Observation of the Borromean three-body Förster resonances for three interacting Rb Rydberg atoms
D.B. Tretyakov, I.I. Beterov, E.A. Yakshina, V.M. Entin, I.I. Ryabtsev, P. Cheinet, P. Pillet
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l Observation of the Borromean three-body F¨orster resonancesfor three interacting Rb Rydberg atoms
D. B. Tretyakov , , I. I. Beterov , , E. A. Yakshina , , V. M. Entin , , I. I. Ryabtsev , , ∗ P. Cheinet , and P. Pillet Rzhanov Institute of Semiconductor Physics SB RAS, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia and Laboratoire Aime Cotton, CNRS, Univ. Paris-Sud, ENS Paris-Saclay, 91405 Orsay, France (Dated: 27 October 2017)Three-body F¨orster resonances at long-range interactions of Rydberg atoms were first predictedand observed in Cs Rydberg atoms by Faoro et al. [Nature Commun. , 8173 (2015)]. In theseresonances, one of the atoms carries away an energy excess preventing the two-body resonance,leading thus to a Borromean type of F¨orster energy transfer. But they were in fact observedas the average signal for the large number of atoms N ≫
1. In this Letter we report on the firstexperimental observation of the three-body F¨orster resonances 3 × nP / ( | M | ) → nS / +( n +1) S / + nP / ( | M ∗ | ) in a few Rb Rydberg atoms with n = 36 ,
37. We have found here clear evidence thatthere is no signature of the three-body F¨orster resonance for exactly two interacting Rydberg atoms,while it is present for N =3 − PACS numbers: 32.80.Ee, 32.70.Jz , 32.80.Rm, 03.67.Lx
Highly excited Rydberg atoms exhibit strong long-range interactions due to their huge dipole moments thatgrow as n with increasing the principal quantum num-ber n [1]. This is especially attractive for the devel-opment of quantum computers and simulators based onqubits represented by single alkali-metal atoms in arraysof optical dipole traps or optical lattices [2-5]. In partic-ular, Rydberg-atom-based quantum simulators can di-rectly model various objects in solid-state physics due totheir ability to mimic various possible interactions be-tween their constituents, if such interactions in a quan-tum simulator are appropriately controlled [6-14].Interactions between Rydberg atoms are flexibly con-trolled by the dc or radio-frequency (rf) electric field viaStark-tuned [15], microwave [16-19], or rf-assisted [4,16]F¨orster resonances corresponding to the F¨orster resonantenergy transfer (FRET). F¨orster resonances have beendemonstrated to be efficient tools in cold Rydberg atoms[20, 21] to tune interactions in strength and distance andcan be either resonant dipole-dipole or nonresonant vander Waals interactions. The interactions are typicallydescribed by a two-body operator of dipole-dipole inter-action for each pair of atoms in the ensemble [1]. Aftersuch an interaction the two atoms are found in an entan-gled state, so that a measurement over one atom deter-ministically predicts the state of the other atom. Thisentanglement is the quantum resource, which is used inquantum computations and simulations [2-13,22].Some exotic quantum simulations demand to simul-taneously control the interactions of three atoms [22-28]. This demands a three-body quantum operator that ∗ Electronic address: [email protected] changes the states of the three qubits simultaneously andmakes them all entangled. Three-body operators are de-scribed by a combination of two-body operators, which infact are reduced to a single effective three-body operator.Such an operator has been proposed and implementedrecently as a Borromean three-body FRET in a frozenRydberg gas of Cs atoms [29]. In these three-body res-onances, one of the atoms carries away an energy ex-cess preventing the two-body resonance, leading thus toa Borromean type of F¨orster energy transfer. Here theBorromean transfer is featured by the strong isolatedthree-body energy transfer with a negligible contributionof the two-body effect. This allows us to characterize thethree-body effect while, it is usually impossible in othersystems because it is imbedded in the strong two-bodyeffect signal. The experiment in Ref. [29] was done withan ensemble of ∼ Cs atoms in the interaction volumeof ∼ µ m in size. Therefore, the three-body F¨orsterresonance was in fact observed as the average signal forthe large number of atoms N ≫ × nP / ( | M | ) → nS / + ( n + 1) S / + nP / ( | M ∗ | ) for N =3 − n = 36 ,
37. We havefound clear evidence that there is no signature of thethree-body F¨orster resonances for exactly two interactingRydberg atoms, while it is present for the larger numberof atoms. We thus demonstrate the possible generaliza-tion of this effect to other Rydberg atoms.The experiments are performed with cold Rb atomsin a magneto-optical trap [4,30]. Our experiments fea-ture atom-number-resolved measurement of the signalsobtained from N =1 − T ≈
70% [31]. It is based on aselective field ionization (SFI) detector with a channelelectron multiplier (CEM) and postselection technique[32]. The electric field for SFI is formed by two stainless-steel plates that are 1 cm apart. These plates have holescovered by meshes for passing the vertical cooling laserbeams and the electrons to be detected. The dc elec-tric field, which is homogeneous due to the meshes, iscalibrated with 0.2% uncertainty using the Stark spec-troscopy of the microwave transition 37 P / → S / at80.124 GHz [30].The CEM output pulses from the nS and[ nP +( n +1) S ] states (the two latter states havenearly identical ionizing fields) are detected with twoindependent gates and postselected over the number ofthe detected Rydberg atoms N =1 −
5. The normalized N -atom signals S N are the fractions of atoms that haveundergone a transition to the final nS state.In this experiment, the detection of N Rydberg atomsmeans that there were N interacting Rydberg atoms with T ≈
70% confidence and N +1 interacting atoms with(1 − T ) ≈
30% confidence [31]. Therefoere, the recordedF¨orster resonance spectra were additionally processed toextract the true multiatom spectra ρ i taking into ac-count finite detection efficiency [33]. As shown in ourpaper [32], for the nonideal SFI detector, which detectsfewer atoms than actually have interacted, various truemultiatom spectra ρ i of the F¨orster resonances for i inter-acting Rydberg atoms contribute to our measured signals S N for N detected Rydberg atoms to a degree that de-pends on the mean numbers of the excited and detectedatoms. The signals S N are thus a mixture of the spectra ρ i from the larger numbers of actually interacted atoms i ≥ N . In order to derive ρ i from S N , we have developeda procedure that solves the system of linear equationsand approximately expresses each ρ i via various S N [33].The excitation of Rb atoms to the nP / Rydbergstates is realized via the three-photon transition 5 S / → P / → S / → nP / by means of three cw lasersmodulated to form 2 µ s exciting pulses at a repetitionrate of 5 kHz [4,35]. A small Rydberg excitation volumeof ∼ µ m in size is formed using the crossed tightly-focused laser beams. The laser intensities are adjusted toobtain about one Rydberg atom excited per laser pulseon average. We use a Stark-switching technique [35,36]to switch the Rydberg interactions on and off. Laser ex-citation occurs during 2 µ s at a fixed electric field of 5.6V/cm. Then the field decreases to a lower value near theresonant electric field, which acts for 3 µ s until the fieldincreases back to 5.6 V/cm. Then, 0.5 µ s later, a rampof the strong field-ionizing electric pulse of 200 V/cm isapplied. The lower electric field is slowly scanned acrossthe F¨orster resonance and the SFI signals are accumu-lated for 10 − laser pulses.Figure 1 presents the numerically calculated Starkstructure of the F¨orster resonance 3 × P / → S / +38 S / + 37 P ∗ / for three Rb Rydberg atoms. The ener-gies W of various three-body collective states are shownversus the controlling dc electric field. The intersec-tions between collective states (labeled by numbers) cor- FIG. 1: Numerically calculated Stark structure of the F¨orsterresonance 3 × P / → S / +38 S / +37 P ∗ / for three RbRydberg atoms. The energies W of various three-body col-lective states are shown versus the controlling electric field.Intersections between collective states (labeled by numbers)correspond to the F¨orster resonances of various kinds. In-tersections 2-7 are in fact two-body resonances that do notrequire the third atom. The intersections 1 and 8 are three-body resonances occurring only in the presence of the thirdatom that carries away an energy excess preventing the two-body resonance. respond to the F¨orster resonances of various kinds. Actu-ally, there are the anticrossings at the intersection pointsdue to Rydberg interactions [29]. In our experiment,however, the average two-body dipole-dipole interactionenergy is small ( ∼ .
25 MHz [35]) and the anticrossingsare not visible in the energy scale of Fig. 1.Intersections 2 − P / stateto the final 37 S / and 38 S / states in two of the threeatoms, while the third atom remains in its initial P statethat does not change.Intersections 1 and 8 are three-body resonances oc-curring only in the presence of the third atom that car-ries away an energy excess preventing the two-body res-onance, leading thus to a Borromean type of F¨orster en-ergy transfer [29]. The three-body resonances are distin-guished from the two-body ones by the fact that the thirdatom does not remain in its initial P state as its initialmoment projection ( | M | =1/2 or | M | =3/2) changes tothe other one ( | M ∗ | =3/2 or | M ∗ | =1/2, correspondingly).Therefore, the three-body resonance corresponds to thetransition when the three interacting atoms change theirstates simultaneously.In our experiments, cold Rb atoms are excited in the dcelectric field either to the initial 37 P / ( | M | =1/2) Starksublevel or to the 37 P / ( | M | =3/2) one. Therefore, notall resonances 1 − P / ( | M | =1/2) only reso-nances 1 and 3 are observable, while for the initial state FIG. 2: Stark-tuned F¨orster resonances in Rb Rydberg atomsobserved for various numbers of atoms i =2 − P / ( | M | =1/2); (b) 37 P / ( | M | =3/2); (c)36 P / ( | M | =1/2); (d) 36 P / ( | M | =3/2). The main peaks aretwo-body resonances, and the additional peaks are three-bodyresonances. The three-body resonance is absent for i =2 in allrecords, evidencing its three-body nature. P / ( | M | =3/2) we can observe only resonances 6 and8. The intermediate resonances 2, 4, 5 and 7 are observ-able only when both | M | =1/2 and | M | =3/2 atoms areinitially excited, as in our earlier paper [37] where weused the excitation by broadband pulsed lasers.Figures 2(a) and 2(b) show the Stark-tuned F¨orsterresonances observed for various numbers of the interact-ing atoms i =2 −
5. In Fig. 2(a) the atoms are in theinitial state 37 P / ( | M | =1/2). The main peak at 1.79V/cm is the ordinary two-body resonance that occursfor all i =2 − i =2 and appears only for i =3 −
5. The two-body andthree-body peak positions well agree with those predictedby Fig. 1.The feature at 1.71 V/cm could in principle be causedby the imperfection of the electric-field pulses used tocontrol the F¨orster resonance, as it was observed anddiscussed in our paper [35]. In order to check for thiseffect, the resonance has also been recorded for atoms inthe initial state 37 P / ( | M | =3/2), as shown in Fig. 1(b).We see that the three-body resonance changes its positionwith respect to the two-body resonance, in full agreementwith Fig. 1. Again, the main peak at 2.0 V/cm is theordinary two-body resonance that occurs for all i =2 − i =2 and appears only for i =3 − i grows due to the increase of the totalinteraction energy and broadening of the two-body res-onance. The overlapping can be reduced if a lower Ry-dberg state is used [29]. For example, if we take atomsin the initial state 36 P / , the Stark structure of theF¨orster resonance is the same as in Fig. 1, but the separa-tion between intersections 1 and 3 is 140 mV/cm insteadof 80 mV/cm for the 37 P / atoms. Figures 2(c) and2(d) present the two-body and three-body resonancesrecorded for atoms in the initial state 36 P / . The res-onances are similar to those in Figs. 2(a) and 2(b), butare better visible due to the larger separation. They ad-ditionally confirm that the three-body resonances reallytake place and can be observed separately from the two-body ones.In our previous experiments [31,35], we used onlyatoms in the initial state 37 P / ( | M | =1/2). Therefore,in the related theoretical analysis [31,35,38] we consid-ered only the two-body resonance 3 of Fig. 1 and ignoredthe possibility of the three-body resonance 1. As a re-sult, the numerically calculated multiatom spectra ρ i for i =2 − i =3 − P / ( | M | =1/2)to another Stark sublevel 37 P / ( | M ∗ | =3/2), but such atransition is not described by the two-body operator ofdipole-dipole interaction. This requires a new theoret-ical model to be developed. It is a rather complicatedproblem, since we should take into account all Stark andmagnetic sublevels of the interacting Rydberg atoms. Inthis Letter we limited our theoretical considerations onlyby the cases of two and three interacting Rydberg atoms.For two Rydberg atoms in the initial state37 P / ( | M | =1/2) only one F¨orster resonance 3of Fig. 1 is possible, which corresponds to theresonant transition between two collective states2 × P / ( | M | =1/2) → S / + 38 S / . Its dipole-dipolematrix element is given by V = d d πε (cid:20) R − Z R (cid:21) , (1)where d and d are the z components of the ma-trix elements of dipole moments of transitions (cid:12)(cid:12) P / (M = 1 / (cid:11) → (cid:12)(cid:12) S / (M = 1 / (cid:11) and (cid:12)(cid:12) P / (M = 1 / (cid:11) → (cid:12)(cid:12) S / (M = 1 / (cid:11) , Z is the z component of the vector R connecting the two atoms( z axis is chosen along the dc electric field), and ε isthe dielectric constant. For the weak interaction, thetwo-body F¨orster resonance amplitude is ρ ∼ V [35]. FIG. 3: Comparison between the theory and experi-ment for the three-atom Stark-tuned F¨orster resonances3 × nP / ( | M | ) → nS / + ( n + 1) S / + nP / ( | M ∗ | ) in RbRydberg atoms for the initial states: (a) 37 P / ( | M | =1/2);(b) 37 P / ( | M | =3/2); (c) 36 P / ( | M | =1/2); (d)36 P / ( | M | =3/2). The theoretical spectra have beencalculated for the cubic interaction volume of 15 × × µ m , 3 µ s interaction time and Monte Carlo averaging over1000 random atom positions. The thick green (gray) lines arethe experiment, the thin black lines are the full theory, andthe thin magenta (dark gray) lines are the theory withoutaccounting for the three-body resonances. For three Rydberg atoms in the initial state37 P / ( | M | =1/2), the two F¨orster resonances 1and 3 of Fig. 1 are possible. The three-body res-onance 1 corresponds to the resonant transitionbetween collective states 3 × P / ( | M | =1/2) → S / +38 S / +37 P / ( | M ∗ | =3/2). This transition is, infact, composed of the two nonresonant two-bodyrelay transitions 3 × P / ( | M | =1/2) → S / +38 S / +37 P / ( | M | =1/2) → S / +38 S / +37 P / ( | M ∗ | =3/2) occurring simultaneously. Thelatter occurs due to non-resonant exchange interaction nP / (M) +n ′ S → n ′ S+nP / (M ∗ ) corresponding to theexcitation hopping between S and P Rydberg atoms[29,38]. Despite the use of a relay, the transfer occursin a single step, implying a Borromean character of therelay atom which absorbs the energy of the finite F¨orsterdefect. The perturbation theory shows that for the weakinteraction the three-body F¨orster resonance amplitudeis ρ ∼ ( V V ∗ / ∆) , where V ∗ is the same as V but for thetransitions (cid:12)(cid:12) P / (M = 3 / (cid:11) → (cid:12)(cid:12) S / (M = 1 / (cid:11) and (cid:12)(cid:12) P / (M = 3 / (cid:11) → (cid:12)(cid:12) S / (M = 1 / (cid:11) , and∆/(2 π )=9.5 MHz is the energy splitting between37 P / ( | M | =1/2) and 37 P / ( | M | =3/2) Stark sublevelsin the electric field of 1.71 V/cm.The three-body resonance is thus less effective thanthe two-body one at the weak dipole-dipole interaction( V < ∆). However, when the three-body resonance isexactly tuned, its contribution to the population trans- fer generally exceeds the contribution from the two-bodyinteraction, which is offresonant in this case. The condi-tion for the three-body resonance to be of the Borromeantype is thus satisfied.We have done numerical simulations of the experimen-tal F¨orster resonances of Fig. 2 for i =3 atoms using themethod described in Refs. [31,38]. It is based upon solv-ing the Schr¨odinger’s equation with subsequent MonteCarlo averaging over the random positions of the threeatoms in a single interaction volume. The Stark andZeeman structures of all Rydberg states are fully takeninto account. The numerical results and their compari-son with the experimental data of Fig. 2 are presented inFig. 3. The thick green (gray) lines are the experimentalthree-atom data, the thin black lines are the full theory,and the thin magenta (dark gray) lines are the theorywithout accounting for the three-body resonances. Thetheoretical spectra have been calculated and averagedover 1000 random atom positions for the cubic interac-tion volume of 15 × × µ m and 3 µ s interaction time,which correspond to our experimental parameters.The overall agreement of the full theory with the ex-periment in Fig. 3 is satisfactory. The calculated lineshapes of the two-body resonances are close to the ex-perimental ones. These are cusp-shaped resonances thatare formed upon spatial averaging in a single interac-tion volume, as discussed in our paper [35] and otherpapers [39,40]. When the three-body resonances are notaccounted for by the theory, the height of the two-bodypeak grows because the population does not leak to theother three-atom states, while the three-body peaks areabsent at all. The three-body resonances are well repro-duced by theory in Figs. 3(a)-3(c), in both their heightsand widths.However, some discrepancy between the experimentand theory is found for the 36 P / ( | M | =3/2) state atomsin Fig. 3(d). This case is distinguished by the largestseparation ∆ between the two-body and three-body reso-nances. The theory predicts weaker two- and three-bodyF¨orster resonances than those observed experimentally.One of the explanations could be that the Schr¨odingerequation model gives incorrect time dynamics of the pop-ulations at large ∆. This discrepancy points towards theneed to build a new model based on the density-matrixequations, as we did for i =2 atoms in Ref. [35]. Com-pared to the Schr¨odinger equation, the density-matrixmodel gives a faster time dynamics of the populations inthe presence of additional dephasing (unresolved hyper-fine structure of Rydberg states and fluctuations of thecontrolling electric field as observed in Ref. [35]). Butbuilding this model is a complicated task which requires adedicated study because of the huge number of collectivestates if the Stark and Zeeman structures are accountedfor.In conclusion, our experiments with a few Rb Rydbergatoms in various initial states have clearly shown theneed for three atoms to obtain a three-body resonancesignature in perfect agreement with expectations. Thethree-body resonance corresponds to a transition whenthe three interacting atoms change their states simulta-neously (two atoms go to the S states, and the third oneremains in the P state but changes its moment projec-tion). Such a Borromean-type transfer displays strongthree-body energy transfer with a negligible contribu-tion of two-body transfer. As the three-body resonanceappears at the different dc electric field with respect tothe two-body resonance, it represents an effective three-body operator, which can be used to directly control thethree-body interactions. This can be especially useful inquantum simulations and quantum information process-ing with neutral atoms in optical lattices [2-15]. It canalso allow us to test and study a quantum system wherethe basic interaction is a three-body interaction.We note that the Borromean trimers of Rydberg atomshave been predicted in Ref. [41], and excitation transferin a spin chain of three Rydberg atoms has been ob- served experimentally in Ref. [42]. We also note that,in principle, it is possible to organize three-body inter-actions for almost arbitrary Rydberg states using theradio-frequency-assisted F¨orster resonances occurring be-tween Floquet sidebands of Rydberg states in a radiofre-quency electric field [43]. Finally, F¨orster resonances ofthe higher orders (four-body etc.) can also be observedin the electric field which is different from the two-bodyone [29,34].The authors are grateful to Elena Kuznetsova andMark Saffman for fruitful discussions. This work wassupported by the RFBR Grants No. 16-02-00383 andNo. 17-02-00987, the Russian Science Foundation GrantNo. 16-12-00028 (for laser excitation of Rydberg states),the Siberian Branch of RAS, the Novosibirsk State Uni-versity, the public Grant CYRAQS from Labex PALM(ANR-10-LABX-0039) and the EU H2020 FET Proac-tive project RySQ (Grant No. 640378). [1] T. F. Gallagher, Rydberg Atoms (Cambridge UniversityPress, Cambridge, 1994).[2] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod.Phys. , 2313 (2010).[3] D. Comparat and P. Pillet, J. Opt. Soc. Am. B , A208(2010).[4] I. I .Ryabtsev, I. I. Beterov, D. B. Tretyakov,V. M. Entin, and E. A. Yakshina, Phys. Usp. , 196(2016).[5] M. Saffman, J. Phys. B , 202001 (2016).[6] H. Weimer, M. M¨uller, I. Lesanovsky, P. Zoller, andH. P. B¨uchler, Nat. Phys. , 382 (2010).[7] J. P. Hague and C. MacCormick, New J. Phys. ,033019 (2012).[8] A. Dauphin, M. M¨uller, and M. A. Martin-Delgado,Phys. Rev. A , 053618 (2012).[9] J. P. Hague and C. MacCormick, Phys. Rev. Lett. ,223001 (2012).[10] A. W. Glaetzle, M. Dalmonte, R. Nath, I. Rousochatza-kis, R. Moessner, and P. Zoller, Phys. Rev. X , 041037(2014).[11] M. Mattioli, A. W. Glatzle, and W. Lechner, New J.Phys. , 113039 (2015).[12] J. Gelhausen, M. Buchhold, A. Rosch, and P. Strack,SciPost Phys. , 004 (2016).[13] A. Dauphin, M. Muller, and M. A. Martin-Delgado,Phys. Rev. A , 043611 (2016).[14] W. Maineult, B. Pelle, R. Faoro, E. Arimondo, P. Pillet,and P. Cheinet, J. Phys B , 214001 (2016).[15] K. A. Safinya, J. F. Delpech, F. Gounand, W. Sandner,and T. F. Gallagher, Phys. Rev. Lett. , 405 (1981).[16] P. Pillet, D. Comparat, M. Muldrich, T. Vogt, N. Za-hzam, V. M. Akulin, T. F. Gallagher, W. Li, P. Tanner,M. W. Noel, and I. Mourachko, in Decoherence, Entan-glement and Information Protection in Complex Quan-tum Systems , edited by V. M. Akulin and G. Kurizki(Springer, New York, 2005).[17] P. Pillet, R. Kachru, N. H. Tran, W. W. Smith, andT. F. Gallagher, Phys. Rev. Lett. , 1763 (1983).[18] P. Pillet, R. Kachru, N. H. Tran, W. W. Smith, and T. F. Gallagher, Phys. Rev. A , 1132 (1987).[19] J. Lee and T. F. Gallagher, Phys. Rev. A , 062509(2016).[20] W. R. Anderson, J. R. Veale, and T. F. Gallagher, Phys.Rev. Lett. , 249 (1998).[21] I. Mourachko, D. Comparat, F. de Tomasi, A. Fioretti,P. Nosbaum, V. M. Akulin, and P. Pillet, Phys. Rev.Lett. , 253 (1998).[22] I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod.Phys. , 153 (2014).[23] H. -W. Hammer, A. Nogga, and A. Schwenk, Rev. Mod.Phys. , 197 (2013).[24] W. Liu, J. Zhang, Z. Deng, and G. Long, Sci. China Ser.G , 1089 (2008).[25] X. Peng, J. Zhang, J. Du, and D. Suter, Phys. Rev. Lett. , 140501 (2009).[26] K. Jachymski, P. Bienias, and H. P. Buchler, Phys. Rev.Lett. , 053601 (2016).[27] W. L. You, Y. C. Qiu, and A. M. Oles, Phys. Rev. B ,214417 (2016).[28] Z. Luo, C. Lei, J. Li, X. Nie, Z. Li, X. Peng, and J. Du,Phys. Rev. A , 052116 (2016).[29] R. Faoro, B. Pelle, A. Zuliani, P. Cheinet, E. Arimondo,and P. Pillet, Nat. Commun. , 8173 (2015).[30] D. B. Tretyakov, I. I. Beterov, V. M. Entin, I. I. Ryabt-sev, and P. L. Chapovsky, J. Exp. Theor. Phys. , 374(2009).[31] I. I. Ryabtsev, D. B. Tretyakov, I. I. Beterov, andV. M. Entin, Phys. Rev. Lett. , 073003 (2010).[32] I. I. Ryabtsev, D. B. Tretyakov, I. I .Beterov, andV. M. Entin, Phys. Rev. A , 012722 (2007); Erratum:Phys. Rev. A , 049902(E) (2007).[33] See Supplemental Material at[http://link.aps.org/supplemental/10.1103/PhysRevLett.119.173402],which includes Refs. [32,34], for details on the derivationof the true many-body F¨orster resonances.[34] J. H. Gurian, P. Cheinet, P. Huillery, A. Fioretti, J. Zhao,P. L. Gould, D. Comparat, and P. Pillet, Phys. Rev. Lett. , 023005 (2012).[35] E. A. Yakshina, D. B. Tretyakov, I. I. Beterov, V. M. Entin, C. Andreeva, A. Cinins, A. Markovski,Z. Iftikhar, A. Ekers, and I. I. Ryabtsev, Phys. Rev. A , 043417 (2016).[36] I. I. Ryabtsev, D. B. Tretyakov, and I. I. Beterov, J. Phys.B , 297 (2003).[37] D. B. Tretyakov, I. I. Beterov, V. M. Entin, E. A. Yak-shina, I. I. Ryabtsev, S. F. Dyubko, E. A. Alekseev,N. L. Pogrebnyak, N. N. Bezuglov, and E. Arimondo,J. Exp. Theor. Phys. , 14 (2012).[38] I. I. Ryabtsev, D. B. Tretyakov, I. I. Beterov,V. M. Entin, and E. A. Yakshina, Phys. Rev. A ,053409 (2010).[39] B. G. Richards and R. R. Jones, Phys. Rev. A , 042505(2016).[40] H. Park, T. F. Gallagher, and P. Pillet, Phys. Rev. A ,052501 (2016).[41] M. Kiffner, W. Li, and D. Jaksch, Phys. Rev. Lett. ,233003 (2013).[42] D. Barredo, H. Labuhn, S. Ravets, T. Lahaye,A. Browaeys, and C. S. Adams, Phys. Rev. Lett. ,113002 (2015).[43] D. B. Tretyakov, V. M. Entin, E. A. Yakshina,I. I. Beterov, C. Andreeva, and I. I. Ryabtsev, Phys. Rev.A , 041403(R) (2014). Appendix: SUPPLEMENTARY MATERIALDerivation of the true many-body spectra from theexperimental multiatom spectra of the F¨orsterresonances
The measured normalized N -atom signals S N are thefractions of atoms that have undergone a transition tothe final nS state (or the population of the nS stateper atom). As shown in our paper [S1], for the non-ideal selective-field-ionization (SFI) detector, which de-tects fewer atoms than actually have interacted, varioustrue multiatom spectra ρ i of the F¨orster resonances for i interacting Rydberg atoms contribute to our measuredsignals S N for N detected Rydberg atoms to a degreethat depends on the mean number of the detected atoms.The signals S N are thus a mixture of the spectra ρ i fromthe larger numbers of actually interacted atoms i ≥ N : S N = ρ + e − ¯ n (1 − T ) ∞ X i = N ρ i [¯ n (1 − T )] i − N ( i − N )! , (A.1)where ρ is a nonresonant background signal due toblackbody-radiation-induced transitions and backgroundcollisions, ¯ n is the mean number of Rydberg atoms ex-cited per laser pulse, and T is the detection efficiency ofthe SFI detector. The value of ρ should be the same forvarious N since it is caused by the parasitic transitionsin each single atom.The mean number of detected Rydberg atoms is ¯ nT .The measurement of this value and of the relationship α = ( S − ρ ) / ( S − ρ ) (A.2) FIG. 4: (color online) (a) Raw data S N recorded for theStark-tuned F¨orster resonance in Rb Rydberg atoms for var-ious numbers of the detected atoms N =1 − P / ( | M | =1/2). Presence of the resonance for N =1 is dueto the finite detection efficiency of 72%. (b) True multiatomspectra ρ i derived from S N . The data are corrected for thedetection efficiency using the procedure described in the text. at zero F¨orster detuning can provide a measurement ofthe unknown values of ¯ n and T . In Ref. [S1] we consideredthe case of the weak dipole-dipole interaction, when thefollowing scaling was assumed to be valid: ρ i ≈ ( i − ρ , (A.3)For this case it was shown that¯ n ≈ [ α/ (1 − α ) + ¯ nT ] . (A.4)This expression, however, is valid only for the very weakdipole-dipole interaction, when multiatom F¨orster res-onances are far below the saturation, as it was in ourexperiment with the Na thermal atomic beam [S1].Now let us consider an example of the multiatomF¨orster resonance for 36 P / ( | M | = 1 /
2) atoms recordedfor the interaction time of 3 µ s in our present experiment,shown in Fig. 4(a). Our aim is to make a decompositionof the experimental records S − S for N =1 − ρ i of the F¨orster resonances for exactly i inter-acting Rydberg atoms, which are defined according toEq. (A.1). For this purpose, we first need to find theunknown values of ¯ n and T .As a starting point, for this experiment we alreadyknow the mean number of the detected Rydberg atoms¯ nT ≈ .
05, which was specially measured and recorded ineach experiment. Then, using Eq. (A.4) we in principlecan find ¯ n and T . However, Eq. (A.4) seems to be invalid FIG. 5: (color online) (a) Numerical simulation of the two-body F¨orster resonance in Rb Rydberg atoms for variousnumbers of the interacting atoms i =2 − P / ( | M | =1/2). The theoretical spectra have been calcu-lated with the Schr¨odinger’s equation for the cubic interactionvolume of 17 × × µ m , 3 µ s interaction time and MonteCarlo averaging over 1000 random atom positions. (b) Ratioof the spectra ρ and ρ is shown by the blue (dark grey)curve, and its fit by the inverted Lorentz function is shownby the green (light grey) curve. for Fig. 4(a), because the spectra are close to the satura-tion and Eq. (A.3) obviously does not work. Therefore weneed first to modify Eq. (A.4) for the case of saturation.Figure 5(a) presents the results of numerical simula-tions for the theoretical multiatom two-body spectra ρ i for the 36 P / ( | M | =1/2) atoms in the cubic interactionvolume of 17 × × µ m for the interaction time of 3 µ s(these parameters are close to the experimental ones). Itis seen that at zero detuning the amplitudes of all res-onances saturate at the 0.25 value. Therefore, at zerodetuning instead of Eq. (A.3) we should now adopt that ρ ≈ ρ ≈ ρ ≈ ρ ≈ ... . Then Eqs. (A.1) and (A.2) give α ≈ − e − ¯ n (1 − T ) , (A.5)¯ n ≈ ln 11 − α + ¯ nT. (A.6)The values of ρ ≈ . S =0.09 and S =0.21 havebeen measured from the spectra in Fig. 4(a). This allowsus to find α ≈ .
34, ¯ n ≈ .
46, and T ≈ .
72. Withthese values we can explicitly write down the expansioncoefficients for the multiatom spectra in Eq. (A.1): S = ρ + 0 . ρ + 0 . ρ + 0 . ρ + 0 . ρ + ... ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ + 0 . ρ + ... ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ + 0 . ρ + ... ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ + 0 . ρ + ... ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ + 0 . ρ + ... . (A.7)In Eqs. (A.7) we should take into account that ρ = 0 in S , because there is no interaction for a single atom.In order to derive ρ and ρ , which are necessary for theanalysis of the three-body F¨orster resonance, we shouldsimplify Eqs. (A.7) to exclude the terms with large num-bers of atoms. First, the terms with the weight of 0.01have small contribution and with a small error can bejust added to the preceding terms as follows: S = ρ + 0 . ρ + 0 . ρ ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ . (A.8)Second, we believe that the multiatom spectra inFig. 4(a) are reliably measured for N =1 −
4, while thespectrum for N =5 can be affected by the nonlinearity ofour channeltron. Therefore, in the further analysis wewill consider only the experimental spectra with N =1 − ρ and ρ in Eqs. (A.8). This can bedone if we approximately express ρ and ρ via ρ usingthe theoretical curves in Fig. 5(a). Figure 5(b) shows asthe blue (dark grey) curve the ratio r = ρ /ρ taken fromFig. 5(a). This ratio depends on the detuning: it is 1 atzero detuning due to saturation and 1.23 at large detun-ings. The fluctuations of r at large detunings in Fig. 5(b)are due to insufficient statistics of the averaging of smallsignals, which can be smoothed if the statistics increasesor using the fitting function.We have found a fitting func-tion for this dependence [green (grey) curve in Fig.5(b)]: r (∆) ≈ . − .
23 0 . . , (A.9)where detuning ∆ is defined by the electric field F (V/cm)for the 36 P / ( | M | =1/2) atoms as∆(MHz) = − .
73 + 2 . F + 25 . F . (A.10)In the further analysis we take ρ ≈ ρ r (∆). We can alsoadopt with some precision that ρ ≈ ρ r (∆) ≈ ρ r (∆)in Eqs. (A.8), although we did not calculate ρ directly.With the above assumptions Eqs. (A.8) are modifiedas S = ρ + 0 . ρ + 0 . ρ ,S = ρ + 0 . ρ + 0 . ρ + 0 . ρ ,S = ρ + 0 . ρ + [0 .
27 + 0 . r (∆)] ρ ,S = ρ + [0 .
66 + 0 . r (∆) + 0 . r (∆)] ρ . (A.11)The straightforward calculations with Eqs. (A.11) giveus the true multi-atom spectra ρ − ρ expressed via themeasured value of ρ and spectra S − S of Fig. 4(a): ρ ≈ S − ρ .
66 + 0 . r (∆) + 0 . r (∆) ,ρ ≈ S − ρ . − [0 .
41 + 0 . r (∆)] ρ ,ρ ≈ S − ρ . − . ρ − . ρ . (A.12)In order to derive Eqs. (A.12) we used only the equa-tions for S − S in Eqs. (A.11). But after calculating ρ and ρ with Eqs. (A.12) we should also check for theidentity ρ ≈ ( S − ρ − . ρ − . ρ ) / . ≈ , (A.13)which means that we correctly decomposed the measuredspectra S − S into true multiatom spectra.Figure 5(b) presents the true multiatom spectra ρ i de-rived from Fig. 5(a) with Eqs. (A.12). The black curvefor ρ represents the identity of Eq. (A.13). We see thatin the 2-atom spectrum the feature at 1.71 V/cm has dis-appeared, while in the 3-atom spectrum it is still present. It indicates that in this experiment we really observe theBorromean three-body resonance. The validity of theabove considerations is confirmed by the fact that theidentity of Eq. (A.13) is well satisfied in Fig. 4(b), beingnearly zero.The other experimental records (Fig. 2 of the main pa-per) have been processed in the same way as the recordsin Fig. 4. We note that the approach we have used hereis similar to the approach we applied earlier to decom-pose the selective-field-ionization signals from four-bodyresonances in Cs Rydberg atoms [S2]. [S1 ] I. I. Ryabtsev, D. B. Tretyakov, I. I .Beterov, andV. M. Entin, Effect of finite detection efficiency onthe observation of the dipole-dipole interaction ofa few Rydberg atoms, Phys. Rev. A , 012722(2007); Erratum: Phys. Rev. A , 049902(E)(2007). [S2 ] J. H. Gurian, P. Cheinet, P. Huillery, A. Fioretti,J. Zhao, P. L. Gould, D. Comparat, and P. Pillet,Observation of a Resonant Four-Body Interactionin Cold Cesium Rydberg Atoms, Phys. Rev. Lett.108