Multiscale quantum-defect theory for two interacting atoms in a symmetric harmonic trap

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1 Multiscal quantum-dfct thory for two intracting atoms in a symmtric harmonic trap Yujun Chn and Bo Gao* Dpartmnt of Physics and Astronomy, Univrsity of Toldo, MS, Toldo, Ohio, USA Rcivd January 7; publishd May 7 W prsnt a multiscal quantum-dfct thory for two idntical atoms in a symmtric harmonic trap that combins th quantum-dfct thory for th van dr Waals intraction B. Gao, Phys. Rv. A, 7 R at short distancs with a quantum-dfct thory for th harmonic trapping potntial at larg distancs. Th thory provids a systmatic undrstanding of two atoms in a trap, from dply bound molcular stats and stats of diffrnt partial wavs, to highly xcitd trap stats. It shows,.g., that a strong p-wav pairing can lad to a lowr nrgy stat around th thrshold than a s-wav pairing. DOI:./PhysRvA.75.5 PACS numbr s :.75.Nt,.. x,.5.g,..pj I. INTRODUCTION Two intracting particls undr confinmnt, dscribd gnrally by a Hamiltonian H = m m + V r + V r + v r r, whr V and V ar th confining potntials and v r is th intraction btwn particls, rprsnts a fundamntal class of problms in physics. On famous xampl is th hliumatom problm that has playd an important rol in our undrstanding of lctron corrlation in atomic physics s,.g., Rf.. Similarly, th problm of two atoms in a harmonic trap, which has attractd considrabl rcnt attntion s,.g., Rfs., is th ky to our undrstanding of atomic corrlation in a trappd many-atom quantum systm. Such corrlation diffrs qualitativly from th lctron corrlation bcaus atoms attract ach othr at larg distancs and can form bound stats. Existing thoris of two atoms in a trap hav rlid mostly upon th psudopotntial modl of atomic intraction, and its gnralizations 7. Whil such modls can work wll in dscribing how th trap stats, spcially th lowst fw, ar affctd by atomic intraction, thy gnrally fail in dscribing how a molcular stat is affctd by trapping, with th only xcption bing th last bound molcular stat with a vry larg scattring lngth. Furthrmor, such thoris do not adquatly addrss nonzro partial wavs, for which naiv gnralizations of th shap-indpndnt approximation, using,.g., th ffctiv rang thory ERT 5, would gnrally lad to incorrct rsults. W prsnt hr a multiscal two lngth scals, to b xact quantum-dfct thory QDT for two idntical atoms in a symmtric harmonic trap. It is a compltly gnral thory that works for diffrnt partial wavs, and from dply bound molcular stats to highly xcitd trap stats. In Sc. II, w xpand our tool box of QDT for diffrnt long-rang potntials, 9 by prsnting a QDT for a symmtric harmonic potntial. It is indpndntly usful byond th scop of two atoms in a trap. For xampl, it may b usd to trat two nuclons outsid of a closd shll. * addrss: bgao@physics.utoldo.du In Sc. III, this thory is combind with th angularmomntum-insnsitiv quantum-dfct thory AQDT for th van dr Waals intraction to formulat a two-scal QDT that provids a systmatic undrstanding of two idntical atoms in a symmtric harmonic trap. Rsults and discussions ar prsntd in Sc. IV, including a discussion of th limitations of shap-indpndnt approximations, and a univrsal spctrum for two atoms in a trap at th van dr Waals lngth scal that shows,.g., that a strong p-wav pairing can lad to a lowr nrgy stat around th thrshold than a s-wav pairing. W will also show that two atoms in a trap has a long-rang corrlation that bcoms important for larg scattring lngths, a rsult that has provn to b th ky for gnralizing th variational Mont Carlo VMC studis of a fw atoms in a trap to th rgim of strong coupling. Conclusions ar givn in Sc. V. II. QUANTUM-DEFECT THEORY FOR A SYMMETRIC HARMONIC POTENTIAL Th goal of a QDT for a symmtric harmonic potntial is to provid a systmatic undrstanding to a class of problms dscribd by th radial Schrödingr quation d dr + l l + r + V r u l r =, with r V r r. Unlik th standard txtbook solution, which rquirs that V r = r for all r, a cas that w shall rfr to as th pur harmonic oscillator, th QDT formulation is applicabl to any V r that is asymptotically a harmonic oscillator, but may diffr from it at short distancs in an arbitrary fashion. As in any QDT formulation,,7, w start by dfining a pair of rfrnc functions that ar two linarly indpndnt solutions for a symmtric harmonic potntial d dr + l l + r + r v r =. Th solutions can b asily found 5, and w will tak 5-97/7/75 5 / Th Amrican Physical Socity

2 YUJUN CHEN AND BO GAO f ho l x = x l+ x / M b,c,x, g ho l x = l + x l x / M +b c, c,x. Hr = / is a scald nrgy. x=r/ ho is a radius scald by ho = / /, which is a lngth scal associatd with th harmonic potntial. M is th conflunt hyprgomtric function 5, b= l+/ /, and c=l+/. In this dfinition, f ho l is rgular at th origin, and g ho l is irrgular. Thy will b calld th rgular solution and th irrgular solution, rspctivly. Thy ar also chosn such that thir Wronskian is givn by W f ho l,g ho ho l f dg ho l ho l g df ho l dx l dx =. 7 With this dfinition of rfrnc functions, th wav function u l for any potntial that is asymptotically a harmonic oscillator can b writtn, at sufficintly larg distancs, as u l r = A l f ho l x K ho,l g ho l x. This dfins th K matrix K ho for a symmtric harmonic potntial, with its valu bing gnrally dtrmind by matching Eq. to th short-rang solution. Making us of th larg r asymptotic bhaviors of f ho and g ho, as givn by 5 r f ho c l b x / +x /, 9 r g ho l l + c +b c x / +x / sin c b c x+ / x /, 5 and nforcing th boundary condition u l r at infinity, w obtain th following quation that givs th nrgy spctrum of Eq. as th crossings points of two functions: Hr l ho = K ho,l. ho l = l+ l + / + l + / / + l + / /, is a univrsal function of th scald nrgy that is dtrmind solly by th harmonic potntial, and K ho is th K matrix that ncapsulats all th short-rang physics. Plots of ho l for th s and p wavs ar shown in Figs. and, rspctivly. Equations and giv a rigorous formulation of nrgy spctrum for any potntial that is asymptotically a harmonic oscillator. It can b implmntd numrically, or usd as a basis for approximat analytic solutions. Th cas of a pur symmtric harmonic oscillator is includd as a spcial cas, corrsponding to K ho,l = for all, with a () χ l (ho) l= FIG.. Th l ho function for th s wav. Th nrgy spctrum for any potntial that is asymptotically a symmtric harmonic oscillator is givn by th crossing points of this function with a short-rang K matrix K ho,l dfind by Eq.. Th spcial cas of K ho,l = for all nrgis corrsponds to a pur harmonic oscillator. wll-known nrgy spctrum of = j+l+/, whr j =,,,... From Eq., it is clar that th ky to finding th nrgy spctrum is to find th K matrix K ho ho,as l is alrady known analytically. For this purpos, w not that th rfrnc functions f ho and g ho hav th following small r asymptotic bhaviors that can b drivd from a propr xpansion of th conflunt hyprgomtric function 5. For, w hav r f ho l ho l+/ l + / k ho l+ kr j l kr, r g ho l ho k ho l l+/ l + / kr y l kr, whr k= / /. For, w hav () χ l (ho) l= FIG.. Th sam as Fig. xcpt that it is for th p wav. 5-

3 MULTISCALE QUANTUM-DEFECT THEORY FOR TWO r ho f ho l+/ l + / l ho l+ r / I l+/ r, 5 r ho g ho l+ ho l l l+/ l + / r / I l / r, whr = / /. For intractions V r that dviat from th harmonic potntial only in th rgion of r ho, ths bhaviors gratly facilitat th matching to th shortrang solution, from which K ho can b dtrmind. This is illustratd in our tratmnt of two idntical atoms in a symmtric harmonic trap, to b prsntd in th nxt sction. III. TWO IDENTICAL ATOMS IN A SYMMETRIC HARMONIC TRAP Two intracting atoms in a symmtric harmonic trap ar dscribd by a Hamiltonian H = m m + m r + m r + v r r, 7 whr v r rprsnts th intraction btwn thm. To b spcific, w rstrict ourslvs hr to a class of problms for which th atomic intraction is charactrizd, at larg distancs, by an attractiv /r van dr Waals potntial r v r C /r, which has an associatd lngth scal of = C / /. For two atoms having th sam trapping frquncis, namly, = =, which includ of cours th cas of two idntical atoms of intrst hr, th cntr-of-mass motion and th rlativ motion ar sparabl, and th solution of two idntical atoms in a symmtric harmonic trap rducs to th solution of Eq. with V r = v r + r = C /r + r, r r, 9 whr r rprsnts th radius insid which th intractions of shortr rang than, such as th C /r corrction, would com into play. Sinc th V r charactrizd by Eq. 9 is asymptotically a harmonic oscillator, it is amnabl for th QDT tratmnt of Sc. II. In particular, th solution for th nrgy spctrum rducs to finding th K matrix K ho for th class of problms dfind by Eqs. and 9. This will b accomplishd hr by taking advantag of th disparat lngth scals in th systm. For r r, th potntial V r as givn by Eq. 9 has two lngth scals. In addition to that is associatd with th van dr Waals intraction, th harmonic trapping potntial has a lngth scal that can b takn ithr as th ho dfind arlir, or as a ho = /m / = ho /. W will us both intrchangably, but will mphasiz a ho for th sak of asir comparison with othr rsults. For atoms in a typical magntic or optical trap, a ho is of th ordr of a micron, which is much gratr than that is of th ordr of a.u. or about 5 nm. Undr this condition of a ho, which w will call th limit of wak confinmnt, th van dr Waals and th trapping potntials oprat on distinctiv lngth scals. In th rgion of r, th van dr Waals intraction is ngligibl, and w hav u II r = A f ho l K ho,l g ho l. In th rgion of r r a ho, th harmonic potntial is ngligibl, and th intraction is dominatd by th van dr Waals intraction. Hr th wav function can b writtn as u I r = B f c s l K c,l g c s l, whr f c s l and g c s l ar th rfrnc functions for th van dr Waals potntial, and K c is th corrsponding short-rang K matrix,,. For wak confinmnt dfind by a ho, thr xists an intrmdiat rgion r a ho in which ithr, or both, th van dr Waals potntial and th trapping potntial can b ignord. In this rgion, th wav function can b writtn ithr in th form of th innr solution as givn by Eq., or in th form of th outr solution as givn by Eq., and thy must agr with ach othr. Sinc r in th intrmdiat rgion, f ho ho l and g l ar givn by Eqs.. In th sam rgion, r/ and th rfrnc functions for th van dr Waals potntial ar givn for by, f c s l = kr Z c k ff j l kr + Z c fg y l kr, g c s l = kr Z c k gf j l kr + Z c gg y l kr, and for by, f c s l = W c f + l W c f+ r/ / I l+/ r + W c f l W c f+ r/ / I l / r, g c s l = W c g + l W c g+ r/ / I l+/ r + W c g l W c g+ r/ / I l / r. 5 Hr th Z c and W c matrics dscrib th propagation of a wav function in a C /r typ of potntial from small to larg distancs, and vic vrsa,,7. Thir lmnts ar all of univrsal functions of a scald nrgy s = /s E, whr s E = / / is th nrgy scal associatd with th van dr Waals intraction. Explicit xprssions for th lmnts of th Z c can b found in Rf. 7. Th W c ma- 5-

4 YUJUN CHEN AND BO GAO trix, which is rlatd to th W matrix dfind in Rf. by a linar transformation, is givn by W c f s = W f+ / G s l X s l + Y s l sin +M s l sin / X s l + M s l cos / Y s l, c s = / G s l cos X s l + Y M s l sin / X s l s l W c g s = + +M s l cos / Y s l, 7 / G s l X s l + Y s l sin M s l cos / X s l +M s l sin / Y s l, c s = / G s l cos X s l + Y +M s l cos / X s l s l W g+ M s l sin / Y s l. 9 Hr M s l=g s l /G s l, with, X s l, Y s l, and G s l, all of which ar functions of th scald nrgy s, bing dfind in Rf.. Comparing, in th intrmdiat rgion, th innr solution givn by Eqs. 5 with th outr solution givn by Eqs. and, w obtain, for, K ho = l + / tan l / l+/, whr tan l = Z c fg Z c gg K c Z c ff Z c gf K c, is th physical K matrix for atomic scattring in fr spac as givn in AQDT. For, w obtain K ho = l l + / c l K c l W c f+ /W c g K c W c g+ /W c g / l+/ c l K c + l W c f+ /W c g K c W c g+ /W g c, whr c l s =W c f /W c g is th function that dtrmins th molcular spctrum in th absnc of trapping. Equations,, and giv a complt dscription of th nrgy spctrum for two idntial atoms in a symmtric harmonic trap, from dply bound molcular stats to highly xcitd trap stats. Th only assumption in th thory is th assumption of wak confinmnt as spcifid by, which is wll satisfid undr all xisting xprimntal conditions. Othr than th two nrgy scaling paramtrs and s E, th only paramtr in th thory is K c,l, which charactrizs th intractions of shortr rang than. It can b rplacd by othr quivalnt short-rang paramtrs such as th quantum-dfct c,l 9 or th K l,l paramtr,. IV. RESULTS AND DISCUSSIONS A. Univrsal spctrum at th lngth scal of Th rsults givn by Eqs.,, and can b writtn in diffrnt forms that ar convnint for diffrnt purposs. For concptual undrstanding, it is bst to rarrang thm so that th ntir nrgy spctrum is givn by th solutions of a singl quation whr l ho,, = K c,l, for, ho, l = Z c fg s Z c ff s ho l Z c gg s Z c gf s ho l ho, l = c l s ho l W c f+ s /W c g s ho l W c g+ s /W c g s for, and w hav dfind and 5 ho l = l / l / / + l + / / + l + / /, ho l = l l ho + ho l. 7 Th ho, l function is a univrsal function that is applicabl to any two idntical atoms in a symmtric harmonic trap, providd thy intract via th C /r typ of van dr Waals potntial at larg intratomic sparations. Th strngths of intractions, as charactrizd by C and, play a rol only through nrgy scaling paramtrs s E and. Spcifically, th ho, l function is mad up of trms that dpnd on nrgy through two diffrnt scald nrgis s = /s E and = / that ar rlatd by s =. In othr words, it is mad up of functions that vary on two distinctiv nrgy scals: th ho and th ho functions that varis on th 5-

5 MULTISCALE QUANTUM-DEFECT THEORY FOR TWO scal of, and th Z c and W c matrix lmnts that vary on a scal of s E. Th solutions, mor prcisly th invrs, of Eq. can b writtn as 9 s wav = i l K c,, whr, similar to ho, l, th l i ar univrsal functions that ar uniquly dtrmind by th xponnt of th van dr Waals intraction n=, and th xponnt of th trapping potntial for th harmonic trap. Instad of th paramtr K c, th sam univrsal functions can also b xprssd in trms of quivalnt paramtrs such as th quantum-dfct c 9, orthk l paramtr, all of which ar wll dfind for all l. Whn th nrgy and angular momntum dpndnc of th short-rang paramtr K c is includd, Eqs. and ar xact, and applicabl to arbitrary nrgy and partial wavs. Ignoring th nrgy and th l dpndnc of K c, which in th cas of a singl channl is du ntirly to intractions of shortr rang than,9, th solutions of Eq., namly, Eq. with K c =K c =,l=, or its variations, giv what w call th univrsal spctrum at lngth scal,,. It is followd by all two-atom systms in a trap with C /r typ of long-rang intraction, ovr a rang of nrgis that is hundrds of s E around th thrshold,, which far xcds all nrgis of intrst in coldatom physics. As an xampl, s E =9. K for Na. Othr than th nrgy scaling paramtr s E that is dtrmind by th C cofficint and th atomic mass, all nrgy lvls in this nrgy rang, including stats of diffrnt l, ar dtrmind, in th cas of singl channl, by two paramtrs, K c =K c =,l= and. This is an xampl of univrsal spctrum at th scond longst lngth scal in th systm, as opposd to th univrsal spctrum at th longst lngth scal, which would hav rquird only a singl paramtr. Dpnding on th physics of intrst, th univrsal spctrum at lngth scal can also b xprssd in trms of othr paramtrs. In particular, it can b xprssd in trms of th s-wav scattring lngth a,as = i l a,, sinc a can b rlatd to K c =K c =,l= by a / n = b +b b b Kc, + tan b/ K c, tan b/, 9 whr b=/ n with n= 9,. Similar rprsntations of univrsal spctrum can also b dfind mor gnrally for N atoms N in a trap. Figur illustrats th univrsal s-wav spctrum at lngth scal for two idntical atoms in a symmtric harmonic trap. It uss th rprsntation of Eq. 9 to facilitat comparison with th shap-indpndnt approximation s Rf. and Sc. IV B. Th l= functions of two variabls a and ar plottd hr as functions of a for diffrnt valus of. In th small rang of nrgis "Shap-indpndnt" β =. β =. - - a FIG.. Color onlin Univrsal s-wav spctrum at lngth scal for two idntical atoms in a symmtric harmonic trap and with an asymptotic intraction of th typ of /r. Th arrow points to th s-wav scattring lngth a x byond which th lastbound molcular s stat is pushd to positiv nrgis. shown in th figur, which corrsponds to s E, th univrsal spctrum approachs that of th shap-indpndnt rsults in th limit of. Figurs and 5 both illustrat th univrsal p-wav spctrum at lngth scal. Thy also srv to illustrat how diffrnt rprsntations of th univrsal spctrum can srv diffrnt purposs in trms of physical undrstanding. In Fig., th univrsal p-wav spctrum is plottd as a function of a / for diffrnt valus of. It givs th bst illustration that, in th cas of singl channl, th s-wav scattring lngth dtrmins not only th s-wav spctrum, but also th spctra of othr partial wavs including th p wav. It also dtrmins th p-wav scattring lngth a also calld scattring volum as it has th dimnsion of a volum through a / = / a ā, ā a whr ā = / / =.77 9 is th man s-wav scattring lngth of Gribakin and Flambaum. 5-5

6 YUJUN CHEN AND BO GAO p wav a / β β =. β =. FIG.. Color onlin Univrsal p-wav spctrum at lngth scal for two idntical atoms in a symmtric harmonic trap and with an asymptotic intraction of th typ of /r, plottd hr vrsus a scald s-wav scattring lngth a /. Figur shows that th p-wav spctrum, in th cas of singl channl, is strongly influncd by th atomic intraction only whn th s-wav scattring lngth is around ā = , which corrsponds to having a p-wav bound stat right at th thrshold 7,9. For a slightly largr than ā, a is larg and ngativ s Eq., th lowst fw trap stats ar strongly affctd by a p-wav shap rsonanc nar th thrshold. For a slightly lss than ā, a is larg and positiv, and thr is a p-wav molcular stat clos to th thrshold. In Fig. 5, th univrsal p-wav spctrum is plottd as a function of a /a ho for diffrnt valus of. Th advantags of this rprsntation ar twofold. First, it facilitats comparison with th p-wav shap-indpndnt approximation to b discussd in Sc. IV B. Scond, whil th rprsntation shown in Fig. is applicabl only in th cas of singl channl, th rprsntation shown in Fig. 5 would apply vn in multichannl cass, providd th nrgy dpndnc of K c du to closd channls is ngligibl in nrgy rang of intrst,. This is bcaus in rprsnting th p-wav spctrum as a function of a, instad of a, on is only making us of th nrgy indpndnc of K c, not its p wav angular indpndnc, which taks on diffrnt charactristics still rlatd for multichannl problms,. Spcifically, this rprsntation is obtaind from solving Eq. using K c =K c =,l=, which is rlatd to th a through th following rlations, a = ā + a =, K l= "Shap-indpndnt" β =. β =. FIG. 5. Color onlin Univrsal p-wav spctrum at lngth scal for two idntical atoms in a symmtric harmonic trap and with an asymptotic intraction of th typ of /r, plottd hr vrsus a scald p-wav scattring lngth volum a /a ho. Th arrow points to th p-wav scattring lngth a x byond which th last-bound molcular p stat is pushd to positiv nrgis. whr ā = / / is a man p-wav scattring lngth, and K l = = c l K c =,l +c l K c =,l, whr c l =tan l +. Comparing ithr Fig. or 5 with Fig. lads to on of th mor important conclusions of this work. That is, unlik th nonintracting particls in a trap for which th lowst p-stat nrgy is always gratr than that of th s stats, a strong p-wav pairing a a ho or gratr for intracting particls can lad to a lowr nrgy stat around th thrsh- 5-

7 MULTISCALE QUANTUM-DEFECT THEORY FOR TWO old than a s-wav pairing. This stat is, for larg and ngativ a,ap-wav-shap rsonanc stabilizd by th trap, and is a p-wav-molcular stat for larg and positiv a. Similar statmnts can also b mad for othr, highr partial wavs. Furthrmor, w xpct th sam physics to prsist in manyatom systms, which will b a subjct of futur invstigations. Othr charactristics of th univrsal spctra ar addrssd sparatly in subsqunt sctions s wav B. Limitations of shap-indpndnt approximations All prvious rsults on two atoms in a symmtric harmonic trap can b asily drivd as various approximations within our thory. In particular, th shap-indpndnt approximation corrsponds to ignoring nrgy dpndnc of K ho, and taking it to b its valu at zro nrgy, namly, K ho l + / ho / l+ lim k tan l k l+. For partial wavs with wll-dfind scattring lngths a l =lim k tan l /k l+, this approximation usd in Eq. lads to th following quation for th nrgy spctrum: l l+/ + l + / / + l + / / = a l /a l+ ho, 5-5 β =. "Shap-indpndnt" a FIG.. Color onlin Univrsal s-stat spctrum at lngth scal for two idntical atoms in a symmtric harmonic trap and with an asymptotic intraction of th typ of /r, plottd to illustrat th failur of th shap-indpndnt approximation away from th thrshold. for both positiv and ngativ nrgis. For l=, Eq. 5 rproducs th rsult of Rf., drivd using a dltafunction psudopotntial. Figur has shown that for th s wav, th shapindpndnt approximation givs a good approximation to th univrsal spctrum undr wak confinmnt and for nrgis s E. Th shap dpndnc is mor important for strong coupling a a ho or gratr, but spcially for nrgis furthr away from th thrshold. In particular, th shap-indpndnt approximation braks down for all nrgis s E or gratr, for which th nrgy dpndnc of K ho du to th long-rang van dr Waals intraction can no longr b ignord,. For xampl, it dos not giv th propr molcular binding nrgy for small positiv scattring lngths, and it fails compltly to dscrib molcular stats of ngativ scattring lngth, as illustratd in Fig.. In rality, ths mor dply bound molcular stats approach thos of a fr molcul in th absnc of a trap, with th trapping potntial srving as a wak prturbation, as to b discussd furthr in Sc. IV D. With th rlativ succss of th shap-indpndnt approximation for th s wav in th thrshold rgion, it is important to mphasiz its svr limitations for any partial wavs othr than th s wav. For atoms with th C /r typ of van dr Waals intraction, th shap-indpndnt approximation clarly fails for l, for which thr ar no wll-dfind scattring lngths,. Evn for th p wav, it is applicabl only at zro nrgy, or for wak p-wav coupling as charactrizd by a a ho or smallr, as illustratd in Fig.. This failur of th shap-indpndnt approximation for l is dirctly rlatd to th failur of th ffctiv rang thory 5 in dscribing th shap rsonancs clos to th thrshold, and mor gnrally to th failur of ERT in dscribing Fshbach rsonancs in nonzro partial wavs. Our thory allows for simpl gnralizations byond th shap-indpndnt approximation. For xampl, for positiv nrgis, Eqs.,, and, that dtrmin th nrgy spctrum can b rwrittn as ho l = tan l. Instad of th ffctiv-rang xpansion for tan l, which lads to th shap-indpndnt approximation, on can simply us th corrsponding QDT xpansion for th C /r typ of potntial. Th rsults would b applicabl from zro nrgy up to s E for th s wav, and ovr a gratr rang of nrgis for highr partial wavs. This nrgy rang, whil much smallr than that dscribd by th univrsal spctrum Eq., alrady xcds th rang of intrst in xisting xprimnts. C. Trap stats and molcular stats Th stats of two atoms in a trap can b classifid into trap stats and molcular stats. Th formr corrsponds to stats that volv into diatomic continuum as th trap is turnd off adiabatically a ho. Th lattr corrsponds to stats that volv into bound molcular stats in th sam limit. Not that th molcular stats would not hav xistd in th hard-sphr atomic modl. Th molcular stat of highst nrgy corrsponds to th ons in Figs. 5 that cross th zro nrgy. It is th lastbound molcular stat that gts pushd up in nrgy by th trapping potntial. Th crossing into positiv nrgy can happn ithr by tuning up th scattring lngth, which has th ffct of making th binding nrgy of th molcul sufficintly small in th absnc of th trap, or by tuning up th trap frquncy. 5-7

8 YUJUN CHEN AND BO GAO Th crossing points, which corrspond to having a stat right at =, can b asily found through Eq. 5, which is xact at zro nrgy for partial wavs with wll dfind scattring lngths. For s and l p wavs, it givs l+/ a lx = / l/ l/ + / a l+ ho. 7 For th s wav with a fixd trap frquncy, it givs a x = / a ho =.9 99 a ho, byond which th last bound molcular stats ar pushd to positiv nrgy. For a fixd a, th sam quation dtrmins th crossing trap frquncy x = / ma, 9 byond which th last bound s-wav molcular stat is pushd into positiv nrgy. For th p wav, it givs a x = / a ho = a ho. 5 For a fixd a, it dtrmins th crossing trap frquncy / x = / / / ma, 5 byond which th last bound p-wav molcular stat is pushd to positiv nrgy. All highr branchs of stats in Figs. 5 ar trap stats. All lowr branchs, which approach thos of molcular stats in th absnc of trapping, as discussd in th nxt sction, ar molcular stats. D. Trapping shift of molcular spctrum Excpt for th last-bound stat with binding nrgy comparabl to or smallr than, th ffcts of trapping on molcular stats ar gnrally wak and can b tratd prturbativly. Th finit rang of such stats is such that th atoms in thm would hardly fl th xistnc of a trap in thir rlativ motion. Th natur of this prturbation is bst undrstood by rwriting Eq. for as c l s = K c + ho l W c f+ K c c W g+ c. 5 W g Comparing this quation to c l s =K c, which dtrmins th molcular spctrum in fr spac, th ffct of trapping is isolatd in this formulation to th scond trm in Eq. 5. From Eq. 7 and ho l + l / l + / l + /, 5 it is clar, as xpctd, that th molcular stats with binding much gratr than ar only wakly affctd by th trapping and can b tratd by solving Eq. 5 prturbativly. To th lowst ordr in, th nrgy shift du to trapping is givn by ls = q l ls a ho, 5 whr q l ls = l l / l + / l + / ls W c f W c g W c g W c, f s = ls 55 and ls = l /s E is th scald bound stat nrgy of a molcul in th absnc of th trap. Equation 5 mans that trapping shift is, to th lowst ordr, proportional to, multiplid by a univrsal function of th scald binding nrgy that gos to zro in th limit of ls. E. Highly xcitd trap stats For an highly xcitd trap stat with, th dtrmination of nrgy spctrum can also b furthr simplifid. From ho l tan l + / /, Eq. rducs, for, to 5 tan l + / / = tan l, 57 or l+/ /+ l = j, whr j is an intgr. This rsult for l= has also bn drivd by Bolda t al. using a gnralizd psudopotntial 7. It could also hav bn drivd by using a smiclassical approximation for th harmonic part of th potntial. For all nrgis of xprimntal intrst, th phas shift l can b accuratly dscribd using AQDT paramtrization of Rf.. F. Long-rang corrlation btwn atoms in a trap In studis of quantum fw-body and quantum many-body systms, it is oftn assumd that th wav function can b writtn in a Jastrow form 5, which is givn,.g., for bosons by N = r i i= N i j= F r ij. 5 Hr N is th numbr of particls, rprsnts an indpndnt-particl orbital, and F is th pair corrlation function. It is commonly assumd that F has th asymptotic bhavior of F r at larg r, maning that th particls bcom uncorrlatd at larg sparations 5. Our thory hr provids an opportunity to chck th validity of ths assumptions, at last for N=. To b spcific, w rstrict ourslvs hr to th lowst s-wav trap stat for which th Jastrow assumptions ar usually applid. Combining th wav function for th rlativ motion, as givn by Eqs., and that for th cntr-of-mass motion, which is a pur harmonic oscillator in its ground stat, it is asy to show that th total wav function for two idntical atoms in 5-

9 MULTISCALE QUANTUM-DEFECT THEORY FOR TWO a symmtric harmonic trap can indd b writtn in th Jastrow form but with a corrlation function that bhavs as F r Cr /, 59 in th limit of larg intratomic sparations. Sinc can dviat significantly from / for strong coupling a or gratr, as illustratd in Fig., this rsult implis that thr xists significant long-rang corrlation for strongly intracting particls in a trap. In a rcnt publication, w hav shown that such long-rang corrlation xists not only for two particls in a trap, but also for N N strongly intracting particls in a trap. V. CONCLUSIONS In conclusion, w hav prsntd a QDT for a harmonic potntial and a two-scal QDT for two idntical atoms in a symmtric harmonic trap. It is a gnral thory that is applicabl to diffrnt partial wavs, and from dply bound molcular stats to highly xcitd trap stats. Th only approximation in th thory,, can b rlaxd if ndd. Th rsult will still b of th form of Eq. xcpt that th ho, function for or gratr will rquir a mor gnral solution of th two-scal potntial dscribd by Eq. 9. If th scattring lngth paramtr usd in our formulation is achivd by tuning around a Fshbach rsonanc,7, th sam rsults would apply, providd it is a broad Fshbach rsonanc with a width much gratr than th nrgy scal s E associatd with th van dr Waals potntial. From a mor gnral prspctiv, th thory w hav prsntd hr dmonstrats th concpt of multiscal QDT, which has potntial implications for a rang of problms. As xampls, w mntion two othr two-scal potntials and V r = C /r C /r, V r = Z/r C /r. Th formr is of intrst for a mor accurat QDT dscription of long-rang molculs ovr a widr rang of nrgis; th lattr is of intrst for a mor systmatic undrstanding of th cor polarization ffct on atomic spctra. Whil such potntials hav bn wll studid in numrical calculations, a mor systmatic approach to such two-scal problms may wll b worthy of futur fforts. ACKNOWLEDGMENTS W thank Eit Tisinga, Eric Bolda, and Paul Julinn for hlpful discussions. This work was supportd by th National Scinc Foundation undr th Grant No. PHY- 57. U. Fano and A. Rau, Atomic Collisions and Spctra Acadmic Prss, Orlando, 9. T. Busch, B.-G. Englrt, K. Rzazwski, and M. Wilkns, Found. Phys., M. Olshanii, Phys. Rv. Ltt., E. Tisinga, C. J. Williams, F. H. Mis, and P. S. Julinn, Phys. Rv. A,. 5 D. Blum and C. H. Grn, Phys. Rv. A 5,. M. Block and M. Holthaus, Phys. Rv. A 5, 5. 7 E. L. Bolda, E. Tisinga, and P. S. Julinn, Phys. Rv. A,. T. Brgman, M. G. Moor, and M. Olshanii, Phys. Rv. Ltt. 9,. 9 E. L. Bolda, E. Tisinga, and P. S. Julinn, Phys. Rv. A, 7. B. E. Grangr and D. Blum, Phys. Rv. Ltt. 9,. H. Moritz, T. Stöfrl, K. Güntr, M. Köhl, and T. Esslingr, Phys. Rv. Ltt. 9, 5. T. Stöfrl, H. Moritz, K. Güntr, M. Köhl, and T. Esslingr, Phys. Rv. Ltt. 9,. Z. Idziaszk and T. Calarco, Phys. Rv. A 7, 7. K. Huang and C. N. Yang, Phys. Rv. 5, J. M. Blatt and D. J. Jackson, Phys. Rv. 7, 99. C. H. Grn, U. Fano, and G. Strinati, Phys. Rv. A 9, C. H. Grn, A. R. P. Rau, and U. Fano, Phys. Rv. A, 9. B. Gao, Phys. Rv. A 5, B. Gao, Phys. Rv. A 59, A. L. Fttr and J. D. Walcka, Quantum Thory of Manyparticl Systms McGraw-Hill, Nw York, 97. B. Gao, Phys. Rv. A, 7 R. B. Gao, J. Phys. B 7, L7. B. Gao, Phys. Rv. Ltt. 95, 5. I. Khan and B. Gao, Phys. Rv. A 7, 9. 5 Handbook of Mathmatical Functions, ditd by M. Abramowitz and I. A. Stgun National Burau of Standards, Washington, D.C., 9. B. Gao unpublishd. 7 B. Gao, Phys. Rv. A, 57 R. B. Gao, E. Tisinga, C. J. Williams, and P. S. Julinn, Phys. Rv. A 7, B. Gao, Eur. Phys. J. D,. B. Gao, Phys. Rv. A 5, 99. B. Gao, J. Phys. B 7, 7. B. Gao, J. Phys. B,. G. F. Gribakin and V. V. Flambaum, Phys. Rv. A, B. R. Lvy and J. B. Kllr, J. Math. Phys., R. Jastrow, Phys. Rv. 9, W. C. Stwally, Phys. Rv. Ltt. 7, E. Tisinga, B. J. Vrhaar, and H. T. C. Stoof, Phys. Rv. A 7, 99. T. Köhlr, K. Góral, and T. Gasnzr, Phys. Rv. A 7,. 9 S. Simonucci, P. Piri, and G. C. Strinati, Europhys. Ltt. 9, 7 5. P. S. Julinn and B. Gao, -print physics/9 unpublishd. 5-9