Long-range interactions between ultracold atoms and molecules including atomic spin-orbit
aa r X i v : . [ phy s i c s . c h e m - ph ] J un Long-range interactions between ultracold atoms and molecules in-cluding atomic spin-orbit
Maxence Lepers ∗ a and Olivier Dulieu a Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XXFirst published on the web Xth XXXXXXXXXX 200X
DOI: 10.1039/b000000x
We investigate theoretically the long-range electrostatic interactions between a ground-state homonuclear alkali-metal dimerand an excited alkali-metal atom taking into account its fine-structure. The interaction involves the combination of first-orderquadrupole-quadrupole and second-order dipole-dipole effects. Depending on the considered species, the atomic spin-orbit maybe comparable to the atom-molecule electrostatic energy and to the dimer rotational structure. Here we extend our generaldescription in the framework of the second-order degenerate perturbation theory [M. Lepers and O. Dulieu, Eur. Phys. J. D,2011] to various regimes induced by the magnitude of the atomic spin-orbit. A complex dynamics of the atom-molecule maytake place at large distances, which may have consequences for the search for an universal model of ultracold inelastic collisionsas proposed for instance in [Z. Idziaszek and P. S. Julienne, Phys. Rev. Lett. , 113202 (2010)].
As illustrated by the present issue of the journal ? , the fieldof cold and ultracold molecules is continuously developing inmany directions of fundamental and interdisciplinary physics.Among many developments, researchers can now create largesamples of ultracold molecules which can undergo elastic,inelastic, and reactive collisions with surrounding ultracoldatoms . The challenge is at least twofold: the achievement ofsympathetic cooling of molecules down to quantum degener-acy , and the control of elementary chemical reaction at coldand ultracold temperatures . As pointed out by Julienne (seefor instance Ref. ), low-energy atom-molecule inelastic colli-sions can be understood as resulting from the dynamics at largedistances controlled by long-ranges forces, combined to the dy-namics induced by short-range chemical forces. The resultingrate can be written as a product involving scattering probabil-ities in both domains. Due to their complex internal structure,molecules most often offer many open channels for inelastic orreactive processes with other systems so that the related proba-bility can be assumed equal to one, so that the entire collisionis uniquely controlled by well-known long-range forces. Justlike for ultracold elastic collisions determined by a single pa-rameter, namely the scattering length, this approach opens theway for elaborating universal models for inelastic and reactiveatom-molecule collisions, as recently proposed by several au-thors . In such models, it is assumed that the inelastic ratesdepend solely on the leading term C n /R n of the long-range a Laboratoire Aim´e Cotton, UPR3321 CNRS, Bˆat. 505, Univ Paris-Sud,91405 Orsay Cedex, France Fax: 33 16941 0156; Tel: 33 16935 2013; E-mail:[email protected] electrostatic interaction between the colliding partners.Most of the cases investigated in the papers above involveatoms and molecules in their electronic ground state. In a se-ries of recent papers we studied the long-range interaction be-tween an alkali-metal dimer in its ground state with an excitedalkali-metal atom, in the perspective of modeling their associ-ation into an excited trimer induced by a properly chosen laser(photoassociation, or PA) at ultracold energies. In Refs. (hereafter referred to as Papers I and II, respectively) wehave characterized the first-order quadrupole-quadrupole andsecond-order dipole-dipole interactions – varying as C /R and C /R (where R is the distance between the colliding part-ners) and associated with the operators ˆ V qq and ˆ V (2) dd , respec-tively – between a homonuclear alkali-metal dimer in the low-est level v d = 0 of its electronic ground state X Σ + g , and analkali-metal atom in the first excited state P . We have locateda range of atom-dimer distances with an upper bound R ∗ p in-side which both kinds of interaction compete with the rotationalenergy of the dimer. Potential energy curves asymptoticallyconnected to different dimer rotational levels N are thus cou-pled, inducing complex patterns further characterized in Ref. (hereafter referred to as Paper III). The lower bound of this re-gion is limited by the so-called LeRoy radius below whichexchange interaction takes place.In this series of three papers we have applied our formalismto the interaction between a Cs dimer and a Cs atom but it canbe applied to all alkali-metal combinations. Considering otherspecies will modify the limits of the (cid:2) R LR ; R ∗ p (cid:3) region but notthe overall aspect of the potential energy curves. This is dueto the moderate variation of the electrostatic and rotational en-ergies from Li to Cs (see Table 1). In contrast, the spin-orbit1 GENERALFORMALISMsplitting of the first excited atomic state n P varies dramati-cally from Li to Cs. Lithium is actually the only species whosefine-structure splitting is comparable to the rotational and elec-trostatic energies. For all the other species, the spin-orbit split-ting is much higher. Following , we define two distinct cou-pling cases in analogy to Hund’s cases: • From Na to Cs, the fine-structure splitting is so large thatthe two fine-structure components are not coupled by theelectrostatic interaction. The spin-orbit Hamiltonian ˆ H SO is part of the zeroth-order energy E p, = E X,v d =0 + E nP + D ˆ H SO E of a given state p , where E X,v d =0 is the en-ergy of the electronic and vibrational dimer ground state,and E nP the energy of the atomic n P state without finestructure. The electrostatic interaction is then obtained bydiagonalizing at each R the Hamiltonian ˆ W ( R ) = B ˆ ~N + ˆ V qq + ˆ V (2) dd , (1)with B the dimer rotational constant in the electronic andvibrational ground state. This situation, denoted as case“1C” in Ref. , will be illustrated with cesium in the P manifold. • For Li in the P manifold, the two fine-structure com-ponents are coupled by the electrostatic interaction, andthe spin-orbit interaction is included in the perturbationHamiltonian ˆ W ( R ) = B ˆ ~N + ˆ H SO + ˆ V qq + ˆ V (2) dd . (2)The zeroth-order energy of state p reads E p, = E X,v d =0 + E nP . Due to the very weak variation of theelectrostatic properties between the P / and P / states, we will consider that those properties do not dependon the fine-structure level. This situation corresponds tocase “2A” in Ref. . Table 1
Some atomic properties relevant for the present study: themean square radius (cid:10) r np (cid:11) on the atomic lowest np orbital, itsspin-orbit splitting ∆ E fs , the rotational constant B of the lowestvibrational level of the corresponding dimer, and the only nonzerotensor component q of the dimer quadrupole moment at itsequilibrium distance (see Paper I)Li K Cs (cid:10) r np (cid:11) (a.u.) 27.1 - 62.7 ∆ E fs (cm − ) 0.335 57.7 554Li K Cs q (a.u.) 10.5 15.7 18.6 B (cm − ) 0.673 0.0567 0.0117 In Section 2 we briefly recall the general formalism that weused in our previous work, emphasizing on the modifications induced by the presence of the atomic spin-orbit. We derive inSection 3 the expressions for the first-order and second-orderlong-range interaction of the atom-molecule system, referringto the two cases “1C” and “2A” above. Then we describe ourresults for the Cs ∗ +Cs case (Section 4) and for the Li ∗ +A case, A being an alkali-metal species (Section 5). Finally wediscuss the implications of our results in the perspective of theuniversal model proposed in Ref. (Section 6). We start from the general form of the electrostatic energy be-tween two interacting charge distributions A and B , whose cen-ters of mass are separated by the distance R , ˆ V el ( R ) = + ∞ X L A ,L B =0 L < X M = − L < R L A + L B × f L A L B M ˆ Q ML A (ˆ r A ) ˆ Q − ML B (ˆ r B ) , (3)where L < is the minimum of L A and L B . Eq. (3) is the well-known expansion on the electric multipoles of A and B . Eachmultipole of order L X (with X = A, B ) is associated with thetensor operator ˆ Q ML X (ˆ r X ) , which can be expressed in a coordi-nate system whose origin is the center of mass of X ˆ Q ML X (ˆ r X ) = r π L X + 1 X i ∈ X q i ˆ r L X i Y ML X (cid:16) ˆ θ i , ˆ φ i (cid:17) , (4)where q i is the value of each charge i composing X , and Y ML X are the usual spherical harmonics. In Eq. (3), the assumptionhas be made that the quantization axis is the one pointing fromthe center of mass of A to the center of mass of B , hence thefactor f L A L B M reads f L A L B M = ( − L B ( L A + L B )! p ( L A + M )! ( L A − M )! × p ( L B + M )! ( L B − M )! . (5)In our particular case (Fig. 1), the system A is the dimer, thesystem B is the atom, and the quantization axis Z joins thedimer center of mass and the atom.In what follows, we want to characterize the first-orderquadrupole-quadrupole interaction (defined by L A = L B = 2 in Eq. (3)) and the second-order dipole-dipole interaction, (de-fined by L A = L B = 1 ), in the jj coupling case. Namely, theatomic state is characterized by the quantum numbers associ-ated with the outermost electron: n the radial quantum number, ℓ the orbital angular momentum, j the total angular momen-tum, and ω its projection on the Z axis, joining the center ofmass of the dimer and the atom. We shall also use s the spin2 FIRST-ORDERANDSECOND-ORDERLONG-RANGEINTERACTIONS Cs CsCs
Z XZ A X A δ ) Figure 1 (Color online) The two coordinate systems, X A Y A Z A (D-CS) and XY Z (T-CS) defined for the dimer and for the trimer,respectively. The Z A axis is along the dimer axis, while Z is orientedfrom the center of mass of the dimer towards the atom B . The Y and Y A axes coincide and point into the plane of the figure. Thesubsystem A in this figure is the Cs molecule, the subsystem B isthe Cs atom. The T-CS is related to the laboratory coordinate system( ˜ x ˜ y ˜ z ) by the usual Euler angles ( α, β, γ ), not represented here. of the electron ( s = 1 / ), its projection σ on the Z axis, aswell as λ , the projection of ℓ . Using these notations, the atomicspin-orbit Hamiltonian reads ˆ H SO = A ˆ ~ℓ. ˆ ~s , (6)where A = 2∆ E fs / , and its matrix elements are diagonal D ˆ H SO E = A j ( j + 1) − ℓ ( ℓ + 1) − s ( s + 1)) . (7)The diatomic molecule is in its ground electronic state (cid:12)(cid:12) X Σ + g (cid:11) and vibrational level | v d = 0 i . The quantum numbers associ-ated with the rotation of the dimer nuclei are the angular mo-mentum N and its projection m on the Z axis. The total angularmomentum J , associated with the mutual atom-dimer rotation,will be considered in a future work. On the contrary, the pro-jection Ω = m + ω (8)of ~J on the Z axis, will be extensively used, as it is a conservedquantity. In comparison to Paper III, the atomic part of theelectrostatic interaction is the only one which is modified. Thatis why we will focus on it in what follows.The atomic multipole moments have now to be expressed inthe jj coupling case. As the electric-multipole-moment opera-tors ˆ Q ML act on orbital part of the atomic state, we first decom- pose the atomicstate, labeled | jω i on the corresponding decou-pled basis {| λσ i} | jω i = X λσ C jωℓλsσ | λσ i , (9)where C cγaαbβ is a Clebsch-Gordan coefficient. Combining thematrix element of ˆ Q ML in the decoupled basis, h n ℓ λ σ | ˆ Q ML | n ℓ λ σ i = − δ σ σ r ℓ + 12 ℓ + 1 (cid:10) ˆ r L (cid:11) C ℓ ℓ L C ℓ λ ℓ λ LM , (10)where (cid:10) ˆ r L (cid:11) ≡ D ˆ r Ln ℓ j n ℓ j E is the matrix element associ-ated with the operator ˆ r L for the outermost electron of the atom,and the formula X αβδ C cγaαbβ C eǫdδbβ C dδaαfφ = ( − b + c + d + f p (2 c + 1) (2 d + 1) × (cid:26) a b ce f d (cid:27) C eǫcγfφ , (11)where (cid:26) a b cd e f (cid:27) is a Wigner 6-j symbol, and using thefact that ℓ + L has the same parity as ℓ , we get to the expres-sion h n ℓ j ω | ˆ Q ML | n ℓ j ω i = ( − ℓ + s + j +1 p (2 ℓ + 1) (2 j + 1) (cid:10) ˆ r L (cid:11) × C ℓ ℓ L (cid:26) ℓ s j j L ℓ (cid:27) C j ω j ω LM . (12)In case “2A” defined in the previous section, the radial matrixelement ˆ r L is independent from j and j . The matrix element corresponding to the quadrupolar interac-tion ˆ V qq is obtained by combining Eq. (12) with L = 2 and ℓ = ℓ for the atomic part, and Eq. (12) of Paper III for thedimer part, which yields D N m j ω (cid:12)(cid:12)(cid:12) ˆ V qq (cid:12)(cid:12)(cid:12) N m j ω E = 24 ( − ℓ + j +3 / r N + 12 N + 1 p (2 ℓ + 1) (2 j + 1) × C N N C ℓ ℓ (cid:26) ℓ j j ℓ (cid:27) q D r nℓj nℓj E R × X M = − C N m N m M C j ω j ω − M (2 + M )! (2 − M )! , (13)3 FIRST-ORDERANDSECOND-ORDERLONG-RANGEINTERACTIONSwhere q is the tensor component of the dimer quadrupole mo-ment along its internuclear axis Z A . The angular factors ofEq. (13) impose strong selection rules: (i) m + ω = m + ω ,which means that the quantum number Ω (see Eq. (8)) is con-served; (ii) N = N , N ± ; and (iii) j , j and L = 2 mustsatisfy the triangle rule. It is important to remark that the latterselection rule is not satisfied for j = j = 1 / , which is ofstrong importance for the Cs + Cs interaction (case “1C”). Ifthe cesium atom is in the P / fine-structure level, it only in-teracts with Cs through the second-order dipolar interaction,which is certainly not favorable for the existence of long-rangevibrational levels below the P / dissociation limit of thetrimer.The second-order dipolar interaction is associated with theoperator ˆ V (2) dd , whose matrix elements can be written as func-tions of the dynamical polarizabilities of the two fragments atimaginary frequencies (see Eq. (12) of Paper II) D N m j ω (cid:12)(cid:12)(cid:12) ˆ V (2) dd (cid:12)(cid:12)(cid:12) N m j ω E = − X M = − X M ′ = − M )! (1 − M )! (1 + M ′ )! (1 − M ′ )! × (cid:20) π Z + ∞ dωα m m MM ′ ( iω ) α ω ω − M − M ′ ( iω )+ X b Θ( − ∆ E b ) α m m MM ′ ( ω = ∆ E b ) × D nℓj ω (cid:12)(cid:12)(cid:12) ˆ Q − M (cid:12)(cid:12)(cid:12) Φ b E D Φ b (cid:12)(cid:12)(cid:12) ˆ Q M ′ (cid:12)(cid:12)(cid:12) nℓj ω Ei , (14)where Θ( x ) is Heaviside function, and the letter b ≡ n ′ ℓ ′ j ′ stands for all the quantum states of the atom accessible throughdipolar transitions. The last two lines of Eq. (14) are contri-butions due to the downward atomic transitions. They dependon the dimer dynamical polarizability α m m MM ′ at the (real) fre-quencies of the atomic transitions, which is given in Eq. (20)of Paper III.In Eq. (14), we have introduced the atomic dynamical polar-izability in the coupled basis α ω ω − M − M ′ ( z )= 2 ( − M X b ( E b − E nℓj )( E b − E nℓj ) − z × D nℓj ω (cid:12)(cid:12)(cid:12) ˆ Q − M (cid:12)(cid:12)(cid:12) Φ b E D Φ b (cid:12)(cid:12)(cid:12) ˆ Q M ′ (cid:12)(cid:12)(cid:12) nℓj ω E , (15)where z can be either real or complex. The first line of Eq. (15)depends on j , but not on j : the subtle reason for this will beexplained in the what follows. By applying Eq. (12) to L = 1 ,and putting the primes in upper indices of the Clebsch-Gordancoefficients using the identity C cγaαbβ = ( − a +2 b − c − β r c + 12 a + 1 C aαcγb − β , (16) we get to the final expression α ω ω − M − M ′ ( z )= 2 X n ′ ℓ ′ j ′ ω ′ ( E n ′ ℓ ′ j ′ − E nℓj )( E n ′ ℓ ′ j ′ − E nℓj ) − z × ( − j + j (2 ℓ + 1) p (2 j + 1) (2 j + 1) × h ˆ r nℓj n ′ ℓ ′ j ′ i h ˆ r nℓj n ′ ℓ ′ j ′ i (cid:16) C ℓ ′ ℓ (cid:17) × (cid:26) ℓ j j ′ ℓ ′ (cid:27) (cid:26) ℓ j j ′ ℓ ′ (cid:27) × C j ′ ω ′ j ω M C j ′ ω ′ j ω M ′ . (17)The key point is now to connect Eq. (17) to the isotropic po-larizability of the atom, which will depend on the consideredcases “1C” and “2A”.In the case “1C” illustrated by cesium, the isotropic polar-izabilities are different for the two fine-structure levels P / and P / . Moreover, as the subspaces associated to those twolevels are fully decoupled, the matrix element of ˆ V (2) dd are zeroif j = j . As pointed out in Paper II, the isotropic polarizabil-ity ¯ α nℓj corresponding to the level nℓ j with sublevels ω (not tobe mixed up with the frequency) is obtained from Eq. (17) bycarrying out a sum over all values of ω ′ and an average over ω ¯ α nℓj ( z ) = α c + 23 X n ′ ℓ ′ j ′ ( E n ′ ℓ ′ j ′ − E nℓj )( E n ′ ℓ ′ j ′ − E nℓj ) − z × (2 ℓ + 1) (2 j ′ + 1) h ˆ r nℓjn ′ ℓ ′ j ′ i × (cid:16) C ℓ ′ ℓ (cid:17) (cid:26) ℓ jj ′ ℓ ′ (cid:27) . (18)Here α c is the polarizability of the atomic core which is thesame as in Paper II, and we used the identity P αγ (cid:16) C cγaαbβ (cid:17) = c +12 b +1 . The situation is thus similar to the one of Paper II: be-cause of angular factors, the quantity α ω ω − M − M ′ cannot be re-lated to the sole polarizability of the nℓ j atomic level, andcontributions from different j → j ′ transitions must be sep-arated. For example, the contributions j = 3 / → j ′ = 1 / , j = 3 / → j ′ = 3 / and j = 3 / → j ′ = 5 / for cesium inthe P / state must be distinguished. Following Eq. (24) ofPaper II, we express the isotropic polarizability as ¯ α nℓj = X n ′ ℓ ′ j ′ α nℓjn ′ ℓ ′ j ′ + α c , (19)where the state-to-state polarizability α nℓjn ′ ℓ ′ j ′ is related to4 CASE“1C”: CS ∗ +CS α ω ω − M − M ′ by α ω ω − M − M ′ = 3 δ j j j +1 X j ′ = j − j + 12 j ′ + 1 × X ℓ ′ =( ℓ − ,ℓ +1) X n ′ α nℓj n ′ ℓ ′ j ′ × + j ′ X ω ′ = − j ′ C j ′ ω ′ j ω M C j ′ ω ′ j ω M ′ . (20)It is important to keep in mind that the quantum numbers ℓ ′ and j ′ are related to each other by the condition ℓ ′ − s ≤ j ′ ≤ ℓ ′ + s . So, the possible transitions are: n P / → n ′ S / and n P / → n ′ D / , for an alkali-metal atom in a n P / state,and n P / → n ′ S / , n P / → n ′ D / and n P / → n ′ D / , for an alkali-metal atom in a n P / state.The case ”1A” corresponds to a lithium atom in the P state. As we have not calculated the dynamical polarizabilityof lithium, we will only present the formalism and no numeri-cal results. We assume that the different fine-structure levels ofthe nℓ manifold can be coupled by electrostatic interaction, butthat the energies of the states nℓ j and n ′ ℓ ′ j ′ , and the transitiondipole moments from state nℓ j to state n ′ ℓ ′ j ′ do not depend on j and j ′ . It is thus more relevant to express the isotropic polar-izability ¯ α nℓ in the Russel-Sanders coupling scheme, which isgiven in Paper II, Eqs. (22) znd (24), ¯ α nℓ ( z ) = α c + 23 X n ′ ℓ ′ ( E n ′ ℓ ′ − E nℓ )( E n ′ ℓ ′ − E nℓ ) − z × h ˆ r nℓn ′ ℓ ′ i (cid:16) C ℓ ′ ℓ (cid:17) . (21)In order to calculate α ω ω MM ′ ( z ) , it is, like in Paper II, necessaryto separate the different ℓ → ℓ ′ transitions (here P → S and P → D ). By introducing the state-to-state polarizability, α nℓn ′ ℓ ′ ( z ) = 23 ( E n ′ ℓ ′ − E nℓ )( E n ′ ℓ ′ − E nℓ ) − z × h ˆ r nℓn ′ ℓ ′ i (cid:16) C ℓ ′ ℓ (cid:17) , (22)we finally get to the relation α ω ω − M − M ′ = 3 X ℓ ′ =( ℓ − ,ℓ +1) X n ′ α nℓn ′ ℓ ′ × j < +1 X j ′ = j > − (2 ℓ + 1) p (2 j + 1) (2 j + 1) × (cid:26) ℓ j j ′ ℓ ′ (cid:27) (cid:26) ℓ j j ′ ℓ ′ (cid:27) × + j ′ X ω ′ = − j ′ C j ′ ω ′ j ω M C j ′ ω ′ j ω M ′ , (23) −0.1 0 0.1 0.2 0.3 100 E ne r g y ( c m − )
40 150(a) N = 0 N = 2 N = 4 0 0.2 0.4 100 E ne r g y ( c m − ) Atom−dimer distance R (a.u.)40 150(b) N = 1 N = 3 N = 5 Figure 2
Long-range potential energy curves between a ground-stateCs and an excited Cs( P j =1 / ) atom, as functions of their mutualseparation R , for | Ω | = 1 / and for: (a) the even values of N and (b)the odd values of N . The origin of energies is taken at theCs ( X, v d = 0 , N = 0)+ Cs (6 P / ) dissociation limit. with j > the maximum of j and j , and j < their minimum. ∗ +Cs In this case the two atomic fine-structure subspaces are fullydecoupled and treated separately. Figs. 4 and 5 present the po-tential energy curves characterizing the interaction between aCs dimer in the lowest rotational levels N and a Cs atom inthe P / level, obtained after diagonalization of ˆ W ( R ) (seeEq. (1)). The curves are sorted by values of | Ω | (here equal to1/2 and 3/2) and parity of N . Table 2 presents the correspond-ing C coefficients for N = 0 and 1, all the C coefficientsbeing zero. The method used to calculate the polarizabilities ofCs and Cs are the same as those described in Paper II.The potential energy curves shown on Figs. 2 and 3 havevery similar features. Most dimer rotational levels split intotwo curves as the two fragments get closer to each other. Typ-ically, for R > a.u., one curve, characterized by a negative C coefficient, is attractive, while the other, characterized bya positive C coefficient, is repulsive. The positive C coeffi-cients are due to the highly-negative parallel polarizability ofthe dimer at the atomic P / → S transition frequency(-5160 a.u.), compared to -3037 a.u. at the P / → S tran-sition frequency. For lower atom-dimer distances, the repulsivecurves turn attractive, due to the coupling with the attractivecurve connected to the higher dissociation limit. The result-ing long-range potential barriers, whose height can go up to0.1 cm − , could prevent collisions in the ultra-cold regime.5 CASE”1A”: LI ∗ +A −0.1 0 0.1 0.2 0.3 100 E ne r g y ( c m − )
40 150(a) N = 2 N = 4−0.1 0 0.1 0.2 0.3 0.4 100 E ne r g y ( c m − ) Atom−dimer distance R (a.u.)40 150(b) N = 1 N = 3 N = 5 Figure 3
Same as Fig. 2, for | Ω | = 3 / , j = 1 / . N | Ω | C (a.u.)0 / -110221 / -199521 / / -19952 Table 2
The C coefficients of theCs ( X Σ + g , v d = 0 , N ) +Cs( P / ) long-range interactioncalculated for N = 0 and 1. In analogy to a diatomic molecule, thestates are sorted by absolute values of the total angular momentumprojection | Ω | on the Z axis. All the C coefficients are zero.
554 554.1 554.2 554.3 100 E ne r g y ( c m − )
45 150(a) N = 0 N = 2 N = 4 554 554.2 554.4 100 E ne r g y ( c m − ) Atom−dimer distance R (a.u.)45 150(b) N = 1 N = 3 N = 5 Figure 4
Same as Fig. 2, for | Ω | = 1 / , j = 3 / . For Cs in the P / level, the potential energy curves(Figs. 4 and 5) look very different from the previous ones. Likethose obtained in Paper III without spin-orbit interaction, mostof them are attractive, and give birth to complex couplings for R < a.u.. Those couplings can consist of the two-curvecrossings, like the ones described in Paper III, but also of three-curve crossings. The most striking example of such a crossingis pointed out by an arrow on Fig. 5(b), but this feature is quitegeneral. They will be described in more details in the next para-graph, with Li + Li as a direct comparison with the curves inthe underlying spinless symmetries will be possible.The corresponding C and C coefficients are presented inTable 3, for N = 0 and 1. The square radius of the p / orbitalof the cesium was calculated in our group with a Dirac-Fockmethod, which gives D r p / E = 78 . a.u.. The quadrupolemoment of Cs is the same as in Paper I ( q = 18 . a.u.). The C and C coefficients are of the same order of magnitude asthose obtained in the LS coupling case (see Papers I and II).The most attractive curve, found for N = 1 and | Ω | = 1 / ,has inherited its behavior from the most attractive Σ + curvediscussed in Paper I and II, although its C coefficient is lessattractive (-1131 a.u. with respect to -1674 a.u.). The associ-ated C coefficient is positive, but much lower the one in LScoupling case (2500 a.u. compared to 51249 a.u.). By contrast,all the other C coefficients are negative. ∗ +A Now the two atomic fine-structure levels are so close to eachother that they can be coupled by the electrostatic interaction.This is illustrated by a lithium atom in the P state and both6 CASE”1A”: LI ∗ +A
554 554.1 554.2 554.3 100 E ne r g y ( c m − )
45 150(a) N = 0 N = 2 N = 4 554 554.2 554.4 100 E ne r g y ( c m − ) Atom−dimer distance R (a.u.)45 150(b) N = 1 N = 3 N = 5 Figure 5
Same as Fig. 2, for | Ω | = 3 / , j = 3 / . On panel (b), thearrow points out the three-curve crossings discussed in the text. N | Ω | C ( D r p / E q ) C (a.u.) C (a.u.)0 / / / − (cid:0) √ (cid:1) -1131 25001 / / (cid:0) √ − (cid:1)
432 -727121 / −√ -73 -397591 / √ / - -349 -23944 Table 3
The C and C coefficients of theCs ( X Σ + g , v d = 0 , N ) +Cs( P / ) long-range interactioncalculated for N = 0 and 1. In analogy to a diatomic molecule, thestates are sorted by absolute values of the total angular momentumprojection | Ω | on the Z axis. As well as in Paper I, the values of C are given in scaled units of D r p / E q . −5 0 5 10 15 20 100 E ne r g y ( c m − )
20 50 (a) N = 0, j = 1/2 N = 0, j = 3/2 N = 2, j = 1/2 N = 2, j = 3/2 N = 4, j = 1/2 N = 4, j = 3/2−5 0 5 10 15 100 E ne r g y ( c m − ) Atom−dimer distance R (a.u.)20 50 (b) N = 1, j = 1/2 N = 1, j = 3/2 N = 3, j = 1/2 N = 3, j = 3/2 Figure 6 (Color online) Long-range potential energy curves (withoutdipolar interaction) between a ground-state Li and an excitedLi( P ) atom, as functions of their mutual separation R , for | Ω | = 1 / and for: (a) the even values of N and (b) the odd values of N . On panel (a), potential curves calculated by neglecting the finestructure of Li( P ) are also displayed: in the Σ + (crosses), Σ − (fullsquares) and Π symmetries (empty squares). The arrow points out thethree-curve crossings discussed in the text. by Li and K dimers. The results presented in this sectionare obtained by diagonalizing ˆ W (see Eq. (2)), but without thesecond-order dipole-dipole interaction. The square radius ofthe p orbital of the lithium atom was calculated in our groupwith an Hartree-Fock method, which gives (cid:10) r p (cid:11) = 25 . a.u..The quadrupole moments of Li and K are equal to 10.73 and15.69 a.u., respectively . First, the potential energy curves be-tween Li and Li are displayed on Fig. 6 for | Ω | = 1 / . Onpanel (a), for the even values of the rotational quantum number N , the curves of symmetries Σ ± and Π , calculated without theatomic fine structure, are also plotted. As the splitting betweentwo subsequent rotational levels of Li , at least B ≈ cm − ,is much higher that the fine-structure splitting (0.3 cm − ), thedescription in LS coupling case seems adequate, especially inthe coupling region, for < R < a.u., where each curvebelonging to the | Ω | = 1 / can be clearly identified with acurve belonging either to the Σ ± or to the Π symmetry. Wecan also see three-curve crossings (see arrow on panel (a)), likein the case Cs + Cs. This crossing concerns one Σ + and two Π states. Outside the crossing, one of the Σ + states and the Π state are almost degenerate, whereas, at the crossing, the two Π states avoid each other. The mechanism is the same for cesium,even if the curves resulting from the calculation in the jj cou-pling case do not have such strong components coming from one given LS-coupling-case symmetry.At last, we consider the interaction between Li( P ) and7 DISCUSSION −1−0.5 0 0.5 1 1.5 2 100 E ne r g y ( c m − ) Atom−dimer distance R (a.u.)30 50 N = 0, j = 1/2 N = 0, j = 3/2 N = 2, j = 1/2 N = 2, j = 3/2 N = 4, j = 1/2 N = 4, j = 3/2 Figure 7
Long-range potential energy curves (without dipolarinteraction) between a ground-state K and an excited Li( P ) atom,as functions of their mutual separation R , for | Ω | = 1 / and the evenvalues of N . K for which potential energy curves are plotted on Fig. 7 for | Ω | = 1 / . This situation is particularly interesting since twoasymptotic channels namely N = 0 , j = 3 / and N = 2 , j =1 / respectively at 0.335 cm − and 0.340 cm − , are almostdegenerate. The two channels can thus be coupled at large dis-tances: for instance at R = 120 a.u., the three curves con-nected to those two limits and taken with increasing energiesdecompose as (0.2;0.8), (1;0), and (0.8;0.2) on the channels( N = 0 , j = 3 / ; N = 2 , j = 1 / ). Moreover, the couplingregion for < R < a.u. is characterized by numerousavoided crossings. The results displayed in the preceding section clearly show thata complicated dynamics is likely to occur at large distances,namely well beyond the range of chemical forces, during thecold collision between an excited atom and a ground statemolecule. This situation represent a new possibility to inves-tigate the range of validity of the universal model for inelas-tic collisions at ultracold energies . Based on MultichannelQuantum Defect Theory (MQDT), this model considers thatthe rate for an ultracold inelastic collision proceeds first fromthe long-range interactions which controls the probability forthe system to reach the range of chemical forces at short dis-tances. A long-range potential with a dominant term C n /R n ischaracterized by an interaction length ¯ a ( n ) ¯ a ( n ) = cos (cid:18) πn − (cid:19) (cid:18) µC n ¯ h ( n − (cid:19) n − Γ (cid:16) n − n − (cid:17) Γ (cid:16) n − n − (cid:17) (24)where µ is the reduced mass of the system, and Γ theusual Gamma function. A characteristic energy ¯ E ( n ) = ¯ h / (2 µ ¯ a ( n ) ) is associated to this length. For collision en-ergies E = ¯ h k / µ dominated by s -wave scattering, i.e. k ¯ a ( n ) << , the inelastic rate is independent of k and writes K in ( n ) = 2 h ¯ a ( n ) /µ . In the present case, the dominantterm for the interaction between a ground state Cs moleculeand a Cs( P / ) atom corresponds to n = 5 , and for a typicalvalue C = − a.u. (see Table 3), one obtains ¯ a (5) =0 . µC / ¯ h ) / ≈ a.u.. For a Cs( P / ) atom( n = 6 ) one has ¯ a (6) = 0 . µC / ¯ h ) / ≈ a.u..In both case, the characteristic length is larger than the inter-mediate range of distances identified above where spin-orbitcoupling, rotational energy and electrostatic energy all competetogether. Therefore such an ultracold inelastic collision will beactually defined by three domains: (i) the large distances wherethe sole C n /R n term controls the scattering, (ii) the intermedi-ate distances above with a specific treatment of the dynamics,(iii) the short distances where one can reasonably assume thatso many channels are open that the reaction probability is equalto unity. The dynamics in the intermediate range could well betreated within a coupled-channel framework in one dimensionas implemented in Ref. for ultracold ground state Rb (or Cs)atoms and RbCs molecules. Thus it is most likely that the va-lidity of the universal model of Refs. will be limited in thepresent case, just like it has been argued for ultracold collisionsbetween ground state KRb molecules in their lowest rovibra-tional levels .Another remarkable expected feature of such collisions willbe the existence of numerous Feshbach resonances inducedby bound levels of the trimer close to one dissociation limitCs ∗ +Cs ( N ) interacting with the continuum related to a disso-ciation limit with a smaller value of N . Such resonances mayenhance the photoassociation probability and the decay downto stable molecules through the R -transfer of the probabilitydensity of the system to smaller distances, as it is well knownfor atom-atom photoassociation .Finally, it is also worthwhile to mention that the energy spac-ing between molecular rotational levels is -at least for Cs - ofthe same order of magnitude than the hyperfine splitting of theCs ground state, which could again induce a complex resonantdynamics at large distances, which will the subject of a furtherstudy. Acknowledgements
Stimulating discussions with N. Bouloufa, V. Kokoouline,and R. Vexiau, are gratefully acknowledged. M.L. acknowl-edges the support of Triangle de la Physique in the frameworkof the contract QCCM-2008-007T Quantum Control of ColdMolecules.8EFERENCES REFERENCES
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