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Feshba h resonan es in ultra old atom-mole ule ollisions Andrea Simoni and Jean-Mi hel Launay Institut de Physique de Rennes, UMR 6251 du CNRS and Université de Rennes 1, 35042 Rennes Cedex, Fran e Pavel Soldán Department of Chemi al Physi s and Opti s, Fa ulty of Mathemati s and Physi s, Charles University in Prague, Ke Karlovu 3, 121 16 Prague 2, Cze h Republi (Dated: January 20, 2009) We investigate the presen e of Feshba h resonan es in ultra old alkali-dialkali rea tive ollisions. 3 Quantums attering al ulations are performedon a new Na quartetpotential energysurfa e. An 9 analysisofs atteringfeaturesisperformedthroughasystemati variationofthenonadditivethree- 0 body intera tion potential. Our results should provide useful information for interpreting future 0 atom-mole ule ollision experiments. 2 PACSnumbers: 34.50.Cx,31.50.B n a J I. INTRODUCTION boththeoreti allyandexperimentallyinultra oldatomi 0 gases; see e.g. [29, 30, 31℄. 2 Inre entyears,therehasbeenagrowinginterestinul- On the other hand, very little is known about atom- ] tra old mole ules [1℄, parti ularly in the produ tion and mole ule resonan es in ultra old ollisions. Similarly to h propertiesofthemole ulesformedfromultra oldatomi atomi s atteringthehyper(cid:28)ne-indu edresonan es ould p gases [2℄. Photoasso iation [3℄ and Feshba h resonan e in prin ipleexist atverylow ollisionenergies. However, - t tuning[4℄aretwomainexperimentalte hniquesfora o- model al ulations have shown that for a general polar- n a herent produ tion of ultra old mole ules from ultra old ization they will be quen hed by inelasti spin-ex hange u alkali-metal atoms. In 2003 long-lived mole ular Bose- transitions forming singlet mole ules [32℄. q Einstein ondensates were reated from weakly bound Alkali-metal dimers on the lowest ele troni triplet [ homonu lear lithium and potassium dimers by exploit- manifold are only stable if they are in a doubly spin- 1 ing magneti ally tunable Feshba h resonan es between polarized state under ollisions with doubly spin polar- v fermioni isotopes [5, 6, 7℄. ized atoms (assuming that the relativisti spin intera - 9 While Feshba h resonan es are always lo ated at the tions are negle ted). Unfortunately, for this spe i(cid:28) po- 2 highest vibrational manifold of the dimer, photoasso i- larizationhyper(cid:28)ne-indu edresonan esarepreventedby 1 ation ould in prin iple allow a ess to low vibrational symmetry (resonan es indu ed by relativisti spin inter- 3 dimer states. In 2005 RbCs mole ules were reated in a tions are still possible). In spite of this, long-lived . 1 their ground vibroni state [8℄. Very re ently, several three-atom omplexes anin prin iple exist and giverise 0 di(cid:27)erent photoasso iation s hemes for mole ular forma- to resonan es. Su h rea tive resonan es have been iden- 9 tion in the ground vibroni state have also been devel- ti(cid:28)ed in rea tive ollisions at room temperature [33℄. 0 2 oped [9, 10, 11, 12, 13℄. In this work, we fo us on ollisions of Na mole ules : v For the orre t interpretation of the forth oming ex- in the triplet ele troni state with ground-state Na Xi periments with old mole ular samples it is essential to atoms3.(1F4Aor′2)this study a new potential energy surfa e understandtheatom-mole uleandmole ule-mole ulein- of Na has been onstru ted. The o urren e of r a tera tions at sub-K temperatures [14℄. Theoreti al re- long-lived three-atom resonan es in su h ollision om- sults have already been published for the homonu lear plex is demonstratedin the ultra old regimefor the (cid:28)rst X+X2 ultra-low-energy ollisions with X=Li,Na,K [15, time. Wealsostudythedependen eof ollision rossse - 16, 18, 19, 20, 21℄. Isotopi ally heteronu lear Li+Li2 tionsonthepotentialenergysurfa eandweshowthatat ultra-low-energy ollisions have also been studied theo- least knowledge of two terms in the ross se tion partial reti ally[21,22, 23, 24, 25℄. Collision rossse tionshave waveexpansionisneededinorderto hara terizefurther been measured experimentally for the Cs+Cs2 ultra old the three-body potential. inelasti pro esses[26,27℄. Veryre entlyrea tiv∗eandin- 2 elasti rate onstants were measured for Li+Li at room temperature [28℄. II. POTENTIAL ENERGY SURFACE Cold ollisionsareknowntobeverysensitivetopoten- tial energy surfa es [16, 21℄, and therefore experimental Ab initio al ulations were performed using a single- information is needed to improvethe orrespondingthe- referen erestri tedopen-shellvariant[34℄ofthe oupled- oreti al models. In parti ular, knowledge of low-energy lustermethod [35℄ withsingle,doubleand non-iterative resonan epatternsoftenallowsdi(cid:27)erentpropertiesofthe triple ex itations [RCCSD(T)℄. A basis set onsisting of intera tion potential to be determined with high a u- [12s,12p,5d,2f,1g℄ basis fun tions [36℄ was used for the ra y. Su h resonan es have been studied in great detail dimer al ulations, and the same basis set without the 2 g fun tions was used for the trimer al ulations. Ele - trons from the 1s orbital on ea h sodium atom were not orrelatedinthe oupled- luster al ulations. Thethree- atomintera tionpotentialwasde omposedintoasumof pair-wise additive and non-additive ontributions V (r ,r ,r )= V (r )+V (r ,r ,r ). trimer 12 23 13 dimer ij 3 12 23 13 Xi<j (1) It has been shown by several authors that in the ase of alkali-metal trimers the non-additive term V (r ,r ,r ) 3 12 23 13 is rather large and annot be negle ted [37, 38, 39℄. Intera tionenergieswere al ulatedwithre- spe t to the separated-atoms disso iation limit, and the full ounterpoise orre tion of Boys and Bernardi [40℄ was employed to ompensate for the basis set superpo- sition error in both the dimer and trimer al ulations. All the ab initio al ulations were performed using the MOLPRO quantum Chemistry pa Vkage [41℄. a3Σ+ 47 dimer intera tion energies dimer on the u manifold were al ulated on an irregular grid overing the range of interatomi distan es from 2.0 Å to 14.0 Å. These points were interpolated using the 1D re ipro al-power reprodu ing kernel Hilbert spa e (RP- RKHS) merth2od [42℄. The interpolation wasmdo=ne2with respe t to using RP-RKHS parameters and n=3 ).rTh=e5r.e1s9u4ltingV urve(rha)d=a−m1i7n2i.m94u6m at−a1pproxi- e dimer e mately Å, m ,whi h is slightlry h=igh5e.r19t2han tVhe pre(vrio)us=ly−re1p7o7r.7ted a−b1initio e dimer e rmin=ima5.20 V Å(,r ) = −176.17 −1 m [43℄, e dimer e r =5.214 Å,V (r )=−174.025 − m1 [37℄, and e dimer e Å, m [38℄. Ivanovetal.[44℄analyzedexperimentaldataontriplet r = 5.16607 2 e NVa an(rd)de=riv−ed17t3h.6e4a9 6 0urate−1position a3Σ+ Å, dimer e m of the u minimum. Therefore our ab initio intera tion energies were shifted and s aled(shifted by -0.02754Åands aledby 1.00407) so that the minimum of the modi(cid:28)ed potential energy urve oin ided with the minimum determined from ex- periment. The RP-RKHS interpolation was then re- peated using the modi(cid:28)ed RP-RKHS method [45℄. Be- yond the last ab initio point, the potential energy was 3 FIG.1: CutsthroughtheNa quartetsurfa einv◦alen e oor- then extrapolated to the form C C C dinates. Upp−er88p1anel:− 1utforra12b=onrd13a=ngrle23o=f640.3;4theglobal 6 8 10 minimumof m isat Å.Lower V (r)=− − − . dimer r6 r8 r10 (2) −pa3n8e2l: u−t1at ollrin12ea=r gr1e3om=e5tr.0ie6s; the ollinear minimum of −1 m is at Å. Contours are labeled in C C 6 8 m . The long-range 1 .o5e6(cid:30)1 ×ien10ts3 E aa6nd 1.w16er×e 1k0ep5tE(cid:28)xae8d h 0 h 0 to the values of and , respe tively[46℄. The value ofthe (cid:16)free(cid:17) long-range oef- C (cid:28) ient 10 was then determined1f.r1o9m×t1h0e7 Eorrae1s0ponding the range of interatomi distan es from 2.5 Å to 10.0 RP-RKHS oe(cid:30) ients[417.℄1t5o8b×e107E a10 h 0 , whi h Å (geometry on(cid:28)gurations were unique up to a permu- omparesverywellwith h 0 fromRef.[46℄. t|ratio−nrof|a≤torms≤anrd s+atirs(cid:28)ed the triangular inequality 12 13 23 12 13 The resulting potential energy urve supports 16 vibra- r r r r; at linear geometries, where 23 12 13 23 tional bound states and gives a s attering length of 67.1 = + , the distan e was permitted to extend a 0, whi h ompares reasonably well (within 10%) with beyond 10.0 Å). 2v 3h published values 65.3 [48℄, 63.9 [49℄,Vatrnimde6r2.51[50℄.14A′2 Thegrid onsiste∞dvof220C points(in lud∞inhg16D 356 trimer intera tion energies on the points) and 136 C points (in luding 16 D points). manifold were al ulated on a regular 3D grid overing From the trimer intera tion energies the ounterpoise 3 V 100 3 orre ted non-additive energies were extra ted using Eq. (1). V 3 The non-additive energy fun tion was represented 50 in the same manner as in the ase of the spin-polarized potassium trimer [19, 51℄. In order to a ommodate a)0 the geometri dependen ies of the long-range multipole s of 0 terms, third-order dipole-dipole-dipole [52℄ and dipole- nit u ndoipno-laed-qduitaivdCeruepnoelregyC[5V3℄3.teTrmhesirw eorrere1ssu.p8bo9tn2rda×i n1tg0ed5loEfnrgoa-mr9anthgee a (s -50 9 11 h 0 oe(cid:30)1 .i4e6n8ts12×a1n0d5 E wa1e1re(cid:28)xedto [46℄ h 0 and respe tively [54℄. The lead- -100 ing term of the remaining multipole asymptoti expansion was the fourth-order dipole-dipole-dipole term [55℄, 0.985 0.99 0.995 1 1.005 1.01 1.015 and after a multipli ation by a suitable fun tion it was λ prepared for an (cid:16)isotropi (cid:17) extrapolation [19, 51℄. The resulting points were then interpolated, using the fully 2(v=0,j =0) s FIG. 2: The Na+Na -wave s attering length λ symmetrized 3D RP-RKHS interpolation method [37℄, asafun tionofthethree-body ontrolparameter (seetext). in ea h interρat=omi rd3istan e with respe t to the redu ed Haso=riz−on1t0a0lal0ines id6e0nat0ify model potentials orresponding to oordinate S and with RP-RKHS parameters and . S =10.0 m=0(cid:0), n(cid:1)=2 Å, . V trimer 14AT′he three-atom intera tion potential for the 2 state of Na3 was then re onst−ru1 ted usring=Eqr. (1=). tsermined by a small number of parameters. In fa t, the 3h 12 13 Irt2s3D= 4.g3l4obal minim∞umh -880.9 m is at −1 -wave s attering lengths and the dominant term in the Å and D saddle point -381.7 m is at long-range multipole potential expansion are su(cid:30) ient r = r = 5.06 12 13 Å. The minimum of our trimer po- in general to predi t all near-threshold s attering and tential is approximately 5% deeper than the minima re- bound state properties. This feature has allowed many ported by Higgins et al. [37℄ and Soldán et al. [38℄. Two systems of experimental interest to be a urately mod- uts through the surfa e are shown as◦ ontour p◦lots in eledbasedon alimited amountofexperimentalinforma- Fig. 1 for values of the valen e angle 60 and 180 . tion [29, 30, 31℄. Observed energy-dependent ross se - tions and Feshba h or shape resonan es often provided a key pie e of information for determining theoreti ally III. QUANTUM DYNAMICS the s attering lengths and the long-range dispersion o- e(cid:30) ients. The s atteringobservablesareobtainedby solvingthe The situation appears to be more omplex for atom- time-independent S hrödinger equation for three atoms. mole ule ollisions due to the additional ro-vibrational Quantumdynami al al ulationareperformedusinghy- degrees of freedom and anisotropi intera tions. As ol- perspheri al demo rati oordinates. This system of o- lisional data may be soon available we begin to study ordinates omprisesofthreeinternal oordinates(twohy- herewhatexperimentalinformationmightbebestsuited peranglesandonehyperradius)des ribingtheshapeand to onstrain the ollision models. For most alkali sys- 2 the size of the mole ular triangle and three Euler an- tems, and for Na in parti ular, the two body poten- gles des ribing the orientation of the mole ular plane in tial is known with high a ura y from a ombination of spa e. The total wavefun tion is expanded on a set of onventionalroomtemperatureandultra oldatomspe - hyperspheri albasis fun tions varyingwith the hyperra- tros opy[50V℄. Therefore, onemay expe t the three-body 3 dius. The resulting losed oupled equations are solved intera tion to represent the largest sour e of un er- using a log-derivative propagator approa h. Details on tainty. We assume that its shape is essentially orre t the method an be found in [56℄. and following the approa h of [16℄, weλsVolve the s atter- 3 The hyperspheri al demo rati oordinates are espe- ing problem with a s aled potential . At varian e ially well adapted in des ribing alkali spe ies rea - with Ref. [16℄ whi h onsidered inelasti s attering we tions that mainly pro eed through an insertion me h- dis uss here s attering resonan es in elasti ollisions. anism [15℄. However, the region of large interparti le We fo us on mole ules in the lowest triplet rovibra- distan es, where the system separates into atom and tional state, whi h is ollisionally stable under two-body mole ule,isnote(cid:30) ientlydes ribedinhyperspheri al o- ollisionswithatomsifboth ollidingpartnershavemax- ordinates. Therefore the s attering wavefun tion in the imal spin proje tion on the quantization axis. We show s a s outerregionis omputedusingJa obi oordinates. State- in Fig. 2the atom-diatom -waves atteringlength as λ to-state probability amplitudes are (cid:28)nally extra ted by a fun tion of the three-body ontrol parameter . Ea h mat hing to the short-rangewavefun tion obtained with time a three-body bound state rosses the disso iation a s the hyperspheri al approa h [56℄. threshold the presents a typi al divergen e, termed a Atom-atom ollisions in the ultra old regime are de- zero-energy resonan e. One may note that a 0.1-1% po- 4 −1 tential variation (1-10 m on the potential depth) is Therewill bealow-energyresonan eonly if the width −∞ +∞ a γ(E) < E s su(cid:30) ient for a omplete to variation of . is su(cid:30) iently narrow . If the more stri t on- γ(E) ≪ E γ(E) In order to investigate the relation between the zero- dition is ful(cid:28)lled an be repla ed with a γ =γ(E ) s r r energyquantity and(cid:28)nite-energys attering,wesele t the onstantquantity andthede omposition λ δ π values of the aontrol parameter orresponding to the in Eq. (3) implies that undergoes a rapid γva∼riaEti1o/n2 s samevalue of and omparethe orrespondingenergy- a ross resonan e. Note that be ause of the ℓ=0 dependent elasti ross se tions. threshold law, s attering at su(cid:30) iently low energy J =0 γ <E We (cid:28)rst onsider total angular momentum . For alwaysviolatesthe ondition and no resonantbe- j = 0 rotational states this implies an angular momen- havior will arise. However, this does not rule out the ℓ = 0 tum . Fig. 3 shows the result of the omparison presen e of resonan es at higher yet very low ollision a J = 0 s for a typi al positive value of . The partial energy (see below). σ a > 0 s ross se tions for show a qualitatively similar Resonan es an also be analyzed in terms of the a s behavior essentially determined by the value of and Wigner time delay [58℄, C by the long-range 6 oe(cid:30) ieσnt→. 4Oπnae2 an remark the Q=2~dδ , well known zero energy limit s. The minimum dE (4) of the ross se tion orresponds to the s attering phase π shiftgoingthroughamultipleof , andintheabsen eof i.e. the average delay of a s attering event ompared ontributionsfrom higherorderpartialwaveswould or- to free transit in the absen e of the potential. In the respond to a Ramsauer-Townsend minimum in the total 10 ross se tion [17℄. 1 1 2) m c 100 22 cm) 0.1 1.5 -12σ (10 0.1 10-1 -1(10 1 0.01 10-2 σ 0.01 0.5 100 101 102 103 0 100 200 300 400 500 100 101 102 103 104 E (µK) 100 101 102 103 104 E (µK) J =0 2(v=0,j =0) FIG.4: Thepartial Na+Na elasti ross se tionas a fun tionof ollision energyfordi(cid:27)erentvaluesof FIG.3: ThepartialJ =0Na+Na2(v=0,j =0)elasti ross the three-body ontrol parameter determinedasby=th−e10sa0ma0e value of the atom-diatom s attering length . se tionas a fun tionof ollision energy fordi(cid:27)erentvaluesof Cross se tions presenting resonant behavior are identi(cid:28)ed by the three-body ontrol parameter determined by the same as = 60a0 dashedlines. Thedottedline denotestheunitaritylimit (see value of the atom-diatom s attering length . The λ text). Sample ross se tions obtained using di(cid:27)erent and inset refers to the typi al energy regime of urrent ultra old presentingverysimilarenergydependen eareemphasizedin mole ule experiments. theinset. Theinsetalso shows(dash-dottedlines) twovirtu- as>0 a <0 ally identi al ross se tions for extra ted fromFig. 3. s The rossse tions al ulatedfor aremoreinter- esting. A large negative s attering length is asso iated threshold regime using Eq. (3) one obtains: wEi0thaboundstateturnedintoavirtualstatewithenergy ~γ dη 2dδbg in the ontinuum. Thissituation anbe onveniently Q = 1− + des ribed by de omposing the elasti phase shift into a (E−Er)2+ γ42 (cid:18) dE(cid:19) v dk ba kground plus a resonan e ontribution [17, 57℄ E−E 1dγ r − γ/2 (E−E )2+ γ2 vdk (5) δ =δbg+δres , δres =−arctan r 4 E−E (3) r v k where is the velo ity in the relative motion and E = E + η γ r 0 where is the resonan e position, with the relative wave ve tor. The (cid:28)rst term is the usual η and the resonan e width andℓshift, respe tively. For Lodrηentzian pro(cid:28)le arising from exponential de ay with the (cid:28)rst few angular momenta low energγy∼s aEttℓe+r1i/n2g a dE orre tion. Nearisolatedresonan esat highenergy iδs d∼eteErmℓ+i1n/e2d by tηh∼e acsoynmstptoti behavior , this isusuallythe dominant ontributηion to the time de- bg and [57℄. lay;seee.g.[59℄. Theresonan eshift isusuallyaslowly 5 a >0 a <0 s s varying fun tion of energy but in the presen e of addi- both and areputforthintheinsetofFig.4. tionals atteringfeaturessu hasshaperesonan esinthe This shows that knowledge of energy dependent ross J =0 ba kground ontinuum [60℄. se tions in the regimewhere only the partialwave The se ond term in Eq. (5) is the lassi al time for is important is in general not su(cid:30) ient to determine the 2dδbg ℓ = 0 the relative parti le to span a distan e dk . For strength of the three-body nonadditivepotential, even if δ ∼ −ka elasti s attering bg bg, and this term redu es to its shape were pre isely (cid:28)xed. −2a /v J = 0 bg orrespondingto anattra tive(repulsive) har- However, model potentials giving equivalent J > 0 a ter of the ba kground potential for negative (positive) ross se tions will not in general be equivalent if a bg ba kground s attering lengths . The third term van- s attering is explored. For instan e Fig. 6 shows the E = E J = 1 (v = 0,j = 0) r ishes for and gives a orre tion of dispersive elasti ross se tions for mole ules. shape a ross resonan e. Cal ulations use the same model potentials as the in- Note that in Fig. 4 one an essentially identify two set in Fig. 4 and an be identi(cid:28)ed a ording to the line lasses of urves. The (cid:28)rst lass (full lines) presents a style. Order of magnitude di(cid:27)eren es are observed for J = 1 J = 0 monotoni allyde reasingbehaviortowardsthe(cid:28)rstmin- ross se tions in ases where ross se - imum,whi h orrespondstononresonants attering. The tions are identi al. One an on lude that the initial se ond lass (dashed lines) present peaks at whi h the hara terization of a theoreti al model based on purely σs a=tt4eπringpartial rossse tionrea hestheunitaritylimit elasti ollisions should take into a ount aJsu>(cid:30) 0iently k2 (dotted line in Fig.3) and anin prin iple beas- broad energy range for the ontribution of par- so iated with a resonant behavior. tial waves to be ome observable. In alternative, in ases 250 where the method for the produ tion of old mole ules allowsthe initial ro-vibrationalstate to be ontrolled, at 200 least one experimental inelasti ross se tion (for some s)150 (v = 0,j > 0) n initial state, for instan e) should omple- ( Q 100 ment the elasti ollision data. 50 102 104 103 1 K) 2δn()0.07.55 2m) 100γµ (r110021 si0.25 -120 c 101 Er1 (0µ2K) 103 1 ( 0 100 200E (µK)300 400 500 σ 10-2 Q sin2δ FIG.5: TheWignertimedelay (upperpanel)andthe J =0 2(v=0,j =0) quantityfor Na+Na elasti ollisionsand λ 400µ sele ted values. Onlythepeaknear K anbe lassi(cid:28)ed 1 10 100 E(µK) as a resonan e (see text). J =1 2(v=0,j =0) Wefo usonthelowest-energypeakofFig.4andmake FIG.6: Thepartial Na+Na elasti ross the three-body potential slightly more attra tive in or- se tion al ulatedwiththesameset ofmodelpotentialsused intheinsetofFig.4. Samelinestyleisusedfor orresponding dertofurtshiner2sδhift tQhisfeaturetowardsthreshold. Fig.5 rossse tionsobtainedwiththesamepotential. Aresonan e showsthe and quantitiesforthreesele tedpoten- 400µ an be observed in the upper dashed urve. The inset shows tials. Thefeaturenear K anbeessentially lassi(cid:28)ed γr Er γ ≃ 0.5E theresonan e width as a fun tion of its position . r r as a resonan e with . As the potential be- γ E γ =E r r r omes more binding be omes larger than ( E ≃200µ r for K)andtheresonantbehaviortends to dis- Please note that one peak is also observed in Fig. 6. appear. As the peak is made to shift loser to threshold Analysisofnumeri alresultsbasedonEq.(3)showsthat γ ≃0.5E r r thetimedelayisfullydominatedbytheba kground on- , i.e. this featurerepresentsa resonan e. The tribution; see upper panel. Alssoinn2oδte=(1lower panel) that inset shows the resonanλ e width as the peak enter is in all ases the unitarity limit is attained. shifted by making the parameter vary. Its position ≃ 400µ ℓ = 1 An additionalinteresting featurethat anbe observed below the maximum K of the entrifugal J = γ =E E ≃500µ r r r by inspe tion of Figs. 3-4 is the near oin iden e of barrier(we(cid:28)nd for K)suggeststhatit 0 elasti ross se tions omputed with di(cid:27)erent three- isashaperesonan e. However,Feshba h ouplingsimilar E < 10 J = 0 body potentials in the whole energy range mK. to the one found for ollisions is not on lusively Sample ross se tions illustrating this ir umstan e for ruled out. 6 In on lusion, we(h1a4vAe′)presented a new potential en- parametrizationsofthepotentialenergysurfa ewillneed 3 2 ergy surfa e for Na . We have demonstrated that to be introdu edin orderto omparequantitatively the- long lived triatomi omplexes exist and give rise to res- ory and experiments. onan e e(cid:27)e ts in rea tive ollisions even at very low ollision energies. General featuresto be expe ted in atom- mole ule s attering in the ultra old regime have also in- A knowledgments vestigated by performing a systemati variation of the three-body part of the intera tion potential. Knowledge J =0 ofenergy-dependent elasti rossse tionsmaynot We wish to thank A. Viel for useful dis ussions. The be su(cid:30) ient to determine the strength of the nonaddi- authors a knowledge support of Egide (PHC Barrande tive three-body intera tion. To this aim, at least one # 13860UA) and of the Ministry of Edu ation, Youth J > 0 additional elasti or inelasti ross se tion needs and Sports of the Cze h Republi (KONTAKT proje t to be experimentally determined. In this work we have Barrande 2-07-3 and Resear h proje t no. 0021620835). studied the sensitivity of s attering observables by in- PSappre iatessupportoftheEuropeanS ien eFounda- trodu ing a global s aling parameter of the three-body tion and the Cze h S ien e Foundation (EUROCORES intera tion. 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