Cesium n D J + 6 S 1/2 Rydberg molecules and their permanent electric dipole moments
Suying Bai, Xiaoxuan Han, Jingxu Bai, Yuechun Jiao, Jianming Zhao, Suotang Jia, Georg Raithel
aa r X i v : . [ phy s i c s . a t m - c l u s ] J un Cesium nD J + S / Rydberg molecules and their permanent electric dipole moments
Suying Bai † , , Xiaoxuan Han † , , Jingxu Bai , , YuechunJiao , , Jianming Zhao , , ∗ Suotang Jia , , and Georg Raithel State Key Laboratory of Quantum Optics and Quantum Optics Devices,Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China and Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120, USA (Dated: June 24, 2020)Cs Rydberg-ground molecules consisting of a Rydberg, nD J (33 ≤ n ≤ S / ( F =3 or 4), are investigated by photo-association spectroscopy in a cold atomic gas.We observe vibrational spectra that correspond to triplet T Σ and mixed
S,T
Σ molecular states. Weestablish scaling laws for the energies of the lowest vibrational states vs principal quantum numberand obtain zero-energy singlet and triplet s -wave scattering lengths from experimental data anda Fermi model. Line broadening in electric fields reveals the permanent molecular electric-dipolemoments; measured values agree well with calculations. We discuss the negative polarity of thedipole moments, which differs from previously reported cases. PACS numbers: 32.80.Ee, 33.20.-t, 34.20.Cf
Recently, molecules formed between a ground-stateand a Rydberg atom have attracted considerable atten-tion due to their rich vibrational level structure andpermanent electric dipole moments, which are uniquefor homonuclear molecules. A Rydberg-ground moleculearises from low-energy scattering between the Rydbergelectron and ground-state atoms located inside the Ryd-berg electron’s wavefunction. This interaction, initiallyinvestigated in [1, 2], has been predicted to lead to molec-ular binding in a novel type of Rydberg molecules, includ-ing the so-called trilobite [3] and butterfly molecule [4, 5].The molecular bond length is on the order of theRydberg-atom size (a thousand Bohr radii a ). Rydberg-ground molecules were first reported in experimentswith Rb nS / ( n = 35-37) states [6] and later withRb nP / , / [7] and nD / , / [8–10] states, as wellas with Cs nS / [11, 12], nP / [13] and nD / [14]states. The permanent electric dipole moment of S-typeRydberg-ground molecules has been measured to be ∼ ( nD J + 6 S / F ) Rydberg-ground molecules for 33 ≤ n ≤ J = 3 / F = 3 or 4. These moleculesare deeply in the Hund’s case(c)-regime, which differsfrom Rb nD / , / -type molecules at lower n , whichare Hund’s case(a) [10] or between Hund’s case(a) and(c) [8, 9]. Using a Fermi model, we calculate molecularpotential energy curves (PECs), vibrational energies and ∗ Corresponding author: [email protected], † These authors con-tributed equally to this work. permanent electric-dipole moments.The scattering interaction between the Rydberg elec-tron and the ground-state atom is, in the reference frameof the Rydberg ionic core [2], b V ( r ; R ) =2 πa s ( k ) δ ( r − R ˆ z )+ 6 π [ a p ( k )] δ ( r − R ˆ z ) ←−∇ · −→∇ (1)where r and R ˆ z are the positions of the Rydberg electronand the perturber atom, a l ( k ) the scattering lengths, k is the electron momentum, and l the scattering partial-wave order (0 or 1 for s -wave or p -wave, respectively).The full Hamiltonian of the system is [23],ˆ H ( r ; R ) = ˆ H + X i = S,T ˆ V ( r ; R ) ˆ P ( i ) + A HF S ˆ S · ˆ I (2)where ˆ H is the unperturbed Hamiltonian, which in-cludes the spin-orbit interaction of the Rydberg atom.The second term sums over singlet ( i = S ) and triplet( i = T ) scattering channels, using the projection opera-tors ˆ P ( T ) = ˆ S · ˆ S + 3 /
4, ˆ P ( S ) = 1 − ˆ P ( T ) (ˆ S and ˆ S are the electronic spins of the Rydberg and ground-stateatom, respectively). The last term represents the hyper-fine coupling of ˆ S to the ground-state-atom nuclear spinˆ I , with hyperfine parameter A HF S . Numerical solutionsof the Hamiltonian in Eq. (2) on a grid of R -values yieldsets of PECs. Figure 1(a) shows four PECs that areasymptotically connected with the atomic 36 D / -state.The PECs for T Σ for 6 S / F =3 and F =4 are practicallyidentical, while the PECs for S,T
Σ are ∼
10 MHz deeperfor F =3 than for F =4. A similar behavior was seen inRb [8, 10] and Cs [13].The experiment is performed in a crossed opticaldipole trap (CODT) loaded from a magneto-optical trap(MOT). The CODT density, measured by absorptionimaging, is ∼ cm − . This is sufficiently denseto excite Rydberg-ground molecules with bond lengths +6S (F=4) 36D +6S (F=3) W ( M H z ) R (a ) (a) (b) +6S (F=4) (c) Signal (arb.units) +6S (F=3)
FIG. 1. (a) PECs for 36 D / + 6 S / ( F = 4) (dashed lines)and 36 D / + 6 S / ( F = 3) molecules (gray solid lines), re-spectively. The deep potentials mostly arise from triplet s -wave scattering ( T Σ) and do not depend on F . The shallowpotentials mostly arise from s -wave scattering of mixed S,T
Σ-states and depend on F ; the PEC for S,T Σ F = 3 is deeperthan that for S,T Σ F = 4. The colored lines show the lowestvibrational wavefunctions on the PECs. (b,c) Experimen-tal photo-association spectra for 36 D / + 6 S / ( F = 4) and36 D / +6 S / ( F = 3) molecules. Energies are relative to therespective 36 D / asymptotes. Filled (open) triangles markthe molecular signals formed by mixed S,T
Σ (triplet T Σ) po-tentials. Gray symbols and error bars show data points, blacklines display smoothed averages. The error bars are the stan-dard error of ten independent measurements. The thin yellowlines display Gaussian fittings. ∼ . µ m (our case). After switching off the trappinglasers, two counter-propagated 852- and 510-nm lasers(pulse duration 3 µ s) are applied to photo-associate theatoms into Rydberg-ground molecules. The lasers areboth frequency-stabilized to the same high-finesse Fabry-Perot (FP) cavity to less than 500 kHz linewidth. The852-nm laser is 360 MHz blue-detuned from the interme-diate | P / , F’= 5 i level. The 510-nm laser is scannedfrom the atomic Rydberg line to ∼
150 MHz below byscanning the radio-frequency signal (RF) applied to theelectro-optic modulator used to lock the laser to the FPcavity. Rydberg molecules are formed when the detun-ing from the atomic line matches the binding energyof a molecular vibrational state. Rydberg atoms andmolecules are detected using electric-field ionization anda microchannel plate (MCP) ion detector. Suitable tim-ing of the MOT repumping laser allows us to prepare theatoms and molecules in either F =4 or F =3. The 510-nm laser can be tuned to excite either nD / + 6 S / or nD / + 6 S / molecules.In Fig. 1 we show photo-association spectra of 36 D / +6 S / molecules for F = 4 (Fig. 1(b)) and F = 3(Fig. 1(c)), respectively. To reduce uncertainties, thespectra are averaged over ten measurements. Both spec-tra display a pair of dominant molecular peaks, markedwith triangles. They correspond to the vibrationalground ( ν = 0) states in the outermost wells of theshallow ( S,T
Σ) and deep ( T Σ) PECs shown in Fig. 1(a), which arise from s -wave scatting. The deep, T Σ PEC cor-responds with a triplet state of the Rydberg electron andthe 6 S / atom. The two S,T
Σ PECs correspond withmixed singlet-triplet states and have a reduced depth,which roughly is in proportion with the amount of tripletcharacter in the molecular states. The binding energiesof the ∗ Σ , ν = 0 states are extracted from Gaussian fitsto the measured molecular peaks, with statistical uncer-tainties on the order of 1 MHz. Systematic uncertaintiesin the molecular line positions are negligible because ofthe high signal-to-noise ratio of the atomic reference linesin the spectra (relative to which the binding energies aremeasured), and because the FP cavity and the RF usedto lock and scan the lasers have no significant drift. (b) (a) + 6S (F=3) + 6S (F=4) S i gn a l ( a r b . un it s ) Detuning from 36D (MHz) Detuning from 36D (MHz) + 6S (F=3) + 6S (F=4)
FIG. 2. Measured spectra of 36 D J + 6 S / molecules for J =3 / J = 5 / F = 3 (top) and F = 4 (bottom).The laser detunings are relative to the atomic resonances, andthe signal strengths are displayed on identical scales. Verticalsolid and dashed lines mark the signals of the T Σ( ν = 0)and S,T Σ( ν = 0) ground vibrational states, respectively. Thesignal strengths of the J = 5 / J = 3 / nD / atoms. We have obtained the photo-association spectra for allcombinations of J and F , for n = 33 to 39. In Fig. 2we show the results for the case n = 36. The T Σ , ν = 0and S,T Σ , ν = 0 states are well-resolved and allow for ac-curate comparison of level energies between experimentand theory. The T Σ, ν = 0 levels, marked by solid ver-tical lines, do not depend on F . Since the PECs for themeasured states are largely due to s -wave scattering, theratio of the binding energies of the T Σ, ν = 0 levels be-tween J = 3 / J = 5 / h J, m j = 1 / | m ℓ = 0 , m s = 1 / i , with J = 3 / /
2, and with magnetic quantum numbers m j , m ℓ and m s for the coupled, orbital and electron spins ofthe Rydberg electron, respectively. For D -type Rydberg-ground molecules in Hund’s case (c), the binding-energyratio is ℓ/ ( ℓ + 1) = 2 /
3, which is close to the binding-energy ratio evident in Fig. 2. The vertical dashed linesof Fig. 2 mark the
S,T Σ , ν = 0 states, which are mixedsinglet-triplet. These are about half as deeply bound as TABLE I. Fitted exponents b (see text) for the scaling ofthe binding energies of S,T Σ ν = 0 and T Σ ν = 0 states of( nD / + 6 S / F ) molecules, for F = 3 and 4, over the range33 n
39. The fit function is a n ∗ b , with effective quantumnumber n ∗ and exponent b .S/T(F=3) T(F=3) S/T(F=4) T(F=4) b Exp. -5.65 ± ± ± ± b Theor. -5.68 ± ± ± ± T Σ , ν = 0, whereby S,T Σ , ν = 0 for F = 3 is about 5 to10 MHz more deeply bound than S,T Σ , ν = 0 for F = 4.For quantitative modeling of the singlet and triplet s -wave scattering length functions a Ts ( k ) and a Ss ( k ), wehave measured the binding energies of the states T Σ , ν =0 and S,T Σ , ν = 0 for nD / + 6 S / molecules with n = 33 −
39, for both values of F . The measured data,listed in detail in the Supplement, are fitted with func-tions a n ∗ b Exp , with effective quantum number n ∗ andexponent b Exp (see Table I). The b Exp. are concentratedaround − .
60, with one exception. Calculated bindingenergies, listed in the Supplement, yield respective fit-ted exponents b T heor. that are within the uncertainty ofthe b Exp. (see Table I), with the exception of the
S,T Σ F = 4 case, where the binding energies are smallest. The b -values generally have a magnitude that is significantlyless than −
6. A value of − b from − n the molecules are less deep in Hund’s case (c) than athigher n . This may diminish the binding of the J = 5 / n and lead to a reduction of the mag-nitude of b . A modification of the scaling may also arisefrom p -wave-scattering-induced configuration mixing atlower n as well as from the zero-point energy of the vi-brational states.The measured binding-energy data are employed to de-termine s -wave scattering lengths via comparison withmodel calculations similar to [10]. The calculations yieldbest-fitting s -wave scattering-length functions for bothsinglet and triplet scattering, a Ss ( k ) and a Ts ( k ), with zero-energy scattering lengths a Ss ( k = 0) = − .
92 a and a Ts ( k = 0) = − .
16 a ; a comparison with previousresults is presented in the Supplement. In our calcula-tion we have included p -wave scattering and found thatit has only a small effect on the lowest vibrational reso-nances in the outermost wells of the PECs [24], withinour n -range of interest. This is because the outermostwells are separated fairly well from further-in wells andare therefore strongly dominated by s -wave scattering,justifying our use of less accurate non-relativistic p -wavescattering-length functions a Sp ( k ) and a Tp ( k ) [25].Homonuclear Rydberg-ground molecules are unusual,in part, because of their permanent electric dipole mo-ment, d , which are caused by configuration mixing. Thevalues of d are usually small in molecules with low- ℓ char- -75 -60 -45 -30 -150.00.10.20.30.40.50.6 -60 -500.400.450.50 -30 -20 -100.400.450.50 ST Detuning (MHz) S i gn a l ( a r b . un it s ) T S i gn a l ( a r b . un it s ) Detuning (MHz)T (Triplet), |d | = 5.70 ea ST (Mixed), |d | = 6.34 ea Detuning (MHz) S i gn a l ( a r b . un it s ) FIG. 3. Spectra of 37 D / + 6 S / ( F = 4) Rydberg-groundmolecules with indicated electric fields, E . The molecularpeaks of T Σ , ν = 0 and S,T Σ , ν = 0 are blue-shifted by E and substantially broadened in fields E ≥ .
27 V/cm. Theright panel shows zoom-ins on the states T Σ( ν = 0) (top) and S,T Σ( ν = 0) (bottom). The red solid lines show model spectrabased on Eq. 3 for dipole moments of magnitude | d | = 5 .
70 ea for T Σ , ν = 0 and 6.34 ea for S,T Σ , ν = 0, respectively. acter, with the notable exception of Cs S -type molecules,where the quantum defect allows strong mixing withtrilobite states [12]. The values of d i,ν , with index i denoting the PEC and ν the vibrational state, can bemeasured via the broadening of the respective molecularline in an applied weak electric field, E . For electric-dipole energies, − d i,ν · E , that are much smaller than themolecular binding energy, the line is inhomogeneouslybroadened about its center by a square function of fullwidth 2 d i,ν E/h in frequency. This model applies if themoment of inertia of Rydberg molecules is very large androtational structure cannot be resolved (our case). Thesquare function is convoluted with a Gaussian profile toaccount for laser line broadening, electric-field inhomo-geneities, magnetic fields etc. The standard deviation σ f of this Gaussian is experimentally determined by fit-ting field-free molecular lines. The overall line profile, S i,ν (∆ f ), as a function of detuning ∆ f from the linecenter then is h dE " erf ∆ f + d i,ν E/h √ σ f ! − erf ∆ f − d i,ν E/h √ σ f ! . (3)Since the field E is accurately known from Rydberg Starkspectroscopy, the values of | d i,ν | follow from comparingmeasured line shapes with profile functions calculated us-ing Eq. (3) over a range of test values for | d i,ν | .In Fig. 3 we show line-broadening measurements for37 D / +6 S / ( F = 4) Rydberg molecules in several elec-tric fields, as well as fit results based on Eq. 3 for thevibrational ground states of T Σ (top) and
S,T
Σ (bottom)PECs for the case E = 0 .
37 V/cm. The obtained dipole-moment magnitudes are 5.70 (6.34) ea for the triplet(mixed) states. Analysis of the spectra for 0.18, 0.27 and0.37 V/cm yields averaged dipole-moment magnitudes of4.79 ± ea for T Σ and 5.49 ± ea for S,T
Σ.For a comparison with theory, we first solve Eq. (2) toobtain the PECs and electronic adiabatic dipole momentsalong the internuclear axis, d i,z ( R ). We then find thevibrational energies and wavefunctions, Ψ i,ν ( R ), on thePECs [23]. The dipole moments of the molecules, d i,ν ,are d i,ν = Z | Ψ i,ν ( R ) | d i,z ( R ) dR. (4)For the T Σ , ν = 0 states we find d i,ν values ranging be-tween -4.85 ea at n = 33 and -4.60 ea at n = 38. For n = 37, the calculated dipole moment is -4.64 ea , whichis in good agreement with the measured result ( | d | = 4.79 ± ea ).We note that the molecular lines also exhibit a DCStark shift due to the electric polarizability, α , of theRydberg atom. The atomic DC Stark shifts, − α m J E / m J due to thetensor component of the polarizability. If the molecularStark shift is less than the molecular binding, it can becalculated perturbatively as an average shift with weights P ( m J ), where m J is in the laboratory frame (definedby the direction of the applied electric field). Figure 3further includes a hint that the molecular lines may splitin stronger electric fields (top curve for 0.55 V/cm). TheDC Stark shifts and possible splittings can result in anoverestimate of the molecular dipole moment; this mayexplain the deviations between measured and calculateddipole moments.While the current measurement method does not givethe sign of d i,ν , the calculations reveal that the d i,ν of Cs nD J -type Rydberg-ground molecules are negative ,which differs from reports on other types of Rydberg-ground molecules [11–13, 22]. Physically, the sign of d i,ν reflects the direction of the electronic charge shift alongthe axis of the Rydberg molecule relative to the locationof the Cs 6 S / atom. The direction of the weak elec-tric field E applied to measure the dipole moment is notrelevant, as long as the field is weak (our case). A nega-tive d i,ν corresponds with a deficiency of electron chargefrom the vicinity of the Cs 6 S / perturber atom. Thissituation can generally be described as destructive inter-ference of the Rydberg electron wavefunction near theperturber or, equivalently, as a possible case of electronicconfiguration mixing near the perturber (LCAO picture).For further illustration, in Fig. 4 we show electronicwavefunctions of Cs D -type and P -type Rydberg-groundmolecules in the outer well of the respective PECs (seeFig. 1 for typical PECs). Since the configuration mix-ing is weak, in the bottom panels in Fig. 4 we plot thedifference of the wavefunction density relative to that ofthe unperturbed atomic state. An analysis of the elec-tronic states by ℓ - and m -quantum numbers shows thatthe D -type molecule mostly mixes with P orbitals andwith a combination of high − ℓ states similar to the trilo-bite state [3], while the P -type molecule mostly mixeswith D orbitals and the trilobite-like state. Admixtures FIG. 4. Densities of adiabatic electronic wavefunctions forCs 31 D / + 6 S / ( F = 4) T Σ (left) and 32 P / + 6 S / ( F = 4) T Σ (right panels), with the perturber located at ≈ a (dot). Top: wavefunction densities. Bottom: differ-ence between electronic wavefunction densities of moleculesand atoms on a linear gray-scale, with white and black in-dicating reductions and increases by amounts shown on thegray-scale bar. The P -type molecular state (right) carries atrilobite-like component that interferes mostly constructivelywith the P -orbital, causing a positive dipole moment of about7 ea . In the case of the D -type molecule (left), the trilobiteorbital predominantly shows destructive interference with the D -orbital, causing a negative dipole moment of about -5 ea . from S - and F -states are smaller. The admixture prob-abilities ∼ − , corresponding to a typical wavefunc-tion density variation on the order of a few percent, asseen in Fig. 4, leading to | d i,ν | -values much smaller thanthe wavefunction diameter. In Fig. 4 it is seen that the P -state molecule exhibits predominantly constructive in-terference near the perturber, corresponding to a positivedipole moment. A similar mixing analysis was reportedfor Rb (35 S +5 S ) molecules with a small positive dipolemoment [15]. Interestingly, for the D -state molecule inCs the mixing near the perturber is predominantly de-structive, corresponding to a negative dipole moment.In summary, we have observed Cs nD Rydberg-groundmolecules involving Rydberg-state fine structure andground-state hyperfine structure. Measurements of thebinding energies for T Σ( ν = 0) and S,T Σ( ν = 0) molecu-lar vibrational states were modeled with calculations. Wehave measured permanent electric dipole moments withmagnitudes of a few ea . Calculations show that thedipole moment is negative . Future work may further elu-cidate this behavior, the exact shifts and splittings due tothe tensor atomic polarizability, as well as the transitionfrom weak to large electric-dipole energy shifts relativeto the molecular binding.The work was supported by the National Key R&DProgram of China (Grant No. 2017YFA0304203), theNational Natural Science Foundation of China (GrantsNos. 11434007, 61835007, 61675123, 61775124 and 11804202), Changjiang Scholars and Innovative ResearchTeam in University of Ministry of Education of China(Grant No. IRT 17R70) and 111 project (Grant No.D18001) and 1331KSC. [1] E. Fermi, Il Nuovo Cimento , 157 (1934).[2] A. Omont, J. Phys. France , 1343 (1977).[3] C. H. Greene, A. S. Dickinson, and H. R. Sadeghpour,Phys. Rev. Lett. , 2458 (2000).[4] E. L. Hamilton, C. H. Greene and H. R. Sadeghpour, J.phys. B: At. Mol. Opt. Phys. , L199 (2002).[5] M. I. Chibisov, A. A. Khuskivadze and I. I. Fabrikant, J.phys. B: At. Mol. Opt. Phys. , L193 (2002).[6] V. Bendkowsky, B. Butscher, J. Nipper, J. P. Shaffer, R.L¨ow, and T. Pfau, Nature , 1005 (2009).[7] M. A. Bellos, R. Carollo, J. Banerjee, E. E. Eyler, P. L.Gould, and W. C. Stwalley, Phys. Rev. Lett. , 053001(2013).[8] D. A. Anderson, S. A. Miller, and G. Raithel, Phys. Rev.Lett. , 163201 (2014).[9] A. T. Krupp, A. Gaj, J. B. Balewski, P. Ilzh¨ofer, S. Hof-ferberth, R. L¨ow, T. Pfau, M. Kurz, and P. Schmelcher,Phys. Rev. Lett. , 143008 (2014).[10] J. L. MacLennan, Y. J.Chen, and G. Raithel, Phys. Rev.A , 033407 (2019).[11] J. Tallant, S. T. Rittenhouse, D. Booth, H. R. Sadegh-pour, and J. P. Shaffer, Phys. Rev. Lett. , 173202(2012).[12] D. Booth, S. T. Rittenhouse, J. Yang, H. R. Sadeghpourand J. P. Shaffer, Science , 6230 (2015).[13] H. Saßmannshausen, F. Merkt, and J. Deiglmayr, Phys.Rev. Lett. , 133201 (2015).[14] C. Fey, J. Yang, S. T. Rittenhouse, F. Munkes, M. Baluk- tsian, P. Schmelcher, H. R. Sadeghpour and J. P. Shaffer,Phys. Rev. Lett. , 103001 (2019).[15] W. Li, T. Pohl, J. M. Rost, Seth T. Rittenhouse, H. R.Sadeghpour, J. Nipper, B. Butscher, J. B. Balewski, V.Bendkowsky, R. L¨ow, T. Pfau, Science, , 1110 (2011).[16] H. Weimer, M. M¨uller, I. Lesanovsky, P. Zoller and H. P.B¨uchler. Nature Physics, , 382-388 (2010).[17] M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D.Jaksch, J. I. Cirac and P. Zoller. Phys. Rev. Lett. ,037901 (2001).[18] D. DeMille, Phys. Rev. Lett. , 067901 (2002).[19] P. Rabl, D. DeMille, J. M. Doyle, , M. D. Lukin, R. J.Schoelkopf and P. Zoller, Phys. Rev. Lett. , 033003(2006).[20] M. A. Baranov, M. Dalmonte, G. Pupillo, P. Zoller,Chem. Rev. , 5012-5061 (2012).[21] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I.Ferrier-Barbut and T. Pfau. Nature , 194-197 (2016).[22] S. Markson, S. T. Rittenhouse, R. Schmidt, J. P. Shafferand H. R. Sadeghpour, Chem. Phys. Chem. , 3683-3691 (2016).[23] D. A. Anderson, S. A. Miller, and G. Raithel, Phys. Rev.A ,062518 (2014).[24] S. Bai, X. Han, J. Bai, Y. Jiao, H. Wang, J. Zhao and S.Jia, J. Chem. Phys. , 084302 (2020);[25] A. A. Khuskivadze, M. I. Chibisov, and I. I. Fabrikant,Phys. Rev. A66