassa

Published Jan 8, 2016• 12 citations

Abstract

In this work, we investigate the dc electric-field dependence of two F\"orster resonant processes in ultracold $^{85}\mathrm{Rb}$, $37{D}_{5/2}+37{D}_{5/2}\ensuremath{\rightarrow}35L(L=O,Q)+39{P}_{3/2}$, as a function of the atomic density. At low densities, the $39{P}_{3/2}$ yield as a function of electric field exhibits resonances. With increasing density, the linewidths increase until the peaks merge. Even under these extreme conditions, where the F\"orster resonance processes show little electric-field dependence, the $39{P}_{3/2}$ population depends quadratically on the total Rydberg atom population, suggesting that a two-body interaction is the dominant process. In order to explain our results, we implement a theoretical model which takes into account the multilevel character of the interactions and Rydberg atom blockade process using only atom pair interactions. The comparison between the experimental data and the model is very good, suggesting that the F\"orster resonant processes are dominated by two-body interactions up to atomic densities of $3.0\ifmmode\times\else\texttimes\fi{}{10}^{12}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$.

DOI: 10.1103/physreva.93.012703

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Two-body Förster resonance involving Rb nD states in a quasi-electrostatic trap

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Effect of Förster resonances on the excitation statistics of many-body Rydberg systems

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Physical Review ADec 18, 200834 citations

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Physical Review A

Role of multilevel Rydberg interactions in electric-field-tuned Förster resonances

Jorge Douglas Massayuki Kondo, Donald W. Booth, Luís F. Gonçalves, James P. Shaffer, Luís Gustavo Marcassa

Published Jan 8, 2016• 12 citations

Publisher

Abstract

In this work, we investigate the dc electric-field dependence of two F"orster resonant processes in ultracold

^{85}\mathrm{Rb}

37{D}_{5/2}+37{D}_{5/2}\ensuremath{\rightarrow}35L(L=O,Q)+39{P}_{3/2}

, as a function of the atomic density. At low densities, the

39{P}_{3/2}

yield as a function of electric field exhibits resonances. With increasing density, the linewidths increase until the peaks merge. Even under these extreme conditions, where the F"orster resonance processes show little electric-field dependence, the

39{P}_{3/2}

population depends quadratically on the total Rydberg atom population, suggesting that a two-body interaction is the dominant process. In order to explain our results, we implement a theoretical model which takes into account the multilevel character of the interactions and Rydberg atom blockade process using only atom pair interactions. The comparison between the experimental data and the model is very good, suggesting that the F"orster resonant processes are dominated by two-body interactions up to atomic densities of

3.0\ifmmode\times\else\texttimes\fi{}{10}^{12}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}

DOI: 10.1103/physreva.93.012703

Related papers

Two-body Förster resonance involving Rb nD states in a quasi-electrostatic trap

Jorge Douglas Massayuki Kondo, Luís F. Gonçalves, Jader Cabral +2 more

In this work, we excite $n{D}_{5/2}$ Rydberg states in a dense and cold Rb atomic sample held in a 10.6 $\ensuremath{\mu}\mathrm{m}$ quasi-electrostatic trap using narrow-bandwidth laser pulses. Our goal is to study the F\"orster resonance process $n{D}_{5/2}+n{D}_{5/2}\ensuremath{\rightarrow}(n\ensuremath{-}2)F+(n+2){P}_{3/2}$ at zero electric field as a function of the total atomic density using pulsed-field ionization in the range $n=37--47$ . Such a process is almost degenerate for $n=43$ . Younge and coworkers studied this process [K. C. Younge, A. Reinhard, T. Pohl, P. R. Berman, and G. Raithel, Phys. Rev. A 79, 043420 (2009)] and attributed the observed saturation to many-body effects. Our results show that as the ground-state atomic density increases, the $n{D}_{5/2}$ state population and the population transfer starts to saturate, which is consistent with the onset of Rydberg atom blockade and previously published results. However, since our experiment allows the independent measurement of the $n{D}_{5/2}$ and $(n+2){P}_{3/2}$ state populations, we were able to obtain the $(n+2){P}_{3/2}$ state population density dependence. Our results clearly show that the $(n+2){P}_{3/2}$ state population depends quadratically on the total Rydberg atomic population, and consequently, the F\"orster resonance is a two-body process for a ground-state atomic density below $3\ifmmode\times\else\texttimes\fi{}{10}^{11}$ ${\mathrm{cm}}^{\ensuremath{-}3}$.In this work, we excite $n{D}_{5/2}$ Rydberg states in a dense and cold Rb atomic sample held in a 10.6 $\ensuremath{\mu}\mathrm{m}$ quasi-electrostatic trap using narrow-bandwidth laser pulses. Our goal is to study the F\"orster resonance process $n{D}_{5/2}+n{D}_{5/2}\ensuremath{\rightarrow}(n\ensuremath{-}2)F+(n+2){P}_{3/2}$ at zero electric field as a function of the total atomic density using pulsed-field ionization in the range $n=37--47$. Such a process is almost degenerate for $n=43$. Younge and coworkers studied this process [K. C. Younge, A. Reinhard, T. Pohl, P. R. Berman, and G. Raithel, Phys. Rev. A 79, 043420 (2009)] and attributed the observed saturation to many-body effects. Our results show that as the ground-state atomic density increases, the $n{D}_{5/2}$ state population and the population transfer starts to saturate, which is consistent with the onset of Rydberg atom blockade and previously published results. However, since our experiment allows the independent measurement of the $n{D}_{5/2}$ and $(n+2){P}_{3/2}$ state populations, we were able to obtain the $(n+2){P}_{3/2}$ state population density dependence. Our results clearly show that the $(n+2){P}_{3/2}$ state population depends quadratically on the total Rydberg atomic population, and consequently, the F\"orster resonance is a two-body process for a ground-state atomic density below $3\ifmmode\times\else\texttimes\fi{}{10}^{11}$ ${\mathrm{cm}}^{\ensuremath{-}3}$.

Physical Review AAug 18, 201411 citations

Effect of Förster resonances on the excitation statistics of many-body Rydberg systems

Aaron Reinhard, Kelly Cooper Younge, Georg A. Raithel

We provide evidence for the dipole blockade of Rydberg excitation by examining the distributions of the number of Rydberg excitations created in ensembles of interacting Rydberg atoms. We show that, over a range of densities, the Rydberg excitation number distributions become significantly narrower when an applied electric field is used to tune the $^{85}\mathrm{Rb}$ interaction channel $45{D}_{5∕2}+45{D}_{5∕2}\ensuremath{\rightarrow}43F+47{P}_{3∕2}$ into F\"orster resonance. Substantially sub-Poissonian distributions that agree with published theory were achieved using very dense atom samples tuned into F\"orster resonance.We provide evidence for the dipole blockade of Rydberg excitation by examining the distributions of the number of Rydberg excitations created in ensembles of interacting Rydberg atoms. We show that, over a range of densities, the Rydberg excitation number distributions become significantly narrower when an applied electric field is used to tune the $^{85}\mathrm{Rb}$ interaction channel $45{D}_{5∕2}+45{D}_{5∕2}\ensuremath{\rightarrow}43F+47{P}_{3∕2}$ into F\"orster resonance. Substantially sub-Poissonian distributions that agree with published theory were achieved using very dense atom samples tuned into F\"orster resonance.

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Coherence of three-body Förster resonances in Rydberg atoms

I. I. Ryabtsev, Ilya I. Beterov, Denis B. Tretyakov +4 more

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.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.

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

I. I. Ryabtsev, I. N. Ashkarin, Ilya I. Beterov +4 more

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 $R$ 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 $R$ . 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 $P$ .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 $R$ 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 $R$. 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 $P$.

arXiv:2507.13658Physical Review ANov 15, 20253 citations

Line shapes and time dynamics of the Förster resonances between two Rydberg atoms in a time-varying electric field

Elena A. Yakshina, Denis B. Tretyakov, Ilya I. Beterov +7 more

The observation of the Stark-tuned F\"orster resonances between Rydberg atoms excited by narrowband cw laser radiation requires usage of a Stark-switching technique in order to excite the atoms first in a fixed electric field and then to induce the interactions in a varied electric field, which is scanned across the F\"orster resonance. In our experiments with a few cold Rb Rydberg atoms, we have found that the transients at the edges of the electric pulses strongly affect the line shapes of the F\"orster resonances, since the population transfer at the resonances occurs on a time scale of $\ensuremath{\sim}100$ ns, which is comparable with the duration of the transients. For example, a short-term ringing at a certain frequency causes additional radio-frequency-assisted F\"orster resonances, while nonsharp edges lead to asymmetry. The intentional application of the radio-frequency field induces transitions between collective states, whose line shape depends on the interaction strengths and time. Spatial averaging over the atom positions in a single interaction volume yields a cusped line shape of the F\"orster resonance. We present a detailed experimental and theoretical analysis of the line shape and time dynamics of the Stark-tuned F\"orster resonances $\mathrm{Rb}(n{P}_{3/2})+\mathrm{Rb}(n{P}_{3/2})\ensuremath{\rightarrow}\mathrm{Rb}(n{S}_{1/2})+\mathrm{Rb}([n+1]{S}_{1/2})$ for two Rb Rydberg atoms interacting in a time-varying electric field.The observation of the Stark-tuned F\"orster resonances between Rydberg atoms excited by narrowband cw laser radiation requires usage of a Stark-switching technique in order to excite the atoms first in a fixed electric field and then to induce the interactions in a varied electric field, which is scanned across the F\"orster resonance. In our experiments with a few cold Rb Rydberg atoms, we have found that the transients at the edges of the electric pulses strongly affect the line shapes of the F\"orster resonances, since the population transfer at the resonances occurs on a time scale of $\ensuremath{\sim}100$ ns, which is comparable with the duration of the transients. For example, a short-term ringing at a certain frequency causes additional radio-frequency-assisted F\"orster resonances, while nonsharp edges lead to asymmetry. The intentional application of the radio-frequency field induces transitions between collective states, whose line shape depends on the interaction strengths and time. Spatial averaging over the atom positions in a single interaction volume yields a cusped line shape of the F\"orster resonance. We present a detailed experimental and theoretical analysis of the line shape and time dynamics of the Stark-tuned F\"orster resonances $\mathrm{Rb}(n{P}_{3/2})+\mathrm{Rb}(n{P}_{3/2})\ensuremath{\rightarrow}\mathrm{Rb}(n{S}_{1/2})+\mathrm{Rb}([n+1]{S}_{1/2})$ for two Rb Rydberg atoms interacting in a time-varying electric field.

Physical Review AOct 21, 201621 citations

Investigation of three-body Förster resonance for various spatial configurations of the three interacting Rubidium Rydberg atoms

I. I. Ryabtsev, I. N. Ashkarin, Ilya I. Beterov +4 more

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 ${\rm 3}\times nP_{3/2} \to nS_{1/2} +(n+1)S_{1/2} +nP_{1/2}$ 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 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.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 ${\rm 3}\times nP_{3/2} \to nS_{1/2} +(n+1)S_{1/2} +nP_{1/2}$ 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 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.

arXiv:2506.22259Jun 27, 20251 citation

In this work, we investigate the dc electric-field dependence of two F"orster resonant processes in ultracold

^{85}\mathrm{Rb}

37{D}_{5/2}+37{D}_{5/2}\ensuremath{\rightarrow}35L(L=O,Q)+39{P}_{3/2}

, as a function of the atomic density. At low densities, the

39{P}_{3/2}

39{P}_{3/2}

3.0\ifmmode\times\else\texttimes\fi{}{10}^{12}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}

In this work, we excite

n{D}_{5/2}

Rydberg states in a dense and cold Rb atomic sample held in a 10.6 $\ensuremath{\mu}\mathrm{m}$ quasi-electrostatic trap using narrow-bandwidth laser pulses. Our goal is to study the F\"orster resonance process $n{D}_{5/2}+n{D}_{5/2}\ensuremath{\rightarrow}(n\ensuremath{-}2)F+(n+2){P}_{3/2}$ at zero electric field as a function of the total atomic density using pulsed-field ionization in the range

n=37--47

. Such a process is almost degenerate for

n=43

. Younge and coworkers studied this process [K. C. Younge, A. Reinhard, T. Pohl, P. R. Berman, and G. Raithel, Phys. Rev. A 79, 043420 (2009)] and attributed the observed saturation to many-body effects. Our results show that as the ground-state atomic density increases, the

n{D}_{5/2}

state population and the population transfer starts to saturate, which is consistent with the onset of Rydberg atom blockade and previously published results. However, since our experiment allows the independent measurement of the

n{D}_{5/2}

and

(n+2){P}_{3/2}

state populations, we were able to obtain the

(n+2){P}_{3/2}

state population density dependence. Our results clearly show that the

(n+2){P}_{3/2}

state population depends quadratically on the total Rydberg atomic population, and consequently, the F\"orster resonance is a two-body process for a ground-state atomic density below $3\ifmmode\times\else\texttimes\fi{}{10}^{11}$ ${\mathrm{cm}}^{\ensuremath{-}3}$.

^{85}\mathrm{Rb}

interaction channel $45{D}_{5∕2}+45{D}_{5∕2}\ensuremath{\rightarrow}43F+47{P}_{3∕2}$ into F\"orster resonance. Substantially sub-Poissonian distributions that agree with published theory were achieved using very dense atom samples tuned into F\"orster resonance.

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.

R

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

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

{\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 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.