Zeeman-insensitive cooling of a single atom to its two-dimensional motional ground state in tightly focused optical tweezers
Pimonpan Sompet, Yin H. Fung, Eyal Schwartz, Matt D.J. Hunter, Jindaratsamee Phrompao, Mikkel F. Andersen
ZZeeman-insensitive cooling of a single atom to its two-dimensional motional groundstate in tightly focused optical tweezers
P. Sompet, Y. H. Fung, E. Schwartz, M. D. J. Hunter, J. Phrompao, and M. F. Andersen ∗ The Dodd-Walls Centre for Photonic and Quantum Technologies,Department of Physics, University of Otago, Dunedin, New Zealand (Dated: March 7, 2017)We combine near–deterministic preparation of a single atom with Raman sideband cooling, tocreate a push button mechanism to prepare a single atom in the motional ground state of tightlyfocused optical tweezers. In the 2D radial plane, we achieve a large ground state fidelity for the entireprocedure (loading and cooling) of ∼ ∼ ∼ ∼ Complete control over individual atoms is vital forgaining a better understanding of the microscopic worldas well as enabling new technological pathways. Exten-sive progress in laser cooled atoms, confined in far off–resonance optical dipole potentials, yields an excellentplatform to observe and manipulate matter at the levelof single atoms. This has already enabled considerableheadway towards quantum logic devices [1, 2] and quan-tum simulations [3], as well as providing detailed insightinto microscopic processes whose features are often hid-den in ensemble averaged measurements. Such examplesare the atomic Hong–Ou–Mandel effect [4, 5] and theemergence of statistical mechanics in a quantum state[6]. The prospect for further developments in initiatinga wide range of effectively zero-entropy quantum statesgives this platform unprecedented potential for futurestudies of few-body physics.A major challenge in the pursuit of this goal is to pre-pare atoms in particular quantum states with near–unityfidelity. In cubic geometry, the BEC to Mott–insulatortransition allows this for sections of optical lattices [7, 8].The flexibility provided by sets of optical tweezers beams[9, 10] makes single atoms in such an ideal building blockfor diverse few–atom quantum states. A number of av-enues are being pursued for high fidelity preparation of asingle atom in a particular quantum state. A controlledspill process, utilizing Pauli’s exclusion principle, allowedfor the isolation of small sets of Fermions from a degen-erate sample [11, 12]. Separating individual Bosonic he-lium atoms using penning ionization, prepared individualatoms in the 2D radial ground state of optical tweezerswith a fidelity of about 0.5. This was primarily limitedby the 50% chance of ending with no atoms in the tweez-ers [13]. An alternative approach to achieving a singleatom in the vibrational ground state of optical tweezersis first to load the atom and subsequently cool it to its2D radial [14] or 3D [15] ground state.In this paper, we present a push button method toprovide a single Rb atom in the motional groundstate of an optical trap. The method combines near– deterministic preparation of single atoms [16, 17], withRaman sideband cooling [18]. We achieve a record fidelityof ∼ | F, m F (cid:105) ≡ | , (cid:105) internalground state while being in the | n (cid:105) state of a harmonicpotential with oscillation frequency ω . A pair of Ramanbeams are tuned to the stimulated Raman transition tothe | , (cid:105) internal ground state while stepping down thevibrational state to | n − (cid:105) . The atom is then opticallypumped back to the original internal state ( | , (cid:105) ), thuslowering the energy of the atom if the vibrational stateremains | n − (cid:105) . The entropy of the trapped atom is re-duced as the spontaneous emission of an optical pumpingphoton carries it away. The process will continue untilthe atom occupies the | n = 0 (cid:105) level in the | , (cid:105) internalstate, where it is dark for both Raman beams and opticalpumping light.The stimulated Raman transition transfers (cid:126) ∆ k of mo-mentum to the atom/trap system, where ∆ k is the wave-vector difference of the two Raman beams. The conse- a r X i v : . [ phy s i c s . a t o m - ph ] M a r … |" − 1⟩ … |" ⟩|" + 1⟩|" ⟩|" − 1⟩|( = 3,, ( = 0⟩|" + 1⟩|0⟩ O p ti ca l pu m p i ng li gh t |( = 2,, ( = 0⟩|0⟩ … … RB D li n e n m (’ = 3( = 3 , ( = 0−1−2−3 321( = 2 , ( = 0−1−2 21 (a) (c) O P O P
35 GHz
RB1RB2 σ + σ +
01 2̂41 (b)
Magnetic field Projection of RB3
FIG. 1: (color online). (a) Energy level diagram showing tran-sitions relevant to Raman sideband cooling. (b) Top view ofthe propagation of Raman beams and optical pumping beamsrelative to an atom trapped in the optical tweezers. (c) Op-tical pumping transitions where | F = 3 , m F = 0 (cid:105) is a darkstate. quent coupling between final ( | m x , m y , m z (cid:105) ) and initial( | n x , n y , n z (cid:105) ) vibrational states in 3D is represented bythe Rabi frequency for the transition [21]:Ω R |(cid:104) m x , m y , m z | e i ∆ k · ˆ R | n x , n y , n z (cid:105)| =Ω R (cid:89) j = x,y,z |(cid:104) m j | e i ∆ k j ˆ R j | n j (cid:105)| . (1)Here, Ω R is the Raman coupling parameter between thetwo internal states, and ˆ R is the position operator. Fromthe Rabi frequency expression, we can change the vibra-tional states ( n j to m j ) for all three dimensions by usingonly a single pair of Raman beams as long as ∆ k has aprojection on all of them.In Fig. 1(b), we present the schematics of our Ramancooling experiment. A strong, linearly polarized, far off–resonance, dipole trap beam ( λ = 1064 nm, ω = 1 . µ m) propagates along the ˆ z direction, and holds an atomat the focal point. The applied magnetic field (7.5 Gaussin the − (cid:113) ˆ x + √ ˆ z direction) defines the quantizationaxis of the atom in its internal ground state. Its direc-tion is not aligned with the polarization axis (ˆ x ) of thetrap beam due to geometric constraints in our experi-ment. We use three beams to drive Raman transitions(denoted RB1, RB2 and RB3 where the beam peak inten-sities are 0.5, 1.7 and 2.4 × mW/cm respectively).RB1/RB2 propagates antiparallel/parallel to the mag-netic field, and both beams are circularly polarized ( σ + ).RB3 propagates orthogonally to the magnetic field (along the − √ ˆ x + √ ˆ y − √ ˆ z direction) and has its linear po-larization perpendicular to it as well ( π ⊥ ), hence it candrive a σ ± transition in the frame defined by the mag-netic field. The ∆ k of the RB1–RB2 pair thereby has aprojection on ˆ x (radial dimension of the trap) and ˆ z (ax-ial), while the ∆ k of a RB1–RB3 pair has a projectionon the ˆ y and ˆ x (radial) directions. The optical pumpinglight, nearly counter propagates with RB3, and is linearlypolarized along the quantization axis.Our cooling scheme uses the | , (cid:105) to | , (cid:105) internalstate transition which is insensitive to the Zeeman effectto first order. This means that the transition frequencydoes not change significantly due to the temporal varia-tions in background magnetic fields that prohibit us fromusing Zeeman–sensitive transitions for Raman sidebandcooling. Using the Zeeman–insensitive transition doeshowever have the drawback, that it typically requires arelatively high number of spontaneous photon–scatteringevents to optically pump the atom back to the initial in-ternal state. This presents a problem in the cooling pro-cess, since spontaneous photon–scattering is a source ofheating due to the recoil kicks that may change the vi-brational quantum number n [22]. In sideband coolingschemes this problem is mitigated by the Lamb–Dickeeffect that suppresses the probability of changing n fortightly confined atoms [23].Figure 1(c) illustrates our optical pumping light whichincorporates two light frequencies matched to the D1 line,denoted OP1 (resonant with the F = 2 to F (cid:48) = 3 transi-tion) and OP2 (resonant with the F = 3 to F (cid:48) = 3 tran-sition). The π -polarized optical pumping light, cause theatoms to accumulate in the | , (cid:105) internal state given thatthe transition from this state to the | (cid:48) , (cid:105) excited stateis forbidden according to selection rules. After opticalpumping, we measure the population of the | , (cid:105) stateto be ∼ m F indepen-dent. Therefore, the magnetic field defines the quantiza-tion axis for optical pumping, even when it is not alignedwith the polarization axis of the trap light. Hence, the | , (cid:105) state remains dark in the presence of the deep opti-cal trap. Since the transfer to the | , (cid:105) ground state relieson random changes of m F and F in the ground state man-ifold (see Fig. 1(c)), it takes an average of ∼ | , (cid:105) state under ideal conditions. This is significantly higherthan the few events required when one uses the maximal m F states, as is conventionally done [14, 15]. The highnumber of photon–scattering events deteriorates the Ra-man sideband cooling process if an atom leaves the | , (cid:105) state for reasons other than undergoing the desired stim-ulated Raman transitions. Moreover, polarization pollu-tion and off–resonant scattering from other excited statesdictate that the | , (cid:105) state will not be completely dark tothe OP2 light. Therefore, during the cooling cycles we in- P r ob a b ilit y i n ( = Detuning from carrier (kHz) Raman pulse duration ( s) Raman pulse duration ( s) Carrier ∆" = +1 ∆" = −1 ∆" = +1∆" = −1 ∆" = +1∆" = −1∆" = −1∆" = +1 ∆" = +1∆" = −1 (a) (b)(c) (d)(e)
FIG. 2: (color online). (a) Raman sideband spectrum before (black) and after (blue) the sideband cooling, obtained fromspectroscopy by using the RB1–RB3 pair for pulse durations of 90 and 180 µ s respectively. The sideband peaks are fitted witha Lorentzian function, with the solid lines showing the fitted curves. The carrier peak measured using the pulse duration of 40 µ sis also plotted as the grey data set. The offset in the spectrum comes from a combination of the spontaneous emission inducedby the Raman beams and the efficiency of the internal state detection. (b)/(c) The transition probability as a function of RB1–RB3 pulse duration at the ∆ n = − n = +1 radial sideband peaks before/after the cooling sequence. The transitionprobability data at an off–resonance (purple triangles) represents the background level. (d)/(e) The transition probability usingthe RB1–RB2 pair, for before/after the cooling sequence. Data is fitted with damped cosine functions, with the solid linesshowing the fitted curves. termittently apply several Raman pulses separated onlyby OP1 light (OP1 depletes the population in | , (cid:105) state)between every optical pumping pulse that contains bothOP1 and OP2 frequencies. This enhances the probabil-ity that an atom undergoes a desired Raman transitionwhile suppressing the likelihood of leaving the | , (cid:105) statedue to the aforementioned imperfections.We start our experimental sequence by laser coolingand preparing a single atom in a tight optical trap us-ing the near–deterministic loading scheme based on en-gineered blue–detuned light–assisted collisions [16]. Inour present configuration, the procedure delivers a singleatom with a probability of 83% into a trap with h × h ×
175 MHz leaving the single atom with a temper-ature of 33 µ K (measured by the release-and-recapture(RR) technique [26]). At this stage, the trap frequenciesare { ω x , ω y , ω z } / π (cid:39) { , , } kHz. Soon after,an optical pumping pulse prepares the atom in the | , (cid:105) state. After Raman pulses, we determine the popula-tion transfer to the | , (cid:105) state by a push–out technique[20] that allows us to distinguish the populations of the F = 2 and F = 3 ground states (with efficiency of 0.96for both).We further characterize the temperature of the atomand the ground state population using sideband spec-troscopy. Figure 2(a) shows the Raman spectrum ob-tained using the RB1–RB3 pair after the initial prepa-ration of the atom. The asymmetry between the heightof the ∆ n = − n = +1 sideband peaks (denoted P − and P +1 respectively) characterizes the populationof the atoms in | n = 0 (cid:105) because this state will not con-tribute to the ∆ n = − n = P − /P +1 − P − /P +1 [22]. Following this, we determine { ¯ n x , ¯ n r (cid:48) , ¯ n z } (cid:39) { . , . , } (ˆ r (cid:48) = √ ˆ x − ˆ y , the di-rection of ∆ k for the RB1–RB3 beam pair) which cor-responds to temperatures of { , , } µ K consistentwith the temperature measured by the RR method.In Fig. 2(a), we also present the Raman spectrum ob-tained after 48 Raman sideband cooling cycles, using thesame RB1–RB3 beam pair. The first 24 cooling cyclesconsist of three Raman beam pulses (50, 90 and 120 µ s)seperated by OP1 light, while the rest consist of a singlepulse (100 µ s). The ∆ n = − n = +1 peak remains, indicating alarge atomic population in the ground state. We chosea Raman detuning corresponding to the ∆ n = − n = +1 sidebands and measured the transition prob-ability as a function of duration of the Raman pulse.Figure 2(b)/(c) shows the result before/after the cool-ing. We see damped oscillations before cooling due tothe fact that the Rabi frequency differs depending on the | n (cid:105) initially populated (see Eq. 1). After cooling (2c),the ∆ n = − n = +1sideband shows coherent Rabi oscillation, showing thatonly the | n = 0 (cid:105) state has a large population.Figures 2(d) and (e) reveal that the RB1–RB3 pairefficiently cools both radial dimensions simultaneously,as also observed in [15]. The figures display similardata to 2(b) and (c) but obtained with RB1–RB2 beampair after RB1–RB3 cooling. We see that the cool-ing also leads to a large sideband asymmetry for theRB1–RB2 pair. In Fig. 2(e), the oscillations are stillhighly damped. The damping arises since the Rabi fre-quency of the radial sideband depends on which axialstate is occupied when ∆ k has a significant projectiononto the axial dimension (as is the case for the RB1–RB2 pair). This axial state dependence can be seen fromEq. 1 which shows that the Rabi frequency of the ra-dial sideband contains the axial carrier matrix element (cid:104) n z | exp( i ∆ k z ˆ R z ) | n z (cid:105) . The high number of axial statesoccupied therefore leads to a large range of different Rabifrequencies and the observed damping in Fig. 2(e) (re-call that the axial dimension is not cooled). This effectwas weak in Fig. 2(c) because the RB1–RB3 pair couplesweakly to the axial dimension. From Figs. 2(c) and (e),we determine the ¯ n values by using the data where the∆ n = +1 transition probabilities are maximal. We find { ¯ n x , ¯ n r (cid:48) } = { . ± . , . ± . } with the correspond-ing ground state population of { . ± . , . ± . } .Such 2D cooling occurs if a trap imperfection breaks theradial symmetry and ∆ k has a projection on both theresulting axes while the resulting frequency difference isbelow the spectral resolution of the Raman pulses.We estimate the 2D radial ground state populationfrom the ground state populations measured by sidebandasymmetry using the RB1–RB2 and RB1–RB3 pairs sep-arately. Since the ∆ k projections of the two pairs on theradial plane are not parallel, they can transfer all non–ground state populations on the ∆ n = − . × .
96 = 0 .
88. How-ever, since the ∆ k of the two pairs are non-orthogonalit is likely that the 2D ground state population is higherthan that. In fact, we saw that the radial symmetry isbroken and the ∆ k of the RB1–RB3 pair has significantprojections on both radial dimensions, as we can achieveefficient 2D cooling using this beam pair alone. There-fore, the ground state population measured by the RB1–RB3 pair (0.96), represents an upper bound of the 2Dpopulation. Similarly, the upper bound from the RB1–RB2 pair is consistent with the RB1–RB3 pair value,within the statistical error.In Fig. 3, we characterize the sideband cooling schemethrough the evolution of ¯ n after different number of cool-ing cycles with the RB1–RB3 pair. The blue line isa fit with a simplified model that assumes the changeof energy (in units of (cid:126) ω ) per cooling cycle, α , is in-dependent of n , except for the ground state, where α (0) = 0. In this model, we further assume an ini-tial thermal population distribution with temperature, T , and calculate the ground state population after agiven number of cooling cycles, c . Consequently, weget ¯ n = [exp (cid:0) (cid:126) ω ( αc + 1) / ( k B T ) (cid:1) − − by assuminga Maxwell-Boltzman distribution. The fit gives α ≈ . | n (cid:105) states, only a portion of the ex- Number of cooling cycles " FIG. 3: (color online). Measured post–cooling ¯ n as a functionof the number of cooling cycles. The blue line is a fit with asimplified model described in the text. cited state population is transferred in a ∆ n = − α from reaching the 0.5 bound. Hence, the measured α value indicates that the cooling is efficient despite thehigh number of photon–scattering events required for op-tical pumping.To extend our cooling to 3D, we added axial coolingusing the RB1–RB2 beam pair. In our experiments sofar, we measure the 3D ground state population to be ∼ ω z / π is 36kHz); it follows that the axial motion of the atom is notdeep in the Lamb–Dicke regime (Lamb–Dicke parame-ter of η ≈ . n z changesduring the optical pumping stage [15]. To identify the re-quirements needed for efficient 3D cooling, we measured¯ n after RB1–RB3 beam pair cooling, as we varied thetrap frequency. Figure 4(a) presents the results along-side an additional point obtained using axial cooling (inred). Additionally we investigated the effects due to thescattering of optical pumping photons. To quantify theperformance of the optical pumping we use a ratio be-tween r in (the rate of pumping the atoms into the | , (cid:105) state due to OP1 and OP2) and r out (the rate of pump-ing the atoms out of the | , (cid:105) state due to OP2). Ideallythis ratio should be as large as possible, indicating theleast number of photon–scattering events during opticalpumping.We varied the r in /r out ratio, by tuning the magneticfield direction, and show the effect of this on ¯ n z after 20axial cooling cycles in Fig. 4(b). Fig. 4 indicates thatour 3D ground state population could be significantlyenhanced by increasing the axial frequency to surpass100 kHz, while a gain from further optimization of theoptical pumping would be marginal. In our apparatus,we could access ω z / π above 100 kHz by changing thedipole trap wavelength to 850 nm yielding a smaller spotsize.The Zeeman–insensitive ground state cooling works " (kHz) : ;< /: >?@ " A (a)(b) FIG. 4: (color online). Measured ¯ n as a function of parame-ters. (a) ¯ n as a function of ω/ (2 π ) after 24 cooling cycles withtriple Raman pulses. Black points represent cooling on the r (cid:48) dimension with varied ω while the red point was obtainedby cooling on the ˆ z axial dimension. (b) ¯ n z as a function of r in /r out ratio after 20 axial cooling cycles. consistently, despite magnetic field fluctuations withinthe experimental region. These fluctuations cause tensof kHz broadening of magnetically sensitive ground stateRaman transitions, which prohibits the use of Zeemansensitive states. Our Raman sideband cooling vari-ation can therefore be implemented in existing non–shielded experiments. Furthermore, cooling by usingthe magnetically–insensitive transitions, avoids the in-ternal state decoherence from using a non–paraxial trapbeam [14] and from motion in spatially varying trap lightshifts [31]. The high fidelity preparation increases thepossibilities for studying few body dynamics. Follow-ing that, the fidelity of our system could be further en-hanced if we optimize the probability for single atom oc-cupancy before cooling. This can be done by variationsof our presently–used near–deterministic loading scheme[16, 17], or through applying atomic sorting [9, 10, 17] torefill the zero occupancies from a reservoir. An alluringoption will be to use Rb atoms, which could provide abetter cooling efficiency as the atoms have a lower num-ber of internal ground states ( ∼ ∼ ∼ ∗ Electronic address: [email protected][1] M. Saffman, J. Phys. B , 202001 (2016).[2] T. Xia, M. Lichtman, K. M. Maller, A. W. Carr,M. J. Piotrowicz, L. Isenhower, and M. Saffman,Phys. Rev. Lett. , 100503 (2015).[3] H. Labuhn, D. Barredo, S. Ravets, S. de L´es´eleuc,T. Macr´ı, T. Lahaye, and A. Browaeys, Nature , 667-670 (2016).[4] A. M. Kaufman, B. J. Lester, C. M. Reynolds,M. L. Wall, M. Foss–Feig, K. R. A. Hazzard, A. M. Rey,and C. A. Regal, Science , 306-309 (2014).[5] R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin,M. Rispoli, and M. Greiner, Nature , 77-83 (2015).[6] A. M. Kaufman, M. Eric Tai, A. Lukin, M. Rispoli,R. Schittko, P. M. Preiss, and M. Greiner, Science ,794 (2016).[7] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau,P. Schau β , S. Hild, D. Bellem, U. Schollw¨ock, T. Gia-marchi, C. Gross, I. Bloch, and S. Kuhr, Nature Phys. ,241 (2013).[8] J. Choi, S. Hild, J. Zeiher, P. Schau β , A. Rubio–Abadal,T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, andC. Gross, Science , 1547-1552 (2016).[9] Y. Miroshnychenko, W. Alt, I. Dotsenko, L. F¨orster,M. Khudaverdyan, D. Meschede, D. Schrader, andA. Rauschenbeutel, Nature , 151-151 (2006).[10] M. Endres, H. Bernien, A. Keesling, H. Levine,E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletic,M. Greiner, and M. D. Lukin, Science , 1024-1027(2016). D. Barredo, S. de L´es´eleuc, V. Lienhard, T. La-haye, and A. Browaeys. Science , 1021-1023 (2016).[11] F. Serwane, G. Z¨urn, T. Lompe, T. B. Ottenstein,A. N. Wenz, and S. Jochim, Science , 336 (2011).[12] B. Zimmermann, T. Mueller, J. Meineke, T. Esslinger,and H. Moritz, New J. Phys. , 043007 (2011).[13] A. G. Manning, R. Khakimov, R. G. Dall, and A. G. Tr-uscott, Phys. Rev. Lett. , 130403 (2014).[14] J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuleti´c,and M. D. Lukin, Phys. Rev. Lett. , 133001 (2013).[15] A. M. Kaufman, B. J. Lester, and C. A. Regal,Phys. Rev. X , 041014 (2012).[16] T. Grunzweig, A. Hilliard, M. McGovern, and M. F. An-dersen Nature Phys. , 951 (2010). A. V. Carpentier,Y. H. Fung, P. Sompet, A. J. Hilliard, T. G. Walker, andM. F. Andersen, Laser Phys. Lett. , 125501 (2013).Y. H. Fung, and M. F. Andersen, New J. Phys. ,073011 (2015).[17] B. J. Lester, N. Luick, A. M. Kaufman, C. M. Reynolds,and C. A. Regal, Phys. Rev. Lett. , 073003 (2015). [18] S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax,I. H. Deutsch, and P. S. Jessen, Phys. Rev. Lett. , 4149(1998). V. Vuleti´c, C. Chin, A. J. Kerman, and S. Chu,Phys. Rev. Lett. , 5768 (1998). X. Li, T. A. Corcovilos,Y. Wang, and D. S. Weiss, Phys. Rev. Lett. , 103001(2012).[19] N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, andS. Chu, Phys. Rev. Lett. , 1311 (1995).[20] S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Mirosh-nychenko, A. Rauschenbeutel, and D. Meschede,Phys. Rev. A , 023406 (2005).[21] S. Blatt, J. W. Thomsen, G. K. Campbell, A. D. Ludlow,M. D. Swallows, M. J. Martin, M. M. Boyd, and J. Ye,Phys. Rev. A , 052703 (2009).[22] D. Leibfried, R. Blatt, C. Monroe, and D. J. Wineland,Rev. Mod. Phys. , 281 (2003).[23] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried,B. E. King, and D. M. Meekhof, arXiv:quant-ph/9710025(1997).[24] Measurment of optical pumping efficiency: after opticalpumping, we introduce OP1 to deplete the F = 2 groundstate population. Then we transfer all | , (cid:105) population to the | , (cid:105) state by using co-propagating Raman beamsand measure the F = 2 state population.[25] A. J. Hilliard, Y. H. Fung, P. Sompet, A. V. Carpentier,M. F. Andersen, Phys. Rev. A , 053414 (2015). M. Mc-Govern, A. J. Hilliard, T. Grunzweig, and M. F. Ander-sen, Opt. Lett. , 1041 (2011)[26] C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sor-tais, and P. Grangier, Phys. Rev. A , 033425 (2008).[27] A. D. Boozer, A. Boca, R. Miller, T. E. Northup, andH. J. Kimble, Phys. Rev. Lett. , 083602 (2006).[28] M. Yeo, M. T. Hummon, A. L. Collopy, B. Yan,B. Hemmerling, E. Chae, J. M. Doyle, and J. Ye,Phys. Rev. Lett. , 223003 (2015).[29] E. B. Norrgard, D. J. McCarron, M. H. Steinecker,M. R. Tarbutt, and D. DeMille, Phys. Rev. Lett. ,063004 (2016).[30] A. Browaeys, D. Barredo, and T. Lahaye, J. Phys. B ,152001 (2016).[31] J. Yang, X. He, R. Guo, P. Xu, K. Wang, C. Sheng,M. Liu, J. Wang, A. Derevianko, and M. ZhanPhys. Rev. Lett.117