$ C 6 $ Coefficients for Interacting Rydberg Atoms and Alkali-Metal

C6 coeﬃcients for interacting Rydberg atoms and alkali-metal dimers

Vanessa Olaya,1 JesúsPérez-R´ıos,2 and Felipe Herrera1, 3, ∗ 1Department of Physics, Universidad de Santiago de Chile, Av. Ecuador 3493, Santiago, Chile. 2Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany 3Millennium Institute for Research in Optics MIRO, Chile. (Dated: December 4, 2019) We study the van der Waals interaction between Rydberg alkali-metal atoms with fine structure 2 1 + (n Lj ; L ≤ 2) and heteronuclear alkali-metal dimers in the ground rovibrational state (X Σ ; v = 0, J = 0). We compute the associated C6 dispersion coefficients of atom-molecule pairs involving 133Cs and 85Rb atoms interacting with KRb, LiCs, LiRb, and RbCs molecules. The obtained dispersion coefficients can be accurately fitted to a state-dependent polynomial O(n7) over the range of principal quantum numbers 40 ≤ n ≤ 150. For all atom-molecule pairs considered, 2 2 6 2 Rydberg states n Sj and n Pj result in attractive 1/R potentials. In contrast, n Dj states can give rise to repulsive potentials for specific atom-molecule pairs. The interaction energy at the LeRoy −5 distance approximately scales as n for n > 40. For intermediate values of n . 40, both repulsive and attractive interaction energies in the order of 10 − 100 µK can be achieved with specific atomic and molecular species. The accuracy of the reported C6 coefficients is limited by the quality of the atomic quantum defects, with relative errors ∆C6/C6 estimated to be no greater than 1% on average.

Rydberg atoms with high principal quantum number neutral particles. For collisions between particles in their have exaggerated properties such as orbital sizes of thou- ground states, the lowest non-vanishing van der Waals co- sands of Bohr radii, long radiative lifetime exceeding mi- efficient is C6. For neutral particles in excited entrance croseconds and are extremely sensitive to static electric channels, the coefficients C3 and C5 can also contribute fields. The latter is due to their giant dipole moment and to the potential. electric polarizability. These exotic features have made In this work, we report a large set of van der Waals C6 Rydberg atoms a widely studied platform for fundamen- coefficients that determine the long-range interaction be- tal studies and applications such as quantum informa- tween selected heteronuclear alkali-metal dimers (LiCs, tion processing [1], quantum nonlinear optics [2], and RbCs, LiRb and KRb) in their electronic and rovibra- precision measurements [3]. Ultracold Rydberg atoms tional ground state (1Σ+, v = 0,J = 0) with 85Rb and 133 2 can be trapped with high densities [4], enabling stud- Cs atoms in Rydberg states n Lj with 15 ≤ n ≤ 150, ies of long-range interactions [5,6] and resonant exci- taking into account spin-orbit coupling. n is the atomic tation exchange (Föster)processes [7]. Fösterenergy principal quantum number, L is the atomic orbital angu- transfer has been proposed as a non-destructive method lar momentum, j is the total electronic angular momen- for detecting cold molecules in optical traps [8,9] that tum, ν the vibrational quantum number of the molecule can simultaneously trap Rydberg atoms and molecules. and J is the rotational angular momentum. For every These hybrid Rydberg-molecule systems have been con- atomic state considered, C5 vanishes exactly by dipole sidered for direct sympathetic cooling of cold molecules selection rules and resonant C3 contributions can be ig- into the ultracold regime through long-range collisions nored. The precision of the computed C6 coefficients is [10, 11]. There is also a growing interest in the prepa- limited by the accuracy of the semi-empirical quantum ration and study of ultralong-range Rydberg molecules defects used. [12–15], which emerges from the binding of a Rydberg We fit the computed atom-molecule C6 coefficients as alkali-metal atom and a neutral molecule in the long a function of n to a polynomial of the form range. Rydberg molecules have extraordinary properties 4 6 7 that can be controlled by driving the Rydberg electron C6 = γ0 + γ4 n + γ6 n + γ7 n , (1) and may lead to promising applications [16, 17]. which is valid in the range 40 ≤ n ≤ 150. The accuracy arXiv:1912.01568v1 [physics.atom-ph] 3 Dec 2019 For an atom-molecule system at low kinetic energies, of the fitting increases with n. We list the fitting coeffi- the relevant scattering properties at distances beyond the cients for all the atom-molecule pairs considered in Table LeRoy radius [18], are determined by the long-range in- I for Cesium andII for Rubidium. The proposed n7 scal- teraction between particles [12]. For R being the relative ing is consistent with the scaling of the static Rydberg distance of the centers of mass of the colliding partners, state polarizability [19]. The data set of computed C6 the long-range interaction potential can be written as an coefficients is provided in the Supplementary Material. P n We describe in Sec.I the theoretical and numerical expansion of the form V (R) = n Cn/R , with n ≥ 3 for methodology used to compute C6 coefficients. In Sec.II we present the dispersion coefficients for selected atom- molecule pairs, and discuss their accuracy in Sec.III. ∗Electronic address: [email protected] We conclude by discussing possible implications of our 2 results. to read [25, 26]

X Cn V (R) = , (5) AB Rn I. METHODOLOGY n

where Cn are the dispersion coefficients. Values of Cn are In this section, we briefly review the theory of long- obtained by defining the zero-th order eigenstates of the range interaction between a heteronuclear alkali-metal collision pair. In the absence of external fields, these are dimer (particle A) and an alkali-metal atom (particle B) given by product states of the form |Φ(0) i = |Ψ(0)i |Ψ(0)i, in an arbitrary fine structure level n2L , in the absence AB A B j where in our case |Ψ(0)i ≡ |X1Σ+i |v = 0,J = 0i is of external static or electromagnetic fields. Our work ex- A the absolute ground state of an alkali-metal dimer and tends the results in Refs. [20–22] to fine structure states (0) 2 with high n, as relevant for Rydberg states. |ΨB i ≡ |(n L)jmi is a general fine structure state of an alkali-metal atom, with m being the projection of the total electronic angular momentum along the quantization axis. A. Interaction potential The non-degenerate rovibrational ground state of a 1Σ molecule has a definite rotational angular momentum, Consider the charge distributions of molecule A and thus parity. Therefore, the lowest non-zero contribution atom B, separated by a distance larger than their corre- 6 to the expansion in Eq. (5) is C6/R . For molecules in sponding LeRoy radii [18]. The long-range electrostatic an excited rotational state J ≥ 1, quadrupole moments interaction between a molecule (A) and atom (B) is given can give non-vanishing C5 coefficients [20–22]. In this by the multipole expansion [23] work, we only consider the ground rotational state (i.e., C5 = 0). The second-order atom-molecule dipole-dipole interaction thus leads to a C6 coefficient of the form ∞ ∞ L< X X X fL L q ˆ A B ˆLA ˆLB 1 VAB(R) = 1+L +L Qq (ˆrA)Q−q (ˆrB), X R A B C6 = −4 LA=0 LB =0 q=−L< (0) (0) (0) (0) 0 0 (E 0 − E ) + (E 0 − E ) (2) A B A A B B " (0) ˆ(1) (0) (0) ˆ(1) (0) where L< is the smallest of the integers LA and LB. The X hΨ |Qq |Ψ 0 i hΨ |Q−q|Ψ 0 i × A A B B multipole moments QˆLX (ˆr ) associated with a particle q X 0 (1 + q)!(1 − q)! X = (A, B) can be written in spherical tensor form [24] qq (0) ˆ(1) (0) (0) ˆ(1) (0) # hΨA0 |Q−q0 |ΨA i hΨB0 |Qq0 |ΨB i × 0 0 , (6) 1/2 (1 + q )!(1 − q )! 4π X QˆLX (ˆr ) = q rˆLX Y LX (θ , ϕ ), (3) q X i i q i i (0) (0) 2LX + 1 i where EA and EB are the molecular and atomic asymptotic energies at R → ∞. All projections of the dipole (1) where qi is the i-th charge composing the X-distribution tensors Qˆq are taken into account. Primed particle la- LX and Yq (θi, ϕi) is a spherical harmonics. Expectation bels refer to intermediate states in the summation. Ev- values of the multipole moments depend on the electronic (0) ery intermediate rovibrational state |ΨA0 i in ground and structure of the particle. For a two-particle coordinate excited electronic potentials is taken into account, as ex- system in which the quantization axis is pointing from plained below. M is the projection of the total angular the center of mass of particle A to the center of mass of momentum of the molecule along the internuclear axis. particle B, the factor fLALB q in equation (2) becomes For alkali-metal atoms, we take into account all possible [22] (0) 02 0 0 0 intermediate states |ΨB0 i ≡ |(n L )j m i up to convergence of C6. Following Ref. [27], we rewrite the sum-over-states in (−1)LB (L + L )! f = A B . Eq. (6) in a more convenient form using the identities LALB q p (LA + q)!(LA − q)!(LB + q)!(LB − q)! (4) 1 2 Z ∞ ab = dω 2 2 2 2 , (7) a + b π 0 (a + ω )(b + ω ) B. Dispersion coefficients and

ˆ In the long range, the interaction Hamiltonian VAB(R) 1 2 Z ∞ ab 2a in Eq. (2) gives a perturbative correction to the asymp- = dω + , (8) a − b π (a2 + ω2)(b2 + ω2) a2 − b2 totic energies of the collision partners. This energy shift 0 is the interaction potential VAB(R), which can be evalu- which hold for positive real parameters a and b. For ated using second-order degenerate perturbation theory molecular states, we set a = Eγ0v0J 0 − EX1ΣvJ , where 3

γ0 labels an intermediate electronic state. For atomic C. Polarizability of atomic Rydberg states states, we set b = En0L0j0 − EnLj for upward transitions (En0L0j0 > EnLj), and b = EnLj − En0L0j0 for downward The dynamic atomic polarizability αmm (ω) in Eq. transitions (En0L0j0 < EnLj). Atom-molecule C6 coeﬃ- −q−q cients can thus be written as (9) can be written for a general atomic state |ki as

Z ωcut dω ˆ† ˆ ˆ ˆ† X 0 mm MM hk|dp|li hl|dp0 |ki hk|dp0 |li hl|dp|ki C6 = − K(q, q ) α 0 (iω) α 0 (iω) kk X 2π −q−q qq αpp0 (ω) = + , (11) 0 0 El − Ek − ω El − Ek + ω q,q l X MM + Θ(−∆En00L00j00 )αqq0 (∆En00L00j00 ) n00L00j00m00 where ω is the frequency of the dipolar response, dˆ is × T (n00L00j00m00)} , (9) p nLjm the p-th component of the electric dipole operator in where K(q, q0) ≡ 4/[(1 + q)!(1 − q)!(1 + q0)!(1 − q0)!] and the spherical basis, Ek is the zero-th order energy of state |ki, and the state summation runs over all other ωcut is a cut-off frequency. The arguments of integral in Eq. (9) are the dynamic atomic states |li in the spectrum, with l 6= k. For alkali- mm metal atoms, the relevant atomic states and energies are atomic polarizability component α−q−q0 (z) and the dy- MM obtained by numerically solving the radial Schrödinger namic molecular polarizability component α 0 (z), each qq equation −∇2/2 + V (r) Φ (r) = E Φ (r) with a evaluated at the imaginary frequency z = iω. The second L B nLj B pseudo potential VL(r) that describes the interaction of term in the square bracket represents downward transi- core electrons with a single valence electron at distance tions in the atom, with ∆En00L00j00 = En00L00j00 − EnLj. r from the core (origin), including spin-orbit coupling. The Heaviside function Θ(x) enforces the downward The angular part of the atomic wavefunctions correspond character of the transitions that contribute to this term. to spherical harmonics Y (θ, φ). We solve for the ra- These terms are weighted by the product of the atomic jm dial wavefunction ΦB(r) as in Ref. [28], with atomic transition dipole integrals 2 energies given by EnLj = −hcR∞/(n − δnLj) , where 00 00 00 00 2 (1) 002 00 00 00 hcR∞ = 1/2 is the Rydberg constant (in atomic units). TnLjm(n L j m ) ≡ h(n L)jm|Q−q|(n L )j m i The fine structure quantum defects δnLj used in this work 002 00 00 00 (1) 2 are given in AppendixA, in terms of the Rydberg-Ritz ×h(n L )j m |Qq0 |(n L)jmi, coefficients for 85Rb and 133Cs atoms. (10) We use the atomic energies and radial wavefunctions to MM and the molecular polarizability function αqq0 (ω) eval- construct the sum-over-states in Eq. (11), for a desired 2 uated at the frequency ω = ∆En00L00j00 /~. In this work, atomic state |(n L)jmi. For convenience, the angular we evaluate both the integral and the downward contri- parts of the dipole integrals are evaluated using angular butions to C6 up to convergence within a cut-off ωcut, as momentum algebra [24]. The non-vanishing elements of described below in more details. the polarizability can thus be written as

" mm X X 2j00+2j−m00−m+q+1 2(En00L00j00 − EnLj) 00 00 α−q−q(ω) = (−1) 2 2 (2j + 1)(2L + 1)(2j + 1)(2L + 1) (En00L00j00 − EnLj) − ω n00L00 j00m00  2  2  2 00 00 # L 1 L  L j s  j 1 j 00 00 2 ×     | hn L | er |nLi | , (12) 0 0 0  j00 L00 1  −m00 q m

where circular and curly brakets correspond to 3j and 6j D. Polarizability of alkali-metal dimers symbols [24], respectively. We use Eq. (12) to compute the non-zero components of the polarizability tensor for 2 atomic Rydberg states |(n L)jmi with n ≥ 15, ensur- The dynamic molecular polarizability needed for the ing the convergence of the sum over intermediate states evaluation of the C6 integral is also given by an expres- for each imaginary frequency iω that is relevant in the sion as in Eq. (11)), but for eigenstates |ki and ener- evaluation of the C6 integral in Eq. (9). gies Ek describing electronic, vibrational and rotational state of alkali-metal dimers. For transition frequencies 4

(El − Ek)/~ up to near infrared (∼ 1 THz), only states frame is convenient since the electronic and vibrational within the ground electronic potential need to be in- eigenfunctions are given in the body-fixed frame by most cluded explicitly in the summation. The contribution quantum chemistry packages. The non-vanishing terms of transitions between rovibrational states in different of the rovibrational polarizability (q0 = q) can thus be electronic states is taken into account separately, as ex- written as plained in what follows. 2(E 0 0 − E ) In the space-fixed frame, Eq. (11) can be given an ex- rv X 0 v J vJ αqq(ω) = (2J + 1)(2J + 1) 2 2 (Ev0J 0 − EvJ ) − ω plicit form by introducing molecular states |γ, v, JMi and v0J 0M 0 energies EγvJ , where γ labels the electronic state. Molec-  2  2 ular polarizability components for a rovibrational state J 0 1 J J 0 1 J |X1Σ, v, JMi in the ground electronic state can thus be ×     written as [29] 0 0 0 −M 0 −q M ˆ(1) 0 0 2 ×| hvJ| Q0 |v J i | , (17) " q X1ΣvJM X 2(−1) (Eγ0v0J 0 − EX1ΣvJ ) where a redundant electronic state label has been omit- αqq0 (ω) = 2 2 (1) 0 0 (Eγ0v0J 0 − E 1 ) − ω ted. The nuclear dipole integrals hvJ| Qˆ |v J i can be γ0v0J 0M 0 X ΣvJ 0 evaluated directly once the rovibrational wavefunctions 1 ˆ(1) 0 0 0 0 × hX ΣvJM|Qq |γ v J M i |vJi are known. These are obtained by solving the cor- # responding nuclear Schrödingerequation (i.e., vibrations 0 0 0 0 ˆ(1) 1 × hγ v J M |Q−q0 |X ΣvJMi . (13) plus rotations) using a discrete variable representation (DVR) as in Ref. [30], with potential energy curves (PEC) and Durham expansions for the energies EvJ given As anticipated above, for transition frequencies up to the in Ref. [31] for the alkali-metal dimers used in this work. mid-infrared, the dominant contributions to the molec- For diatomic molecules, the electronic contribution to ular polarizability come from rovibrational transitions the polarizability in Eq. (14) is fully characterized in within the ground electronic state. Therefore, Eq. (13) el the body-fixed frame by the components α00(ω) and can be rewritten as el el α1,1(ω) = α−1,−1(ω)[32], which define the parallel po- el larizability αk(ω) = α00(ω) and the perpendicular polar- X1ΣvJM rv el izability α (ω) = −αel (ω), with respect to its sym- αqq0 (ω) = αqq0 (ω) + αqq0 (ω), (14) ⊥ ±1,±1 metry axis. For frequencies up to the near infrared, the rv el dynamic electronic polarizability of alkali-metal dimers where αqq0 and αqq0 are the rovibrational and electronic polarizabilities, respectively given by do not deviate significantly from their static values αk(0) and α⊥(0). Accurate static electronic polarizabilities for several alkali-metal dimers can obtained from Ref. [33]. " q rv X 2(−1) (EX1Σv0J 0 − EX1ΣvJ ) Explicitly, the space-fixed polarizability tensor for alkali- 1 αqq0 (ω) = 2 2 metal dimers in the Σ state is given by (E 1 0 0 − E 1 ) − ω v0J 0M 0 X Σv J X ΣvJ 1 ˆ(1) 1 0 0 0  2 × hX ΣvJM|Qq |X Σv J M i 0 el X 0 J 1 J # αqq(ω) = (2J + 1)(2J + 1)   (1) 1 0 0 0 ˆ 1 0 0 0 × hX Σv J M |Q−q0 |X ΣvJMi (15) J M −M −q M  2  2  0 0 and  J 1 J J 1 J  ×   αk + 2   α⊥ . " q 0 0 0 1 −1 0 el X X 2(−1) (Eγ0v0J 0 − EX1ΣvJ ) α 0 (ω) = qq 2 2 (18) (Eγ0v0J 0 − EX1ΣvJ ) − ω γ06=X1Σ v0J 0M 0 1 ˆ(1) 0 0 0 0 For the rovibrational ground state Eq. (18) reduces to its × hX ΣvJM|Qq |γ v J M i el isotropic value αiso = (αk +2α⊥)/3 for all q-components. # 3 0 0 0 0 ˆ(1) 1 It was shown in Ref. [31], that for frequencies up to ∼ 10 × hγ v J M |Q 0 |X ΣvJMi . (16) −q THz, the isotropic electronic molecular polarizability can be accurately approximated by

For evaluating the molecular dipole integrals in Eqs. 2 2 (15) and (16), the space-fixed q-component of the el 2ωΣ dΣ 2ωΠ dΠ αiso(ω) = 2 2 + 2 2 , (19) dipole operator is written in terms of the body-fixed p- ωΣ − ω ωΠ − ω (1) components through the unitary transformation Qˆq = where the parameters ωΣ and dΣ are the effective transi- P ∗(1) ˆ(1) ∗(1) p Dqp Qp , where Dqp is an element of the Wigner tion energy and dipole moment associated with the low- rotation matrix [24]. Transforming to the molecule-fixed est Σ → Σ transition. The parameters ωΠ and dΠ are 5

II. RESULTS

Motivated by recent atom-molecule co-trapping experiments [34, 35], we focus our analysis on the long-range interaction between two sets of collision partners: (i) 133Cs Rydberg atoms interacting with LiCs and RbCs molecules; (ii) 85Rb Rydberg atoms interacting with KRb, LiRb and RbCs molecules. We use Eq. (9) to compute the C6 coeﬃcient of each atom-molecule pair considered, as a function of the principal quantum num- 2 ber n of the atomic Rydberg state n Lj. We restrict our calculations to atomic states with L ≤ 2. The total angular momentum projection along the quantization axis

Ω = m + M, (20)

is a conserved quantity for an atom-molecule collision. For molecules in the rovibrational ground state (J = 0), we thus have Ω = m. Below we present C6 coeﬃcients for each allowed value of |Ω|. C6 < 0 correspond to attractive interactions and C6 > 0 describe repulsive potentials.

A. Cesium + Molecule

133 In Fig.1, we plot the C6 coefficients for Cs Ryd- 2 berg states n Lj interacting with LiCs molecules in the rovibrational ground state, as a function of the atomic principal quantum number n, for all allowed values of Ω. For concreteness, we restrict the atomic quantum numbers to the range 15 ≤ n ≤ 150, for L ≤ 2. 2 2 2 For Cs Rydberg atoms in S1/2, P1/2 and P3/2 states, the interaction with LiCs molecules is attractive over the entire range of n considered. As discussed in more detail below, this is due to the positive character of the atomic and molecular polarizability functions at imaginary fre- FIG. 1: C6 dispersion coefficients as a function the atomic principal quantum number n for the Cs-LiCs collision pair. quencies α(iω), which determine the value of the integral 2 2 Results are shown for several atomic Rydberg states n Lj . term in Eq. (9). On the other hand, Cs atoms in D3/2 2 6 LiCs is in the electronic and rovibrational ground state. Pan- and D5/2 Rydberg states give rise to repulsive 1/R po- els show results for different total angular momentum projec- tentials. This repulsive character of the atom-molecule tions Ω = m+M: (a) |Ω| = 1/2; (b) |Ω| = 3/2; (c) |Ω| = 5/2. interaction is due to the predominantly negative atomic 7 Solid lines correspond to a fitted n scaling. polarizability function α(iω), while the molecular polarizability function remains positive. This is consistent with n2D Rydberg states having negative static polarizabili- associated with the lowest Σ → Π transition. For the ties α00(ω = 0) [36]. For both attractive and repulsive 7 alkali-metal dimers used in this work, we take the pa- interactions, the magnitude of C6 scales as ∼ n over a rameters listed in Ref. [31] to estimate the electronic wide range of n, as shown explicitly in Fig.1c. contribution to the molecular polarizability in over the The C6 coefficients for the Cs-RbCs collision pair ex- frequencies of interest. hibit the same qualitative behavior as the Cs-LiCs case, 2 Finally, we compute directly the downward transition with repulsive potentials for Dj states and attractive 2 2 terms that contribute to C6 in Eq. (9) by evaluating the interaction for Sj and Pj Rydberg states. We provide total molecular polarizability in Eq. (14) at the relevant the complete list of all C6 coefficients computed for the atomic transition frequencies, with an explicit evaluation Cs-LiCs and Cs-RbCs collision partners in the Supple- of the atomic dipole integrals in Eq. (10). mentary Material. 6

FIG. 3: Molecular polarizability function at imaginary frequency α00(iω) for selected alkali-metal dimers in the electronic and rovibrational ground state X1Σ+, ν = J = M = 0.

atoms with RbCs and LiRb molecules exhibit the same qualitative behavior as the Rb-KRb case, giving attrac- 2 2 2 tive interaction for Sj, Pj, and Dj Rydberg states. We provide the complete list of all C6 coeﬃcients computed for the Cs-LiCs and Cs-RbCs collision partners in the Supplementary Material.

C. Scaling of C6 with n

For all the atom-molecule pairs considered in this work, we ﬁt the computed C6 coeﬃcients as a function of the atomic principal quantum number n to the polynomial

4 6 7 C6 = γ0 + γ4 n + γ6 n + γ7 n . (21)

This scaling is valid in the range n ≈ 40 − 150, with a fit quality that improves with increasing n. We list the fit- FIG. 2: C6 dispersion coefficients as a function the atomic principal quantum number n for the Rb-KRb collision pair. ting coefficients for Cs-LiCs and Cs-RbCs pairs in TableI 2 Results are shown for several atomic Rydberg states n Lj . for all atomic angular momentum states considered. The KRb is in the electronic and rovibrational ground state. Pan- corresponding fitting coefficients for the collision pairs els show results for different total angular momentum projec- Rb-KRb, Rb-LiRb, and Rb-RbCs, are given in TableII. 7 tions Ω = m+M: (a) |Ω| = 1/2; (b) |Ω| = 3/2; (c) |Ω| = 5/2. The n scaling found for C6, is the same scaling of the Solid lines correspond to a fitted n7 scaling. static polarizability of Rydberg atoms [19]. This suggests that the long-range interaction potential is dominated by the giant Rydberg polarizability, as expected. B. Rubidium + Molecule

85 In Fig.2, we plot the C6 coefficients for Rb Rydberg III. DISCUSSION 2 states n Lj interacting with KRb molecules in the rovibrational ground state, as a function of the atomic princi- Since the C6 coefficients for the atom-molecule pairs pal quantum number n, for L ≤ 2. The results resemble listed in TablesI andII have yet to be experimentally 2 2 2 those of the Cs-LiCs pair with S1/2, P1/2 and P3/2 measured, we estimate their accuracy from other con- atomic Rydberg states giving rise to attractive 1/R6 po- siderations. The first question to address is the impor- 7 tentials that scale as ∼ n , as show explicitly in Fig.2b tance of the contribution to C6 of the downward transi- 2 2 for P3/2 states. In this case, Dj states do not give tion terms in Eq. (9). We find that for all the atomic 2 repulsive potentials. states n Lj considered, the downward transition terms The C6 coefficients for the interaction of Rb Rydberg represent a negligible contribution to C6 in comparison 7

Molecule L j |Ω| γ0 γ4 γ6 γ7

S 1/2 1/2 1.518[11] -1.035[5] -30.94 0.1630

1/2 1/2 2.104[12] -1.984[6] 162.1 -1.606

P 1/2 2.307[12] -2.366[6] 238.2 -2.375 3/2 3/2 2.149[12] -2.186[6] 225.5 -2.236

LiCs 1/2 -1.428[12] 1.708[6] -226.6 1.879 3/2 3/2 -5.864[11] 7.756[5] -134.4 1.090

D 1/2 -1.431[12] 1.909[6] -289.5 2.469

5/2 3/2 -1.081[12] 1.462[6] -234.1 1.973

5/2 -3.809[11] 5.716[5] -123.2 0.9800

S 1/2 1/2 -2.907[10] 2.757[4] -13.25 0.05058

1/2 1/2 -3.542[11] -1.737[5] -3.808 -0.4723

P 1/2 4.866[11] -2.753[5] 18.15 -0.7826 3/2 3/2 4.425[11] -2.588[5] 19.43 -0.7408

RbCs 1/2 -3.906[11] 2.610[5] -31.40 0.5847 3/2 3/2 -2.156[11] 1.565[5] -26.86 0.3579

D 1/2 -4.818[11] 3.501[5] -53.47 0.8316

5/2 3/2 -3.846[11] 2.821[5] -45.57 0.6657

5/2 -1.901[11] 1.458[5] -29.63 -0.3334

4 6 7 133 TABLE I: Parameters for the fitting C6 = γ0 + γ4 n + γ6 n + γ7 n , for selected atom-molecule pairs involving Cs atoms 2 in Rydberg states n Lj , interacting with LiCs and RbCs molecules in the ground electronic and rovibrational state. Ω = m is 3 the total angular momentum projection of the collision pair. C6 is in atomic units (a0). The fitting is accurate in the range n = 40 − 150. The notation A[x] means A × 10x. with the integral term that involves the atomic Rydberg transitions (dipole) in Eq. (9) can thus scale at least as polarizability function. n3, which gives a ratio of order 104 for n = 50 and 106 This conclusion is valid provided we exclude reso- for n = 100. nant contributions to the downward transition term that involve evaluating the molecular polarizability at the A. Error bounds on C values atomic transition frequency ∆Ea = 2Be, where Be is 6 the rotational constant. High-n Rydberg states with transition frequencies that are resonant with rotational After safely ignoring the atomic downward transition excitation frequencies may instead contribute to energy contributions to C6 for n > 15, we focus now on estimat- exchange processes that scale as 1/R3, which can be ing the accuracy of the frequency integral contribution avoided by careful selection of n and L quantum numbers. to Eq. (9). The rovibrational structure and electrostatic 1 After removing resonant contributions (∆Ea = 2Be) response of most alkali-metal dimers in the ground X Σ from the summation, the contribution of the integral state is well-known from precision spectroscopy experi- term to C6 in Eq. (9) was found to be at least three orders ments and accurate ab-initio studies [33, 37, 38]. There- JM of magnitude larger than the contribution of downward fore, the molecular polarizability function αqq in Eq. transition terms, for all the atom-molecule pairs studied (14) is assumed to be known with very high precision in the range n ≥ 15. One way to qualitatively understand in comparison with the atomic polarizability function this result is by comparing the n7 scaling of the static [45]. In Fig.3 we plot the molecular polarizability func- 2 JM atomic polarizability α(0) versus the n scaling of the tion evaluated at imaginary frequencies αqq (iω) up to radial dipole integrals hr2i1/2 for Rydberg states. The the microwave regime, for KRb, RbCs, LiRb and LiCs ratio between the integral (polarizability) and downward molecules. The figure shows the decreasing monotonic 8

Molecule L j |Ω| γ0 γ4 γ6 γ7

S 1/2 1/2 4.340[9] -1905 -0.5274 4.695[-4]

1/2 1/2 1.155[10] 27.48 -6.706 0.02190

P 1/2 1.429[10] -840.9 -7.419 0.02381 3/2 3/2 1.227[10] 9.555 -7.089 0.02286

KRb 1/2 4.907[9] -3284 0.1596 -1.883[-3] 3/2 3/2 6.531[9] -2307 -2.039 6.199[-3]

D 1/2 3.298[9] -2486 0.4228 -2.302[-3]

5/2 3/2 4.481[9] -2337 -0.5691 1.173[-3]

5/2 6.782[9] -2026 -2.521 8.018[-3]

S 1/2 1/2 1.381[11] -1.085[5] 6.408 -0.04418

1/2 1/2 5.353[11] -4.224[5] -13.06 0.01194

P 1/2 6.063[11] -4.839[5] -12.77 1.718[-3] 3/2 3/2 5.598[11] -4.420[5] -14.57 0.01386

LiRb 1/2 1.091[11] -1.040[5] 11.58 -0.06377 3/2 3/2 2.100[11] -2.024[5] 3.214 -0.03349

D 1/2 8.086[10] -7.309[4] 10.89 -0.05510

5/2 3/2 1.317[11] -1.237[5] 7.670 -0.04539

5/2 2.309[11] -2.225[5] 1.254 -0.02591

S 1/2 1/2 1.316[10] -3897 -2.743 -7.242[-3]

1/2 1/2 -1.576[10] 5.049[4] -39.63 0.07148

P 1/2 -1.225[10] 5.381[4] -44.40 0.07713 3/2 3/2 -1.640[10] 5.347[4] -41.94 0.07316 RbCs 1/2 2.285[10] -1.490[4] 1.265 -0.01663 3/2 3/2 9.039[9] 7469 -12.55 0.01839 D 3/2 5.808[10] -1.796[4] -1.099 -0.01126 5/2 5/2 5.978[10] -2261 -13.17 0.01667

4 6 7 85 TABLE II: Parameters for the ﬁtting C6 = γ0 + γ4 n + γ6 n + γ7 n , for selected atom-molecule pairs involving Rb atoms in 2 Rydberg states n Lj , interacting with RbCs, LiRb and KRb molecules in the ground electronic and rovibrational state. Ω = m 3 is the total angular momentum projection of the collision pair. C6 is in atomic units (a0). The ﬁtting is accurate in the range n = 40 − 150. The notation A[x] means A × 10x. character of all molecular polarizability functions stud- and can be considered to be bounded from above by its ied. As the frequency ω reaches the THz regime (not static value. shown), all molecular functions αJM (iω) tend asymptot- qq The accuracy of our computed atomic polarizability el ically to their isotropic static polarizabilities αiso [Eq. mm functions αqq (iω) is limited by the precision of the (19)], and remain constant over a large frequency range quantum defects used, which we take from spectroscopic up to several hundred THz. In other words, over a broad measurements [28]. For the atomic Rydberg states con- frequency range up to ∼ 100 THz, the contribution of the sidered, the polarizability functions obtained from Eq. molecular polarizability to C6 in Eq.9 is always positive, (12) are predominantly monotonic as a function of fre- 9

FIG. 4: Dynamic polarizability function at imaginary fre- 85 133 2 quencies (iω) for Rb (a) and Cs (b), for selected n Lj states with n = 80. FIG. 5: Static polarizability of 133Cs atoms in selected an- 3 gular momentum states (in units of a0), as a function of the principal quantum number n, for 133Cs atoms in selected 2 2 quency, although we found specific states n Lj with angular momentum states: (a) α00 component for S1/2; (b) 2 non-monotonic frequency dependence. We illustrate this α00 component for D5/2, m = 5/2; (c) α00, α11, and α−1−1 2 in Fig.4, where we show the polarizability functions components for D3/2 m = 1/2. Results for α00 from Ref. mm 2 α00 (iω) for several n Lj Rydberg levels of Rb and Cs [36] are also shown. atoms in the n = 80 manifold. Panel4a shows that the 2 projection m = 1/2 of the 80 D3/2 Rydberg state of Rb, the function α(iω) is negative in the static limit, then obtained in our calculations, and the error is approxi- has a maximum at ω ≈ 6.5 GHz, from where it decays mately given by to a positive asymptotic value up to a cutoff frequency K(q, q0) Z ωcut dω ωcut of a few THz. This value at cutoff is five orders of X mm MM ∆C6 ≈ − ∆α 0 (iω) α 0 (iω), magnitude smaller (not show) than the maximum in the 2π 2π qq −q−q q,q0 0 microwave regime. In general, for all the atomic states (22) mm considered, we find that |αqq (iω)| is always bounded mm where ∆αqq0 (iω) is the error in the atomic polarizability from above by its value at ω = 0. function evaluated at imaginary frequencies. We can as- The error of the computed C6 coefficients can thus be sume that the order of magnitude of C6 and ∆C6 is domi- estimated for n ≥ 15 as follows. Ignoring the down- nated by the 00-components of the atomic and molecular ward transition terms, and the error in the molecu- polarizability functions. If we also assume that the rela- mm mm lar polarizability function, Eq. (9) can be written as tive error ∆αqq (iω)/αqq (iω) remains constant over all C˜6 = C6 ± ∆C6, where C˜6 is the dispersion coefficient frequencies up to the cutoff ωcut, and use the fact that 10

|α(iω)| is bounded from above by its static value in the atomic and molecular cases, we can estimate an approx- imate error bound for C6 as

mm ∆C6 ∆α00 (0) . mm . (23) C6 α00 (0)

In other words, the accuracy of our C6 calculations can- not expected to be better than the accuracy of the static atomic polarizability. The static polarizabilities of several Rydberg states of 85Rb and 133Cs are known from laser spectroscopy measurements in static electric ﬁelds [39–41], and also from precision calculations using state- of-the-art ab-initio pseudo-potentials [36]. Therefore, we mm can estimate ∆α00 (0) for several atomic Rydberg states 2 n Lj, by comparing with available data. It proves convenient for comparisons to rewrite the atomic polarizability in Eq. (12) such that the Stark shift ∆E(nL)jm of the Ry- dberg state |(n2L)jmi in the presence of the electric ﬁeld E in the Z-direction can be written in the standard form [32]

1 3m2 − j(j + 1) ∆E = − α (j) + α (j) E2, (nL)jm 2 0 2 j(2j − 1) (24) where α0(j) is the scalar polarizability and α2(j) the tensor polarizability. The factor in square brackets is equals mm to α00 (0) in Eq. (12). In Fig.5, we plot the the static polarizability of 133Cs atoms in selected angular momentum states, as a function of the principal quantum number n. As a standard, we use ab-initio results from Ref. [36]. Our computed values agree with the standard with very high accuracy. For example, the average relative errors over the range 2 15 ≤ n ≤ 50 are −0.02% for S1/2 states (panel5a), 2 +0.27 % for D5/2 states with m = 5/2 (panel5b), and 2 +0.13 % for D3/2 states with m = 3/2 (panel5c). For FIG. 6: Static polarizability of 85Rb atoms in selected an- 133 3 other Rydberg states of Cs we obtain similar accura- gular momentum states (in units of a0), as a function of the 2 cies. principal quantum number n: (a) α00 component for S1/2; 85 2 In Fig.6, we plot the the static polarizability of Rb (b) α00 component for D5/2, m = 5/2; (c) α00, α11, and 2 atoms in selected angular momentum states, as a func- α−1−1 components for D3/2 m = 1/2. Results for α00 from tion of n. For 85Rb atoms, all the static polarizabil- Ref. [36] are also shown. ities we compute show excellent agreement with refer- ence values (errors smaller than 1%), except for the Ry- 2 dberg state n D3/2 with m = 1/2. For this atomic state, the atomic polarizability calculations unreliable for this our results for α00(0) have large relative errors around particular tensor component (α00) and atomic quantum n = 45, as Fig.6c shows. We can understand this numbers. Errors can be traced to the empirical quantum by noting that for j = 3/2 and m = 1/2, Eq. (24) defects used. We also show in Fig.6c that over the same 2 2 reads ∆E = −(α0 − α2)E /2. For the n D3/2 states of range of n in which α00(0) exhibits large relative errors, 85 Rb, experiments show that α0 ≈ α2 over in the range other polarizability components that do not change sign n = 30 − 60 [42], with α0 . α2 in the higher end of behave smoothly. this range. This is confirmed by the ab-initio results in Another possible source of error in our C6 calculations Ref. [36], which predict a change of sign in the static is the choice of the high frequency cutoff ωcut in the nu- polarizability at n = 46, from positive to negative. By merical integration of Eq. (9). For every atomic Rydberg separately comparing our results with experimental and states considered, we tested the numerical convergence of theoretical values for α0 and α2 (not shown), we observe the integration by increasing the value of the cutoff until that our errors are of the same magnitude as the differ- the relative change δC6/C6 was smaller than a predefined ence α0 − α2 in the range 30 < n < 60, which makes tolerance value ε. For atom-molecule pairs involving both 11

85 133 Rb and Cs atoms, the polarizability integral con- alkali-metal dimer is the LeRoy radius RLR [18], given by verges faster with increasing cutoff for intermediate and the average root-mean-square electron radius of the col- high values of n & 30, in comparison with low-n states. lision pair. For alkali-metal atoms in low-lying Rydberg The latter result in slower integral convergence. We con- states (n ≈ 15 − 20), the typical mean atomic radius can verged all our n ≈ 15 integrals at ωcut = 3 THz with be on the order of 100 − 1000 a0, where a0 is the Bohr a tolerance ε = 0.01, which ensures convergence over an radius, thus exceeding by orders the typical size of the entire range of n. electron radius of ground state molecules of only a few Bohr radii. The ratio between atomic and molecular radial distances further increases with n. For the atomic B. Effect of the molecular dipole moment states considered in this work, the van der Waals length is thus dominated by the LeRoy radius of the Rydberg 2 2 2 1/2 2 In Fig.7 we show the increase in the magnitude of C6 atom RnL ≡ hn L|r |n Li . Given the n scaling of the as the permanent dipole moment of alkali-metal dimers Rydberg radius and the n7 scaling of the atom-molecule 2 85 increases, for selected states n P1/2 of Rb. The C6 C6 coefficients, the van der Waals energy should thus ap- coefficient for the Rb-LiRb pair is larger than the corre- 6 −5 proximate scale as UvdW ≡ C6/RnL ∼ n . We find this sponding values for RbCs and KRb, which have a smaller scaling to be most accurate for n & 50. dipole moment. The same trend also holds for other From the values of C6 listed in TablesI–II, the van der 2 85 n Lj states of Rb, and for atom-molecule pairs involv- Waals energy UvdW can be estimated in absolute units. ing 133Cs atoms. For example, for the LiCs–Cs system with 133Cs in the 2 n D5/2 state and Ω = 5/2, the van der Waals potential is repulsive (Fig.1c), with a collisional barrier reaching UvdW ≈ 29 MHz for n = 20. This should be sufficient to avoid short-range collisions for atom-molecule pairs with relative kinetic energy up to 150 µK. By increasing the atomic quantum number to n = 40, the potential barrier drops to UvdW ≈ 0.41 MHz for the same collision 14 2 pair. Our results thus suggest that given a specific atom- 15 P1/2 13 2 molecule system of experimental interest, it is possible to 50 P1/2 2 find an atomic Rydberg state that gives an attractive or 12 80 P1/2 repulsive potential with a desired interaction strength.

11 We can extend the formalism in this work to also ob- | 6 tain van der Waals coeﬃcients for excited rovibrational 10 states of alkali-metal dimers. In this case, C5 coeﬃcients Log |C 9 do not vanish in general [22]. The interplay between C5 and C6 with opposite signs at long distances can possi- 8 bly lead to long-range potential wells that can support Rydberg-like metastable bound states accessible in pho- 7 toassociation spectroscopy [12, 20–22]. 6 Repulsive van der Waals interactions may be used for KRb(0.6D) RbCs(1.2D) LiRb(4.1D) sympathetic cooling of alkali-metal dimers via elastic collisions with ultracold Rydberg atoms. Since inelastic and reactive ultracold collisions [43, 44] can lead to sponta- neously emitted photons carrying energy away from a trapped system [10], it should be possible to measure the elastic-to-inelastic scattering rates and follow the ther- malization process of a co-trapped atom-molecule mix- ture. Attractive van del Waals potentials can be ex- FIG. 7: Bar plots log|C6| for n = 15, 50, 80 for atom-molecule 85 2 ploited to form long-range alkali-metal trimers via pho- pairs involving Rb atoms in the n P1/2 state with KRb, RbCs and LiRb molecules in the rovibrational ground state. toassociation [15]. The permanent dipole moment of each molecule is shown in parenthesis on the horizontal axis [33]. Acknowledgments

VO thanks support by Conicyt through grant Fonde- IV. CONCLUSION cyt Regular No. 1181743. FH is supported by Conicyt through grant REDES ETAPA INICIAL, Convocatoria The characteristic length scale for the van der Waals 2017 REDI170423, Fondecyt Regular No. 1181743, and interaction between a Rydberg atom and a ground state by Iniciativa Cient´ıﬁcaMilenio (ICM) through the Mil- 12

Lj a b c d nmin

S1/2 3.1311804(10) 0.1784(6) -1.8 − 14

P1/2 2.6548849(10) 0.2900(6) -7.9040 116.4373 11

P3/2 2.6416737(10) 0.2950(7) -0.97495 14.6001 13

D3/2 1.34809171(40) -0.60286(26) -1.50517 -2.4206 4

D5/2 1.34646572(30) -0.59600(18) -1.50517 -2.4206 4

Fj 0.016312 -0.064007 -0.36005 3.2390 4

TABLE III: Rydberg-Ritz coeﬃcients for 85Rb.

Lj a b c d nmin

S1/2 4.049352(38) 0.238(7) 0.24044 0.12177 6

P1/2 3.5916(5) 0.36(1) 0.34284 1.23986 6

P3/2 3.5590(7) 0.38(1) 0.28013 1.57631 6

D3/2 2.475454(20) 0.010(4) -0.43324 -0.96555 5

D5/2 2.466308(30) 0.015(6) -0.43674 -0.74442 5

F5/2 0.033587 -0.213732 0.70025 -3.66216 4

TABLE IV: Rydberg-Ritz coeﬃcients for 133Cs. lennium Institute for Research in Optics (MIRO).

Appendix A: Quantum defects

The quantum defects used in this work are collected from Ref. [28] in terms of the expansion

b c d e δ = a + + + + , nLj (n − a)2 (n − a)2 (n − a)2 (n − a)2 (A1) where the Rydberg-Ritz coeﬃcients (a, b, c, d) are given in TableIII for 85Rb and TableIV for 133Cs atoms, to- gether with the minimum value of n for which the expansion is estimated to be valid. 13

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