C6 coefficients for interacting Rydberg atoms and alkali-metal dimers Vanessa Olaya,1 Jes´usP´erez-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 (F¨oster)processes [7]. F¨osterenergy 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 jΦ(0) i = jΨ(0)i jΨ(0)i, in an arbitrary fine structure level n2L , in the absence AB A B j where in our case jΨ(0)i ≡ jX1Σ+i jv = 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. jΨB i ≡ j(n L)jmi is a general fine structure state of an alkali-metal atom, with m being the projection of the to- tal 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 1 1 L< X X X fL L q ^ A B ^LA ^LB 1 VAB(R) = 1+L +L Qq (^rA)Q−q (^rB); 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) ^(1) (0) (0) ^(1) (0) where L< is the smallest of the integers LA and LB. The X hΨ jQq jΨ 0 i hΨ jQ−qjΨ 0 i × A A B B multipole moments Q^LX (^r ) associated with a particle q X 0 (1 + q)!(1 − q)! X = (A; B) can be written in spherical tensor form [24] qq (0) ^(1) (0) (0) ^(1) (0) # hΨA0 jQ−q0 jΨA i hΨB0 jQq0 jΨB i × 0 0 ; (6) 1=2 (1 + q )!(1 − q )! 4π X Q^LX (^r ) = q r^LX Y LX (θ ;' ); (3) q X i i q i i (0) (0) 2LX + 1 i where EA and EB are the molecular and atomic asymp- totic energies at R ! 1. All projections of the dipole (1) where qi is the i-th charge composing the X-distribution tensors Q^q are taken into account. Primed particle la- LX and Yq (θi;'i) 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 jΨA0 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 jΨB0 i ≡ j(n L )j m i up to conver- gence 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 1 ab = d! 2 2 2 2 ; (7) a + b π 0 (a + ! )(b + ! ) B.