Ab initio interaction potentials and scattering lengths for ultracold mixtures of metastable helium and alkali-metal atoms Dariusz K¸edziera∗ and L ukasz Mentel Department of Chemistry, Nicolaus Copernicus University, 7 Gagarin Street, 87-100 Torun´, Poland Piotr S. Z˙uchowski† Institute of Physics, Faculty of Physics, Astronomy and Informatics, 5 Nicolaus Copernicus University, ul. Grudziadzka 5/7, 87-100 Torun´, Poland 1 0 2 Steven Knoop‡ LaserLaB, Department of Physics and Astronomy, VU University, n De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands u (Dated: June29, 2015) J 6 We have obtained accurate ab initio 4Σ+ quartet potentials for the diatomic metastable triplet 2 helium + alkali-metal (Li, Na, K, Rb) systems, using all-electron restricted open-shell coupled clustersinglesanddoubleswithnoniterativetriplescorrections[CCSD(T)]calculationsandaccurate ] calculations of the long-range C6 coefficients. These potentials provide accurate ab initio quartet s scattering lengths, which for these many-electron systems is possible, because of thesmall reduced a g masses and shallow potentials that results in a small amount of bound states. Our results are - relevant for ultracold metastable triplet helium + alkali-metal mixtureexperiments. t n a PACSnumbers: 31.15.A-,34.20.Cf,34.50.Cx,67.85.-d u q . I. INTRODUCTION theories. Forexample,precisemeasurementoftransition t a frequenciesandthelifetimesareusedtotestthequantum m Interactions and collisions involving helium in the electrodynamic (QED) calculations [14–16]. Similarly, - metastabletriplet23S1 state(denotedasHe∗)havebeen thetwo-bodyinteractionpotentialforspin-polarizedHe∗ d regarded for many years as one of the most fascinating atoms is so far the only system in ultracold physics for n in atomic and molecular physics. The enormous amount whichit is possible to calculate the aforementionedscat- o c of internal energy (19.8 eV) allows for Penning ioniza- teringlengths[12]withanaccuracysurpassingtheexper- [ tion, which has been extensively exploited in crossed- imental value [17]. This contrasts with other systems, beam studies [1]. More recently, molecular-beam tech- such as e.g. the interaction potential for alkali-metal 2 niques involving He∗ have achieved sufficient resolution atoms, for which predicting the scattering length with v 4 inkinetic energy[2–4]toobserveforshaperesonancesin such accuracy is impossible, and without experimental 8 single partial waves for sub-Kelvin collisions, which are data, it is essentially unknown. 8 a sensitive probe of the interaction potential. Recently we have challenged this situation for the in- 0 Elastic andinelastic collisionsat sub-milliKelvinener- teractionbetweenHe∗ andRb, anddemonstratedavery 0 giesarerelevantforultracoldtrappedHe∗ gases[5]. Pen- good agreement between theoretical predictions based . 1 ning ionization greatly limits the lifetime of the trapped on all-electron restricted open-shell coupled cluster sin- 0 gas. However, for a spin-polarized gas of He∗ Penning gles and doubles with noniterative triples corrections 5 ionizationisstronglysuppressedduetospinconservation [CCSD(T)]calculationsandthescatteringlengthderived 1 [6]. Bose-Einstein condensates of 4He∗ have been real- from thermalization measurements of an ultracold mix- v: ized[7–11], benefiting froma sufficiently largescattering ture of 4He∗ and 87Rb [18]. The fact that even for so Xi length of 142 a0 [12] that allows for efficient evaporative many electrons it is possible for ab initio calculations to cooling. Degenerate Fermi gases of 3He∗ are obtained quantitatively predict scattering lengths results not only ar by sympathetic cooling with 4He∗ [13], which is efficient from the simple electronic structure of He∗, but mostly duetoaverylargeinterspeciesscatteringlengthof496a0 comes from the small reduced mass and shallow interac- [12]. tion potential that give a small number of bound states, One of the unique features of He∗ is its simple elec- makingthescatteringlengthlesssensitivetouncertainty tronic structure. Thus, it is being explored extensively in the calculated potential (see Ref. [19] for the opposite in atomic physics to confront the state-of-the-art, ultra- case). precisespectroscopyandthemostadvancedfundamental Motivated by our previous work on He∗+Rb [18], we consider the relevant interaction potentials for He∗ and the other alkali-metal atoms. Similarly as in the case of ∗ [email protected] homonuclear He∗ collisions, Penning ionization: † pzuch@fizyka.umk.pl ‡ [email protected] He∗+A→He+A++e−, (1) 2 is suppressed by spin-polarizing He∗ and alkali-metal 1.9864455×10−23 J as an energy unit. atom A both in the spin-stretched states, such that the total spin and its projection is maximum [20]. While the s characterof the valence electron of He∗ and alkali- II. THEORY metal atoms are alike, we expect the amountof suppres- siontobe similarto thatin He∗+He∗ (incontrastto the A. General considerations other metastable noble gas systems [5]). Here we have assumed that the product ion A+ has zero spin, which The s-wave scattering length a is obtained by solving is true for the ground state. However, if excited, non- the1DradialSchr¨odingerequationwithzeroangularmo- zero spin, (A+)∗ states are energetically available, Pen- mentum and vanishing kinetic energy E: ning ionization is not spin forbidden, even for the dou- 2µ bly spin-stretched state combination. Among the alkali- ψ′′(r)+ [E−V(r)]ψ(r)=0, (2) metal atoms the (A+)∗ states cannot be reached, except ~2 forCs,forwhichtheexcitationenergyfromtheneutralto where µ is the reduced mass, r the internuclear dis- the first excited ionic state is 17.2 eV [21]. Therefore we tance and V(r) is the interaction potential. Current willdiscardCsinthis workasastable ultracoldmixture quantum chemistry ab initio methods are able to de- withHe∗ seemsexperimentallynotfeasibleandcomputa- termine the short-range part of V(r) with an accuracy tionallythepresenceoftheenergeticallyavailableexcited on the order of few percent, which translates into few ionic channelwill complicate the calculations. Note that cm−1 for dispersion-bound systems, like spin-stretched the abovementionedcriteriato suppressPenningioniza- alkali-metal dimers or He∗+alkali-metal atom systems. tion exclude most other atomic species as well (except The long-range part of V(r) for alkali-dimers and sys- He∗+H). tems with helium atomcanbe describedvery accurately The collision properties for He∗ + alkali-metal atom through the van der Waals expansion: the correspond- are determined by a 2Σ+ doublet and a 4Σ+ quartet ing coefficients for such systems can be obtained with potential, however, for the doubly-spin stretched state sub-percent accuracy. combination scattering only occurs in quartet potential. With an appropriate analytical form of V(r) it is Thereforefor realizinga stable mixture the propertiesof possible to explore separately the influence of short- the quartetpotentialare the mostrelevant,in particular and long-range potential modifications on the scattering the quartet scattering length. This scattering length de- length. A good choice of such a function is the so-called termines interspecies thermalization rates and therefore Morse/Long-Range(MLR) potential [28], which has the whether sympathetic cooling is efficient or not. Also, form: forquantumdegeneratemixturesitdetermines,together 2 withtheintraspeciesscatteringlengths,whetherthemix- V(r)=D 1− uLR(r) exp[−φ(r)y (r)] −D , (3) e p e ture is miscible or immiscible in case of Bose-Bose mix- (cid:18) uLR(re) (cid:19) tures [22, 23], or whether a Fermi core or shell is formed where incaseofBose-Fermimixtures[24]. Similartothetriplet C6 C8 C10 potential in alkali-metal + alkali-metal interactions, the uLR(r)= r6 + r8 + r10, (4) quartet potential is quite shallow, which in combination rk−rk with the small reduced mass leads to a small number of y (r)= e, (5) k rk+rk bound states in the range of 11 to 15. Note that pre- e vious experimental work on these kind of collisions sys- 4 tems was based on measuring electron emission spectra φ(r)=[1−yp(r)] φj[yq(r)]j +yp(r)φ∞. (6) incrossedbeamexperiments[25–27],whichinherently is Xj=0 only sensitive to the doublet potential. Therefore ultra- ThefreeparametersoftheMLRpotential,determined cold mixture experiments are the first to explore these by fitting, are the φ (j = 0,...,4) coefficients, while j quartet potentials. the potential well depth D , equilibrium distance r and e e In this paper we present ab initio calculations of the φ∞ =log[2De/uLR(re)]aredirectlyobtainedfromtheab quartet potentials of He∗ + alkali-metal (Li, Na, K and initio calculations of the short-range potential. For the Rb) systems, where the results for He∗+Rb are taken long-range part of the potential, V(r) r→→∞ −uLR(r) = from Ref. [18]. In Sec. II we describe the methodology −C6r−6−C8r−8−C10r−10. Notethatthestatisticalerror of the calculations, while in Sec. III we present the ob- introducedby the analyticalfit is much smallerthan the tained potentials and give a detailed discussion on the systematic uncertainty in the ab initio calculations. accuracy of those potentials. In Sec. IV we provide the correspondingscatteringlengthsforalltheisotopologues and discuss possible implications for future experiments. B. Short-range potential Finally, in Sec. V we conclude and give an outlook. Throughout this paper we use Bohr radius, a0 = For the short-range potential we have used the cou- 5.2917721 × 10−11 m, as a length unit and cm−1 = pled cluster method [29] in which the correlated elec- 3 tronicwavefunctionisrepresentedbytheexponentialop- A proper choice of helium basis set is also essential: erator exp(T) acting on the Slater determinant, where the optimal basis sets for the metastable triplet state T = T1 + T2 + ... is the so-called cluster operator have entirely different character compared to basis sets which includes single-, double- and higher-order excita- for ground state helium [12]. Hence, we have decided tions. Agoldstandardforweaklyboundsystemshasbe- to build a new basis set that takes into account diffused cometheapproximatecoupledclustermethoddenotedas character of He∗. To this end we have optimized a new CCSD(T),inwhichonefullyincludessingle-anddouble- setofexponentsaccordingtoafollowingprocedure. The excitations and treats triple-excitations approximately. starting point was an uncontracted ANO-RCC basis set In this paper we use the open-shell version of CCSD(T) for ground-statehelium, i.e. 9s4p3d2f. Exponents were introduced by Knowles et al. [30], implemented in the re-optimized to minimize the total energy in the DKH5 molpro program [31]. relativistic method (at the full configuration interaction The accuracy of CCSD(T) for predicting binding en- level)andextendedto15s8p5d4f2g,whichgiveslessthan ergies for weakly bound systems is typically on the or- 1 µHartree convergence. Then the lowest exponents for der of 2-3% percent [32]. Here we confirm this accu- eachshellwereagainaugmentedandre-optimizedforthe racybycalculatingtheD parameterusinganevenmore totalenergyofheliumdimer quintetstateatthe equilib- e sophisticated method, namely coupled cluster with full rium distance. The final helium basis set has an angular tripleexcitations,CCSDT.Inaddition,wehaveincorpo- momentum structure of 17s10p7d5f4g. rated scalar relativistic effects using the Douglas-Kroll- With such basis sets we have performed test calcu- Hess approximationup to the fifth order in external po- lations for the spin-polarized homonuclear systems He∗2, tential (DKH5) [33]. All electrons have been correlated Li2, Na2 and K2 to assess their performance near the in these calculations. Interaction energies were calcu- equilibriumdistance. Wehaveobtainedverygoodagree- lated using the counterpoise correction scheme of Boys ment of De parameters with experimental results: for and Bernardi [34], in which the total monomer energies He∗2 we have obtained 1042.3 cm−1 which is very close calculated in dimer basis sets are substracted from total to estimated limit of complete basis set for CCSD(T) energy of dimer. method (1042.9 cm−1) [12] and about 5 cm−1 shal- lower than the theoretical potential obtained with a Theaccuracyofemployedquantumchemistrymethod- ology is also determined by the choice of appropriate full configuration interaction method. For Li2, Na2 gaussian basis set describing the orbitals. Due to com- and K2 systems CCSD(T) values calculated with ba- sis sets described above are respectively, 330.5 cm−1, putational limitations, the basis set commonly used in 172.3cm−1and247.5cm−1,whichcanbecomparedwith quantum chemical calculations are truncated. However, experimental values of 333.69 cm−1, 174.96 cm−1 and the basis sets that are used in this paper are tailored 255.017 cm−1 [37–39]. Clearly CCSD(T) with current to systematically improve the correlation energy and al- basis sets systematicallyunderestimates the well depths, low approximate extrapolation to the complete basis set but the deviation from the benchmark values are small. limit (CBS). By increasing the maximum angular mo- mentum in the basis sets the errorsin correlationenergy are expected to decrease. Hence, in order to converge C. Long-range potential thequantumchemicalcalculationsoneshouldobtainthe resultsforaseriesofbasissetswithincreasingmaximum angular momentum. The C6 van der Waals coefficients of a general A+B systemcan be obtainedby integrationof the dipolar dy- While for most atoms corresponding to the first- and namic polarizabilities α over the imaginary frequencies second rows in periodic table a variety of various fam- [40]: ilies of well-optimized, systematically convergent basis sets are available, for heavy alkali-metal atoms (K, Rb, CAB = 3 αA(iω)αB(iω)dω. (7) Cs) the choice of basis sets is limited. We have used a 6 π Z sequence of core-valence correlation consistent basis sets For He∗ we have used the polarizabilities that are ob- developed by Prascher et al. [35] for Li and Na, which tained from explicitly correlated gaussian wavefunctions we will denote as TZ, QZ and 5Z. For K we have used with an accuracy on the order of 0.1% for the zero- an uncontracted atomic natural orbital relativistic basis frequency [18, 41]. In case of alkali-metalatoms we have sets [36] (denoted as ANO-RCC) with the exponents of used the dynamic polarizabilities at imaginary frequen- hfunctionstakenfromg ofthesamebasis(wefollowthe cies given by Derevianko et al. [42]. These polarizabili- usualconventiontolabelthegaussianfunctionswithap- tiesgivehomonuclearC6 coefficientswithanaccuracyof propriate orbital angular momentum). In each case the 0.14%,0.25%and0.4%for the Li+Li,Na+NaandK+K basissethavebeenaugmentedbyasetoftwoadditional systems, respectively. Hence, the inaccuracy of C6 in diffused functions per shell generated as an even tem- heteronuclear systems determined as 1(δ2 +δ2) [42] pered set according to prescription implemented in the 2 A B molpro program. The angular momentum structure of (where δA and δB are unsigned errors oqf pertinent static largest basis sets for Li, Na and K is 20s14p10d8f6g1h, polarizabilitiesofmonomersAandB,respectively)isal- 22s20p10d8f6g2h,23s18p7d4g2h,respectively. ways smaller than 0.25%. Therefore the uncertainty in 4 TABLEI.Keyparametersofpotentialenergycurvesofquar- tet (4Σ+) He∗ + alkali-metal systems, including equilibrium distance re, the potential depth De and the C6 long-range coefficient. The uncertainty of the CCSD(T) calculations is reflected in theerror bars on De (see text). system re (a0) De (cm−1) C6 (cm−1a60) He∗ + Li 7.53 575+4 4.5782×108 −1 He∗ + Na 8.57 361+4 4.7723×108 −1 He∗ + K 9.08 470+9 7.7314×108 −1 He∗ + Rb 9.41 453+8 8.4673×108 −1 larger radius. Increase in exchange energy is reflected in monotonic increase of equilibrium distance of dimers. FIG. 1. (Color online) Results of the ab initio calculations On the other hand, the dispersion interaction gives rise of the potential energy curves of the quartet (4Σ+) He∗ + to the attraction of atoms in the system and its magni- alkali-metal systems. tude correlates with the C6 coefficients. Li and Na have comparable dispersion interactions, but a much larger exchange repulsion in the case of Na results in a much thelong-rangepartofthe potentialismuchsmallerthan smaller well depth. On the other hand, K and Rb ex- that of the short-range part obtained from CCSD(T), hibitmuchstrongerdispersioninteraction,thustheirwell and one can treat the long-range part of the interaction depths are noticeably larger than that of Na. as fixed by theory. Note that our C6 coefficients agree with the values obtained by Zhang et al. [43] to better than 0.1%, which suggests that their estimated uncer- B. Accuracy of the ab initio potentials tainty of 1-5% is too conservative. FortheC8 andC10 coefficientswehaveusedthevalues To provide tests of the potential accuracy in the min- calculated by Zhang et al. [43] for which the authors imum region we have performed additional calculations estimated an uncertainty of about 1-10%. However, our forHe∗+(Li, Na)usingacoupled-clustersapproachwith studyofHe∗+Rbsystemhasshownthattheerrorbounds for the C8 coefficient given by Zhang et al. was also far full triple excitations [44] (CCSDT) for basis sets with maximum angular momentum limited to f (for Li and too conservative [18]. Na)andd(forHe∗)functions. CCSDTcalculationswere performedusingthecfourquantumchemistrycode[45]. The results are shown in the Table II, and are discussed III. AB INITIO POTENTIALS below. Note that a detailed description of the accuracy of the He∗+Rb potential is given in Ref. [18]. A. Recommended ab initio potentials TheresultsofourcalculationsforHe∗+(Li,Na,K,Rb) 1. Basis set convergence quartet potentials are shown in the Fig. 1; the values of the potential well depth De, the equilibrium distance re We have studied the basis set convergence for all sys- and the C6 coefficient are given in Table I. A complete tems. To this end we studied the dependence of the De list of the parameter values of the MLR potential are parameter in family of basis sets obtained by taking out given in Appendix A (Table IV). The uncertainty of the from our actual basis sets two- and one- highest angular ab initio calculations is predominantly translated in an momentumfunctions. Suchbasissetsroughlycorrespond uncertainty in De, while the uncertainty in re is smaller totriple-andquadruple-zetaquality(whichwedenoteas than 0.01 a0. TZ and QZ, respectively), while our best basis set is of Theoverallqualitativepatternofthepotentialsresem- 5-zeta quality (5Z). bles the triplet states of homonuclear alkali-metal sys- We have found that even for the smallest basis sets tems: the one containing Li has the deepest well, K and of TZ quality the binding energies are very close to the Rb are slightly shallower than Li, and Na is anomaly values obtained from those obtained with 5Z basis sets: shallowerthan the other potentials. This pattern can be for the He∗+(Li, Na) systems the CCSD(T) interaction explained by subtle interplay between attractive disper- energies calculated for the recommended r ’s (Tab. IV) e sion forces and Pauli repulsion (exchange energy). The are, respectively 569.2 cm−1 and 350.2 cm−1, respec- Pauli repulsion increases with system size as the atoms tively, that is 6.2−1 and 10.6 cm−1 below the recom- aresystematicallymorediffusedandhavesystematically mended D parameters. Such rapid convergence might e 5 ergy at equilibrium as a probe of post-CCSD(T) effects TABLE II. Interaction energies of quartet states of the nearthe equilibriumdistance. Moreover,forthe He∗+Li He∗+(Li,Na)systemscalculatedforthedistancecorrespond- case CCSDT is exact if the 1s electrons of Li are kept ing to re of recommended potential (7.53 a0 and 8.57 a0, frozen, and is exact also for isolated Li and He∗ atoms. respectively) using CCSD(T) and CCSDT levels of coupled- We have calculated the CCSDT interaction energy for clusters theory. Wereport also test calculations with various basis sets and levels of frozen-core approximation. See Sec. all electrons active and for the case when 1s orbital is IIB for notations regarding thebasis sets. frozen. The results are given in the Table II. In a basis setofTZ-qualitytheCCSDTinteractionenergyisdeeper basis set level active electrons V(re) (cm−1) bymerely3.4cm−1. TheTZbasissetisremarkablyclose He∗+Li to the complete basis set convergence limit (CBS), and TZ CCSDT all 572.6 wecansafelyassumethatthedifferencebetweenCCSDT TZ CCSD(T) all 569.2 and CCSD(T) is also nearly converged. It is interesting TZ CCSDT 3 578.3 to notice that CCSDT for 3-electron calculations (i.e. with frozen 1s shell of Li) gives actually an even smaller TZ CCSD(T) 3 576.6 differencebetweenCCSDTandCCSD(T)interactionen- QZ CCSD(T) all 574.5 ergies (1.7 cm−1), which shows that the core-relaxation 5Z CCSD(T) all 575.4 effects in this case are comparable to the contributions CBS CCSD(T) all 576.1 beyond the CCSD(T) model. He∗+Na For He∗+Na we were able to calculate the CCSDT TZ CCSDT 7 352.0 interaction energies for 7 active electrons. It turns out TZ CCSD(T) 7 350.2 that the difference between interaction energies CCSDT QZ CCSD(T) all 362.2 and CCSD(T) is only 1.8 cm−1 in TZ-quality basis set. We might assume that with converged basis set and all 5Z CCSD(T) all 360.8 electrons correlated the error might be at most twice as CBS CCSD(T) all 359.3 large. For He∗+K we were not able to converge the CCSDT calculations. Hence, to estimate bounds on D we have e result from the fact that in the high-spin He∗+alkali- compared how the analytical long-range potential given metal systems same-spin electronic pairs are the main by van der Waals series compares to the interaction en- contribution to the interaction energy and it is known ergies from the actual calculations. It turns out that C6 thatthecorrelationenergyforsuchpairssaturatesfaster coefficient extracted from the ab initio CCSD(T) inter- than for the opposite-spin pairs. While we have esti- actionenergies(fittedfrom20a0 to35a0)isabout1.5% mated the CCSD(T) complete basis set (CBS) limit for smaller compared to the value obtained from perturba- the He∗+(Li, Na) interaction energies to be 576.1 cm−1 tion theory (i.e. Eq. 7). If this value is treated as an and 359.3 cm−1, respectively, we rather prefer to treat estimate of ab initio potential uncertainty it translates the difference between 5Z basis set and CBS interaction to about +7 cm−1. energy as (unsigned) uncertainty attributed to the basis setincompletenessandtotaketheresultsfor5Zbasisset asrecommendedvalueswithbasisseterrorsof±0.6cm−1 and ±1.3 cm−1, respectively. 3. Bounds on De parameters Our basis set for He∗+K is not correlation consistent, As mentioned in Sec. IIB, our methodology predicts so instead of performing the CBS estimate we have sim- plycomparedtheresultinextendedANO-RCCbasisset thatthewelldepthsofthehomonucleardimersHe∗2,Li2, (see Sec. IIB) to the interaction energy obtained with Na2, K2 are systematically shallower compared to the experimental values by 0.53%, 0.92%, 1.52% and 2.94%, basisset withremovedh functions. The value ofthe lat- ter is 468.6 cm−1, hence the uncertainty due to basis set respectively. Because of the simple structure of the He∗ incompleteness is about ±1.3 cm−1. atom (for isolated He∗ atom CCSD(T) method is exact) we expect that within the He∗+alkali-metal atom sys- tems the errors should be even smaller. By taking an average of the appropriate percentage uncertainties for 2. Post-CCSD(T) contributions to the interaction energy homonuclear alkali dimers, i.e. δAB = (δA +δB)/2 we might expect that our ab initio potentials are deeper by Using the CCSDT method in reduced basis sets we about0.7%, 1%and1.7%, whichtranslates to 4.0 cm−1, have explored the performance of ab initio methods be- 3.7 cm−1, 8.1 cm−1 for the He∗+(Li, Na, K) potentials, yond the CCSD(T) model for He∗+(Li, Na) systems us- respectively. These uncertainties are consistent with our ingtheTZ-qualitybasissets. Forthesystemscontaining estimate of post-CCSD(T) interactionenergies,which in non-polar species quadruply excited configurations give eachcase predicts smalland systematically positive con- substantially smaller contribution to the binding ener- tributions beyond CCSD(T). We can conservatively as- gies [32], hence we might treat CCSDT interaction en- sumethattherealpotentialshaveawelldepthparameter 6 TABLEIII.Scattering lengthsfor all He∗ +alkali-metal iso- topologues, showing thescattering length a corresponding to therecommendedDe,andthebounds[a−;a+]corresponding to the bounds on De. Also the number of bound states N is given. Note that for the alkali-metal atoms the even iso- topesare fermions and theoddones arebosons, while4Heis a boson and 3Heis a fermion. system isotopes a [a−;a+] N He∗+Li 3 + 6 +26 [+23;+26] 11 3 + 7 −17 [−27;−15] 11 4 + 6 +22 [+19;+23] 12 4 + 7 −193 [−607;−161] 12 He∗+Na 3 + 23 +58 [+52;+59] 11 4 + 23 +7 [−2;+9] 12 He∗+K 3 + 39 +51 [+42;+52] 13 FIG. 2. (Color online) Scattering length as function of re- 3 + 40 +49 [+41;+51] 13 duced mass for theHe∗+Li potential, indicating the reduced 3 + 41 +48 [+39;+49] 13 massesforthefourisotopologues(reddashedlines). Thegray area gives the boundson the scattering length related to the 4 + 39 +97 [+74;+101] 15 uncertaintyin thepotential. 4 + 40 +91 [+70;+94] 15 4 + 41 +86 [+67;+89] 15 He∗+Rb a 3 + 85 +5 [−17;+7] 13 tio scattering lengths solely by scaling the D parameter e 3 + 87 +3 [−19;+5] 13 withinitsestimatedbounds. Suchscalingdoesnotaffect 4 + 85 +16 [−4;+18] 15 the long-range part of the MLR potential, which is kept 4 + 87 +15 [−6;+17] 15 fixed. The results are presented in Table III, showing a and the bounds [a−;a+] related to the bounds on the a Thescatteringlengthvaluesareshiftedbyabout1a0 fromour calculated potentials, where the lowest possible value of earlierreportedvaluesinRef. [18],whichsufferedfromasmall De corresponds to a+ and the highest one to a−. Also computational error. the number of bound states N is indicated, whichdiffers between the two He isotopes by one or two units. Thesensitivityofthescatteringlengthaforthepoten- D that is: i) not smaller than D from CCSD(T) mi- e e tial well depth D is very non-linear, and a diverges at nus basis set uncertainty;ii) not largerthan our D plus e e each value of D at which the potential supports a new basisset uncertainty,plus the post-CCSD(T) correction. e Hence, for He∗+(Li, Na, K) systems the D parameters boundstate. Inmostcaseshereaisfarawayfromapole e for quartetpotentials haveuncertainties of–1/+4cm−1, andthe boundsonaarequitetight. Anoticeableexcep- –1/+4 cm−1 and –1/+9 cm−1, respectively. tion is 4He∗+7Li, which lies very close to a pole, leading to a broader range of possible scattering length values. Finally, we note that nonadiabatic effects can be ne- This is also illustrated in Fig. 2, showing a as function glected here despite the relatively small reduced masses. ofthe reducedmassµforthe He∗+Lipotential. The ap- For the He∗+Li system in the van der Waals mini- pearanceofpolesinawhenvaryingD orµissimilar,as mum the diagonal Born-Oppenheimer correction calcu- e µ and V(r) appear only as a product in the Schr¨odinger latedwiththeHartree-Fockelectronicwavefunctions(us- ingthecfourprogram[45])isabout0.02cm−1. Forthe equation (see Eq. 2). For He∗+Li the four isotopologues have quite different reduced masses such that the corre- He∗+(Na, K) systems these errors will be even smaller. sponding scattering lengths can be very different. The similarity between the scattering lengths of 3He∗+6Li and4He∗+6Liispurelyaccidental. Fortheheavieralkali- IV. SCATTERING LENGTHS metal atoms (K and Rb) the scattering lengths for the same helium isotope are nearly the same, as the reduced With the recommended MLR potentials the quartet mass hardly changes for the different alkali isotopes. scatteringlengthaforallisotopologuescanbecalculated, The 16 isotopologues of He∗+(Li, Na, K, Rb) contain for which we have used the 1D renormalized Numerov 6 Bose-Bose, 8 Bose-Fermi and 2 Fermi-Fermi mixtures. propagator [46] for a kinetic energy of 10 nK. Within From the calculated scattering lengths we find that the the Born-Oppenheimer approximation it is only the dif- 4He∗+(23Na, 39K, 41K, 87Rb) Bose-Bose mixtures are ferent reduced masses that give rise to different scatter- miscible, while the Bose-Fermi mixtures provide both ing lengths for the isotopologueswithin eachsystem and Fermi core and Fermi shell situations. The 3He∗+40K we have verified that non-adiabatic terms are negligible. Fermi-Fermi mixture has already been proposed as its We can conveniently explore the bounds on the ab ini- mass ratio is very close to a narrow interval where a 7 purely four-body Efimov effect is predicted [47]. It is He∗ + alkali-metal systems, as both atoms have elec- interesting to note that both 3He∗ and 40K have an in- tronspinandatleastonehasnuclearspin,howeverthey verted hyperfine structure, such that the high-field seek- would require spin-state combinations in which at least ingdoublyspin-stretchedstatecombinationisthelowest one of the atom is not in the spin-stretched state, and channel within the 3He∗+40K manifold. scattering has both doublet and quartet character. Un- From the calculated scattering lengths one can also fortunately, accurate ab initio calculations of the dou- find whether alkali-metal atoms can be used to sym- blet potentials are far more challenging. First of all, the pathetically cool He∗, as an alternative to evaporative doublet potentials need multiconfigurational treatment cooling of 4He∗ and sympathetic cooling of 3He∗ by and these methods are at present far less accurate than 4He∗. Although both these schemes are successfully ap- CCSD(T).Secondly,forthedoubletstatescouplingwith plied,sympatheticallycoolingwithanotherspecieswould continuum states of ionized channels is possible, which put a less stringent requirements on the initial num- mightcomplicatethecalculations. Inaddition,themuch ber of laser-cooled 4He∗ atoms. Here one has to take largerwelldepths[26],andthereforemuchlargeramount into account that the thermalization rate scales with of bound states, results in a much stronger sensitivity of ξ =4mHe∗mA/(mHe∗+mA)2,anda2 inthezerotemper- thedoubletscatteringlengthtotheunderlyingpotential. ature limit (see for instance Ref. [18]). Na might be the Therefore,experimentalinput, suchaspositionsofFesh- mostsuitablecandidatefor sympatheticcoolingof3He∗, bachresonances,willbeneededtodeterminethedoublet while7Liisthebestcoolantfor4He∗(althoughonlyuntil scattering properties. quantumdegeneracyisreachedbecauseofimmiscibility). ACKNOWLEDGMENTS D. K., L . M. and P. S. Z˙. acknowledge support from NCN grant DEC-2012/07/B/ST4/01347 and generous V. CONCLUSIONS AND OUTLOOK amountofCPUtime from WroclawCentre for Network- ing and Supercomputing, grant no. 218. S. K. acknowl- We have obtained accurate ab initio 4Σ+ quartet po- edges financial support by the Netherlands Organisation tentials for He∗+(Li, Na, K, Rb), using CCSD(T) calcu- for Scientific Research (NWO) via a VIDI grant (680- lations and accurate calculations of the C6 coefficients, 47-511). We thank Mariusz Puchalski for providing the andhavecalculatedthe correspondingscatteringlengths dynamicpolarizabilitiesofmetastableheliumandDaniel for all the isotopologues. An accurate predictionof scat- Cocks for corrections on our initial manuscript. tering lengths for these many-electron systems is possi- ble,incontrasttonearlyallothertypesofultracoldmix- tures, because of the small reduced masses and shallow Appendix A: Parameters of MLR potential potentialsthatresultsinasmallamountofboundstates (N =11−15),and thereforea reducedthe sensitivity of InTableIVwegivetheparametervaluesforthe MLR the scattering length to the properties of the potentials. potentials, where D and r are obtained directly from e e So far, we have only considered the quartet potential, the CCSD(T) calculations, C6 is calculated from the which is the only relevant potential for the doubly spin- dynamical polarizabilities, C8 and C10 are taken from stretched state combinations, for which Penning ioniza- Ref. [43], andthe φ parametersareobtainedfromfitting tion is suppressed. Feshbach resonances that allow to the MLR potential to the CCSD(T) data. 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