Asymptotic approximations to the energy of dispersion interaction between rubidium atoms in Rydberg states

Irreducible components determining the dependence of the van der Waals coefficient C 6 ( nlJM ) on the angle θ between the interatomic and the quantisation axes of two Rb atoms in their identical Rydberg states ∣ nlJM 〉 are evaluated with account of the most contributing terms of the spectral resolution for the bi-atomic Green’s function. Asymptotic polynomials in powers of the Rydberg-state principal quantum number n are derived for the C6 irreducible components. Numerical values of the polynomial coefficients are determined for Rb atoms in their n 2 S 1 / 2 , n 2 P 1 / 2,3 / 2 , n 2 D 3 / 2,5 / 2 and n 2 F 5 / 2,7 / 2 Rydberg states of arbitrary high n. The transformation of the van-der-Waals interaction law − C 6 / R 6 into the dipole–dipole law C 3 / R 3 in the case of close two-atomic states (the Förster resonance) is considered. Numerical values, the dependences on the magnetic quantum numbers M and on the angle θ of the constant C3 are determined together with the ranges of interatomic distances R, where the interaction law R − 6 transforms into the law of R − 3 .


Introduction
Highly excited atoms in their Rydberg states attract much attention as worthwhile candidates for designing highperformance quantum processors of extremely fast logic operations [1][2][3]. In absence of external fields, the energy of interaction between two neutral atoms A and B separated by a distance R, significantly exceeding the total linear dimension of interacting atoms r r R 2 LR A 2 1 2 B 2 1 2 = á ñ + á ñ ( ) (the Le Roy radius [4]), follows the van der Waals law E C R vdW 6 6 D = -. The constant coefficient C 6 may be determined from the second-order perturbation theory for the dipole-dipole interaction of atoms. If spin effects may be neglected, the van-der-Waals constant C nS 6 ( )is a scalar number dependent on the energies and matrix elements of dipole transitions between S-and P-states (see, e.g. [5]): where the summation involves the complete set of two-atomic states n n P P The distribution of the P-state energies E n P 1 2 ( ) , both above and below the energy of the nS-state, makes calculation of the two-fold sum rather complicated. The terms of the closest to n values of n 1 and n 2 may have the largest magnitudes, but the alternating signs. In the sum of the opposite-sign terms several digit numbers may be cancelled out, thus reducing the precision of calculations. For high principal quantum number n the specific 'resonance' effects appear when one or a few of the denominators of fractions in (1) achieve nearly zero value. In the case of n 1 and n 2 close to n, these 'resonance' terms provide the principal contributions to the two-fold sum and make the absolute value of C n n S; S 6 ( )so large that in a definite region of interatomic distances R R R LR F < < the magnitude of E vdW D becomes comparable or even exceeds that of the energy difference in the denominator. In this case (known in the literature as the 'Förster resonance' [1,2,6,7]) the perturbation theory for isolated states may become inapplicable. Therefore a perturbation theory for close energy states should be used in the region of distances below R F (the 'Förster radius'). So the resonance states should be removed from the Green's function Hilbert space (from the sum over n n 1 2 in (1)) and considered as a separate subspace of degenerate states where the interaction operator should be diagonalized. The reduced Green's function will appear in higher-order matrix elements of the interaction [8]. The second-order diagonal matrix elements are added to the detuning E E E 2 n n n 1 2 d = + of the two-atomic states of infinitely separated atoms. The corresponding sum becomes the distance-dependent detuning R D( ), which for a definite R R 0 = may vanish, thus enhancing the role of the non-diagonal element of the R 3 dependence and consequently enhancing the long-range interaction energy in the above indicated region of distances (see section 5). Thus, at R ≈ R 0 the opposite-parity states of each atom may be mixed, creating the degenerate state with large permanent electric dipole moment similar to those produced with the use of electric and/or magnetic field [9]. Detailed analysis on the basis of the data for quantum defects of Rb Rydberg states detects the indicated type of degeneracy for 38P 3/2 , 39D 3/2 , 43D 5/2 and 58D 3/2 states [10][11][12][13][14][15][16][17][18][19]. The energy of two atoms in these states is separated from the closest dipole-coupled two-atomic levels by an energy gap δ of only few Megahertz, at least two orders smaller than the separation on the order of Gigahertz from any other state. The energy of the resonance interaction is usually considered as the first-order dipole-dipole interaction energy of the R 3 dependence [3,11,18,19]. The van-der-Waals energy, estimated as the squared dipole-dipole energy divided by the independent of the interatomic distance R detuning δ [19], corresponds to account of only a single resonance term in the infinite series of the second-order perturbation theory, presented on the right-hand side of equation (1). Evidently, this situation appears inevitably for large distances R R F > , where the dipole-dipole energy is essentially smaller than δ. The resonance-term estimate for the C 6 value holds only, if the contribution of non-resonant terms to the infinite summation in (1) is inessential.
In this paper, we determine the energy of asymptotic interaction with the use of the higher-order perturbation theory for close states [8]. The transformation between the van-der-Waals law E C R D = of the interaction-induced energy shift may be determined both qualitatively and quantitatively from the general equations, which take into account the variation of the resonance detuning caused by the interaction of atoms. As a numerical example, in section 5 the case of closeenergy two-atomic states of Rb atoms in their 38P 3/2 states, only 4 MHz above the joint energy of 39S-and 38Sstates [1], is considered in detail numerically.
For states of nonzero angular momentum the constant C 6 is a tensor quantity [20] dependent on the magnetic quantum numbers and consequently on the relative orientation of the quantisation and interatomic axes determined by the unit vectors a and n respectively (see figure 1). In heavy alkali-metal atoms Rb and Cs the tensor properties of C 6 may arise also in n S 2 1 2 states due to considerable fine-structure splitting between n P J 1 2 2 ( ) states of the total momenta J 1 2 = and J 3 2 = , which should appear on the right-hand side of equation (1) and determine the double-axial (double-vector) component R aa of the C 6 tensor (see section 3). Experimental investigations of the long-range interaction dependence on orientation of the interatomic axis were performed in [18] for Rb atoms in 32D states with the use of a static electric field providing resonant energy exchange with opposite-parity states 34(33)P and n 30 31 = ( )manifold states of angular momenta l values from 5 to n 1 -.
The van-der-Waals interaction between two Rydberg atoms in identical states is the principal object under considerations of this paper. This interaction may shift Rydberg levels from resonance with a laser excitation radiation, thus prohibiting simultaneous excitation of nearby atoms [21]. This effect, called in the literature the 'blockade effect' (see e.g. [2] and references therein), may be useful for processing quantum information. The shift of a Rydberg state nlJMñ | is mainly determined by the C 6 constant which for highly excited states (usually for n 20 > ) is proportional to n 11 . The principal contribution to C 6 comes from the terms in the right-hand side of equation (1) with nearest to n values of the principal quantum numbers n n , 1 2 , providing the smallest energy denominators and the largest values of the dipole-transition matrix elements. In this case of so-called 'Förster resonance' the interaction between two atoms may transform from the usual van-der-Waals form C R 6 6 into Figure 1. Long-range-interacting atoms A and B in their identical Rydberg states, the radius-vector R R n = apart. The unit vector n points from A to B at the angle θ to the unit vector a of the quantisation axis for the total momenta J A B ( ) .
the dipole-dipole interaction of the form C R 3 3 , which appears despite of absence of permanent electric-dipole moments in atoms. The situation is quite similar to a resonance for a frequency-dependent long-range susceptibility of ground-state atoms: the constant C 6 may be also enhanced by a laser radiation of a frequency corresponding to a two-atomic resonance on states of opposite parity [22], on analogy with the effect of the Förster-type resonance between opposite-parity Rydberg states npnp nsn f ñ -¢ ñ | | caused by the dipolequadrupole interaction, which was observed experimentally for ultracold Cs atoms [23].
The states of the angular momenta l 4 > in many-electron atoms are similar to degenerate Rydberg states of a hydrogen atom. These states may be presented as superpositions of states with the momenta from l=5 to l n 1 = -. Therefore, atoms in these states have no definite parity and therefore possess a constant electric dipole, quadrupole, octupole and higher-order (up to 2 n 2 2 --pole) negative-and positive-parity multipole moments [24]. So the diagonal matrix elements of the first-order correction to energy include the dipole-dipole interaction, providing the most important contribution to the long-range interaction-induced shifts of energy levels, inversely proportional to the cube of the distance R: Evidently, the first-order matrix element also includes the dipole-quadrupole, , and other higher-multipole terms of dispersion interaction [25]. However, since the increase of multipolarity accompanies corresponding increase of the R 1 power, the account of the indicated terms for R R LR > may introduce only small corrections. The coefficients C 3 are determined by products of electric-dipole moments of the given Rydberg states (each proportional to the square of the principal quantum number) and depend on the orientation of the dipole-moment vectors relative to the interatomic axis. For nlJMñ | states of l 5 < in many-electron atoms the permanent electric dipole moments and all 2 q -pole moments of odd q are zero. The first-order interaction energy does not vanish for nP 3/2 states and for states of the angular momenta l 1 > . The lowest-order in R 1 term corresponds to the interaction of the electric quadrupole moments: E C R , where the tensor constant C 5 depends on the magnetic quantum numbers M and on the relative orientation of the interatomic and quantisation axes. For identical Rydberg states of atoms C 5 is proportional to n 8 .
The n-dependence of the van-der-Waals constant C 6 determining the second-order energy E vdW D , is essentially stronger, thus providing the principal contribution to the long-range interaction, enabling extremely sensitive control of the blockade effect and finally, ensuring high-efficiency logic operations with the use of Rydberg atoms. Therefore we consider in this paper analytical properties of the van-der-Waals energy E n R vdW 11 6 D µ determining asymptotic (both in n and R) interaction of atoms in identical Rydberg states with low angular momenta l 3  .
The structure of the paper is as follows. In section 2, the basic equations are presented for the energy of longrange interaction between two Rydberg-state atoms. The formulae are derived from the first-order and higherorder perturbation theory for the interaction operator resolved in power series of R 1 -, the inverse distance between two Rydberg-state atoms. Rigorously speaking, the value of van-der-Waals constant C 6 for highlyexcited atoms is complex with imaginary part determining the rate of the interaction-induced ionisation [16,26]. The relations between the real and imaginary parts of the constant C 6 are discussed at the end of section 2. The dependence of the interaction between atoms in identical Rydberg states on the magnetic quantum number M and on the orientation of the interatomic axis is described in section 3 in terms of the irreducible parts of C 6 . Equations for the components of C 6 are presented in terms of the second-order radial matrix elements for doublet Rydberg states nlJMñ | of low angular momentum l. In section 4, asymptotic approximations are proposed for tensor components of C 6 determining its dependence on relative orientations of interatomic and quantisation axes. Coefficients of asymptotic polynomials in powers of the principal quantum number are determined numerically on the basis of standard curve fitting polynomial procedures for nS-, nP-, nD-and nF-states of highly excited Rb atoms.
The case of close two-atomic states with the difference of energies in denominator of equation ( ) comparable or smaller in magnitude than E vdW D is considered in detail in section 5. The possibility of transformation of the van-der-Waals law into the law of the dipole-dipole interaction is discussed and corresponding ranges of the distance R, where this effect may appear, are determined explicitly.
Atomic units e m 1  = = = are used throughout the paper, unless otherwise specified.

General formulae
The operator of electrostatic interaction between two neutral atoms A and B may be presented in the form of asymptotic series of interaction between 2 L -pole electric moments which account for the contribution of each of Z A (Z B ) electrons determined by its position vector r r n i i i = relative to the atomic nucleus (n i is a unit vector, which points from the nucleus to the ith electron) [20] A separate term of this sum is the operator of asymptotic interaction of electric 2 2 General notations of the quantum theory of angular momentum [27] are used here for scalar and tensor products. L C n Y n 4 2 1 is the modified spherical function of the vector n i angular variables. The first term of the two-fold series (3) V R 11 ( ) describes the interaction between virtual electric dipole moments of atoms and determines in the first-order perturbation theory the dipole-dipole interaction of atoms in degenerate two-atomic states [11,18] and the van-der-Waals interaction in the second order for atoms in nondegenerate two-atomic states [19]. The modified spherical function C n L ( ) determines the dependence of the 2 2

L L
A B --pole interaction on the angular variables of the unit vector R n R = which points from atom A to B. Thus the energy of interaction between two atoms depends on the magnitude and orientation of the relative position vector R. The orientational dependence finally transforms into the dependence on the angle n a cos 1 q = -( · )between the vector n and a unit vector a pointing in the positive direction of the quantisation axis for atomic total momenta J A B ( ) (see figure 1). The presentation of the interaction operator (4) seems the most convenient since the variables of the 'external' vector R, collected in the tensor It is useful to note, that the dipole-dipole interaction operator is usually presented in terms of the electricdipole operators Q d The first expression seems more convenient for analytical calculations and for analysing explicitly the R-dependence of the long-range interaction between atoms in Rydberg states. In particular, the determination of the dependence on the orientation of the vector R R n = , related to a fixed external axis, is distributed over numerous components of dot-products in the second expression, which required laborious calculations of different dipole matrix elements already in the first-order perturbation theory [7,18,19,28]. In contrast, the n-dependence is accumulated in a single factor C n 2 ( ) of the first expression, which may be used straightforwardly not only in the first order, but also in the second and higher orders of perturbation theory.
In the first-order perturbation theory the dipole-dipole interaction may contribute to the shift of energy levels in atoms A and B, but only in two cases: (i) if the states of interacting atoms represent a superposition of dipole-coupled states of opposite parity or (ii) if the identical atoms A and B are in different but dipole-coupled states [5]. For identical atoms in identical states of definite parity the contribution of dipole-dipole interaction (and of all odd-parity interactions, dipole-dipole, octupole-octupole, etc) in the first-order perturbation theory for V R AB ( ) vanishes. In this case the even-parity interactions may become important for states of non-zero angular momenta. Besides that, the higher multipolar interaction between atoms (quadrupole-quadrupole, etc) should be taken into consideration in order to control applicability of the dipole-dipole approximation in higher orders of perturbation theory.

First-order perturbation theory for asymptotic interaction of two Rydberg atoms
The system of two identical infinitely separated atoms in their nlJMñ | Rydberg-states is a multiple degenerate state of the J 2 1 2 + ( ) multiplicity. Evidently, the operator of interaction (3) is not diagonal in the two-atomic system of eigenfunctions of different magnetic quantum numbers M. Nevertheless, the problem of evaluating the energy shift in a state of fixed magnetic quantum numbers does not aim at the search of the operator (3) eigenstates and eigenvalues. The principal challenge to the theory is to evaluate the magnitude of the interactioninduced shift which could forbid the simultaneous excitation of two atoms into identical Rydberg states. Therefore, the perturbation theory for a non-degenerate state may be used for determining this kind of detuning from the resonance excitation of the two-atomic system.
If the wave function r r , AB A B á ñ | (the Dirac's notations) determines the state of an isolated system of two noninteracting atoms A and B ( r r = + , as in equation (4), and common notations are used for the Clebsch-Gordan coefficients Evidently, the principal contribution to the first-order interaction energy (6) comes from the lowest non-vanishing order in powers of R 1 determined by the electric quadrupolar moments Q 2 A B ( ) . The next term, described by the interaction of term. This ratio, equivalent to the ratio of the mean-squared radius of the Rydberg-electron orbit nl r nl n 2 2 á ñ µ | | and the squared distance R between atoms, is sufficiently small to ensure applicability of the long-range approximation (3) and (4) for the interaction between atoms in the region of R R n 5 where the principal contribution to the first-order energy (6) comes from the term L L 2 2 2 A B = = , which depends on the distance as R 5 -. The number of terms in the right- ). For both atoms in their nP-states N 1 AB = , hence only Q Q 2 2 term remains in the right-hand side of (6) an estimate for which may be written as E n R . For n=100 the long-range approximation holds at R 5 10 4 >´a . However, the shift (6) will vanish in the nodes of the polynomial P n a 4 ( · )at the angles between the vectors n and a, equal to 30.6 , 70.1 , 109.9 vanishes after averaging over orientations of the vector R or after averaging over orientations of the total momentum J A B ( ) (over magnetic quantum numbers M A B ( ) ) of the atom A(B).

Second-order perturbation theory for the asymptotic interaction
In the second-order perturbation theory the shift of the two-atomic energy is determined by the matrix element with two dispersion-interaction operators (3) and a reduced two-atomic Green's function which accounts for the sums over bound states and integrals over continua of non-interacting atoms [29][30][31]: The summation spreads over the complete basis of eigenvectors , independently of the values of angular momenta l A B ( ) , since the Green's function contains all states and allows arbitrary second-order multipolar transitions between states conforming to the parity conservation law. Therefore the second-order shift (7) involves infinite series of terms arising from the resolution in powers of R 1 for the interaction operator (3): is determined by the dipolequadrupole interaction, which for high-n states is on the order of n R 4 2 in comparison with the first nonvanishing term. The general relation C C n q , 0, 1, 2

Here the infinite sum accounts for all virtual multipole moments of atoms A and B in the operators
holds for coefficients. So, the ratio n R 1 2 < ensures convergence of the series (9).

Higher-order perturbation theory for the asymptotic interaction
Higher-order perturbation theory for the dispersion interaction of ground-state atoms was already considered in 1980s (see, for example [22,30,32] and references therein). Numerical data were calculated for coefficients determining asymptotic resolution of the third-order and fourth-order dispersion interaction energy for neutral hydrogen and alkali-metal atoms. Corrections to the energy of asymptotic interaction between ground-state hydrogen atoms up to the tenth order of perturbation theory were calculated in [33]. Specific properties of Rydberg atoms, in particular, the increase of contributions from higher multipole interactions already in the second order term E R AB 2 D ( ) ( ) stimulate the analysis of higher-order terms of perturbation theory in the asymptotic interaction (3).
The third-order energy shift involves matrix elements with three operators (3) and two Green's functions G AB ¢ [32,33]: Corresponding asymptotic resolution involves only odd powers of R 1 , starting for states of definite parity from R 1 11 , and may be written as follows [32,33]: where the coefficients C n q 11 2 3 + ( ) ( ) may be resolved in tensor components dependent on the direction of the interatomic axis n.
By analogy with (7) and (10), the fourth order perturbation theory will present the asymptotic interactioninduced energy E R AB 4 D ( ) ( ) in terms of a superposition of the fourth-order matrix elements and products of the first-order energy (6) with the third-order matrix element and the second-order energy (9) with the secondorder matrix element of the product of two Green's functions (8) between two operators (3). Each term of the superposition effectively includes four operators V R AB ( ) and three Green's functions [32,33]. The asymptotic resolution of the fourth-order energy may be written as a series in even powers of R 1 similar to (9), starting from R 1 12 . Generally speaking, the energy of asymptotic interaction between neutral atoms in their bound states with definite orbital quantum numbers may be presented as an infinite series of terms of the Nth-order perturbation theory for the energy of asymptotic interaction (3) In its turn, each term of these series may be resolved in power series of an inverse interatomic distance R 1 : where q runs positive integer numbers starting from q 0 0 = for even N, q 1 0 = for odd N. Thus the series (12) finally transforms into power series of R 1 : where the coefficients C n where the notation a [ ]is used for the integer part of a positive number a. The first-order energy (6) does not vanish only for states with non-zero angular momenta l 1 A B  ( ) , and is also described by equation (13) terms are taken into account, is slightly stronger: Simultaneously, comparison of terms from the Nth and N k 2 demonstrates negligible contributions of the lower-order perturbation-theory terms into the constant C s tot of the asymptotic resolution (14). This regularity follows from the asymptotic nature of the interaction operator (3) and was observed already for ground-state hydrogen atoms [33] but for rather remote terms of s 30  in the series (14). It is important to note different signs in the right-hand sides of equations (15) and (16) for even-and oddorder sums of coefficients C n s N ( ) ( ) determining the resultant resolution (14) in powers of R 1 for the long-range interaction energy. This rule follows from the signs of the highest-order matrix elements in corresponding expressions of the perturbation theory (compare (7) and (10)). In particular, C C n n s s In what follows, we confine our considerations mainly to the secondorder corrections described by C 6 2 ( ) for which we use the notation C 6 , thus omitting superscripts.

Van-der-Waals ionisation of Rydberg atoms
Evidently, the cooperative energy of two atoms is always sufficient to put the Rydberg electron of the atom B(A) into continuum while the electron of the atom A(B) falls down to a lower-energy state . In this case the integral over continuum of the atom B(A) in the Green's function (8) has a singularity at the energy . According to Sokhotski theorem, the imaginary part of the Green's function is given by the sum of products of corresponding bound-state and continuum wave functions: G n n n n r r r r r r r r r r r r Im , ; , is the energy of a state from atom B(A) positive-energy continuum. So the second-order energy (7) of interaction between two Rydberg atoms and constants of corresponding resolution (13) are complex values, the imaginary parts of which determine the rate of ionisation for one of the atoms accompanied by simultaneous de-excitation of another one with transition to lower-energy levels [16].
This effect is analogous to an autoionization process of two-electron excited states in many-electron atoms, if the energy of the two-electron excitation exceeds the single-electron ionisation potential. Also, the van-der-Waals interaction-induced ionisation of Rydberg atoms may be compared with the so-called 'Penning ionisation' of an atom of a low ionisation potential by a metastable atom of rather high excitation energy, which was considered in [30] as a result of the long-range interaction between inert-gas atoms in metastable states and ground-state alkali atoms. The rate of that radiation-less ionisation was estimated as rather low and rapidly vanishing with distance (at least, as R 12 -, since the first-order dipole transition between metastable and ground states is strictly forbidden). The imaginary parts of constant factors C s of the resolution (14) are rather small in comparison with real parts C Re s { }. They appear already in the second-order perturbation theory from the imaginary part of the twoatomic Green's function (17) in the integral over continuum of atom A(B) at the energy . Simple estimates for the van-der-Waals constant demonstrate that the ratio between the imaginary and real parts of C 6 may be determined as where the exponent p varies from 10 to 3 in the region of the principal quantum numbers from n 10 » to n 1000  , thus maintaining the imaginary part of the van-der-Waals energy 9-10 orders smaller in absolute value than the real part in all the indicated region of states. So small values of C Im 6 { }allow to neglect the broadening of Rydberg levels, caused by the long-range interaction-induced ionisation. Therefore, we confine ourselves to calculating only real parts of the van-der-Waals constants.
An opposite to the long-range case of Rydberg-Rydberg autoionization was considered in [34] for a system of two Rydberg atoms, separated at distances R R LR  , where the ionisation rate increases almost exponentially with the overlap of Rydberg orbits.
3. Irreducible components of van-der-Waals constants 3.1. C 6 dependence on the relative orientation of the quantisation and interatomic axes The general equation for the van-der-Waals constant describing the interaction of two identical atoms, both in one and the same Rydberg state, say n l J M n l J M nlJM , may be derived on the basis of an equation for the van-der-Waals constant determining the long-range interaction of two arbitrary (identical or different) atoms in their arbitrary (also identical or different) excited states [20]. Thus, the van-der-Waals constant for the like Rydberg-state atoms may be presented as a function of the magnetic quantum numbers M and of the angle θ between the unit vectors of interatomic n and quantisation a axes (see figure 1), as follows: We consider below the case of doublet states (spin S 1 2 = and total angular momentum J l 1 2 =  ). The irreducible parts R ss , R aa , R sT , R Ts , R TT may be presented as for the radial matrix elements with two-atomic radial Green's function g l J l J , ; , 1 1 2 2 . These terms correspond to different angular-momentum channels, some properties of which were discussed, in particular, for Rb and Cs atoms in [1].
The results for states of J M = | |(orbit in the plane perpendicular to the quantisation axis) turns out to be interesting due to the orientation-dependent factors sin 2 q arising in several terms of the C 6 constant. As can be seen from (20) to (23), the contribution to C 6 of the radial matrix elements l J l J  (1)) and M 3 2 = | | (plot (2)). The negative (repulsive) values of C 6 for 38P 3 2 states exceed by more than one order in magnitude those of the 37P 3 2 and 39P 3 2 states. This property follows from the resonance behaviour of the spectrum of two-atomic nS-and nP-states: the difference of the energy of two atoms in 38P 3 2 states from the total energy of the 38S 1 2 and 39S 1 2 states -»is more than one order smaller in magnitude than the difference from the energies of the other closest dipole-coupled two-atomic states. Therefore, the interaction of Rb atoms in 38P 3 2 states in a definite region of interatomic distances R R R , should be considered on the basis of the perturbation theory for close states (see below section 5). In the region above the 'Förster radius' R R F > the interaction-induced shift is essentially smaller than the 'resonance detuning' δ and the usual perturbation theory for non-degenerate states is applicable with the vander-Waals constants C 38P M ( )more than one order exceeds those for n = 37, 39, since the resonance energy defect d | | is the smallest for n=38 states.

Numerical evaluations of irreducible components of the van-der-Waals constant C 6 for Rydberg atoms
The radial part of the Green's function (8) in the radial matrix elements (24) may be presented as a spectral resolution where the summation is performed over a complete set of radial eigenfunctions of isolated atoms A and B in a Hilbert subspace of bound and continuum states of fixed values of the orbital and total angular momenta. The substitution of equation (28) for the Green's function turns the right-hand side of the radial matrix element (24) into the twofold infinite sum over complete set of states, including continua, of products of the radial parts of the first-order dipole-transition amplitudes of atoms A and B. The principal contributions to the second-order matrix element come from the terms with closest principal quantum numbers of the intermediate states n n , 1 2 to that of the Rydberg state n and simultaneously, of the lowest absolute values of the energy differences for both atoms E E n n l J A B is at least one order smaller than the magnitudes of the differences n 1 2 d ( ) . The contribution of remaining terms of the spectral resolution in (28) rapidly decreases with the increase of the absolute values of the differences between the principal quantum numbers. Numerical calculations for the radial matrix elements reveal rather rapid convergence of the series over bound states. Therefore, the account of terms from the region of n n 8 where the usual notations for the Γ function and the Pochhammer symbol a a n a n = G + G ( ) ( ) ( )are used [35]. Here n r is the radial quantum number, n 1 r n l = + + is the effective principal quantum number, Z is the net charge of residual ion (Z = 1 for a neutral atom).
The generalised hypergeometric function (Appel function) F 2 of five parameters and two variables, may be calculated in terms of the Gauss hypergeometric functions F 2 1 . But the precision of evaluation of the hypergeometric functions rapidly decreases with the growth of n r , due to the strong cancelation of digits in corresponding sums of sign-alternating terms.
turns out to be the most useful for Rydberg states because the last few terms p n n r r~¢ ( ) are of one and the same sign and provide the main contribution to the sum.
The model-potential parameters of equation (30) are determined from the most precise data for the energy levels of atoms and ions available, in particular, from the Internet resources [36]. The basic role for evaluating the model-potential parameters plays the so-called quantum defect n nlJ nlJ m n =of the energy level, which determines the difference between the principal quantum number n and the effective principal quantum number Z E 2 nlJ nlJ n = determined from the value of the nlJñ | -state energy. Since available data for energies in every series of states are confined to a finite number of levels, the quantum defect allows to extend the information on the energies and therefore on the wave-function parameters, up to n  ¥. The data for quantum defects from [10] (Sand D-states), [12,13] (P-states) and [14] (F-states) were used for determining the model-potential parameters of the Rydberg-state wave functions and the two-atomic frequencies (29) according to the relation ) are equivalent to degenerate states of hydrogen atoms (spin-orbit effects neglected). These states have no definite parity and therefore possess both even and odd permanent electric multipole moments, which can also contribute to the sum (14) [24]. The electric dipole moments point along the quantisation axis and may be written in terms of the parabolic quantum numbers n n , 1 2 [5] and the unit vector a, as , with fixed parabolic quantum numbers nn n 1 2 ñ | , determines the long-range energy shift, which for R R LR > may be written as nn n V nn n C nn n R ; , where the coefficient C nn n n q ; 9 4 1 3cos demonstrates the interaction-induced shift and splitting of the Rydberg manifold proportional to squared product of the principal n and dipole q n n 1 2 =quantum numbers. The orientational dependence is proportional to the second-order Legendre polynomial P cos 3 cos 1 2, as is usual for the interaction of two permanent electric dipole moments (33), parallel to each other and pointing at the angle θ relative the separation vector R.

Asymptotic presentation of C 6 for Rydberg-state Rb atoms
The presentation of the calculated data for the van-der-Waals constant and its tensor components (20)-(23) may be reduced to tabulating numerical data for a few constants determining an asymptotic dependence of the calculated values on the principal quantum number of a Rydberg state. Results of numerical calculations reveal general regularities in the dependence of C 6 on the Rydberg-state quantum numbers. First of all, we start from the dependence (32) of the quantum defects n lJ m ( ) of different nlJseries used for determining energies and their differences (29) in the spectral resolution of the Green's function (28). For Rydberg series in Rb atoms, the energy defect (29) is a monotonically decreasing function of n. However, in the series of states of the angular momenta l 0 ¹ the defect δ for the most contributing states may tend to zero and change its sign between 'resonance' states of the principal quantum numbers n res and n 1 res  . This situation appears for the C 6 components of P-, D-and F-states.
Nevertheless, the basic behaviour of the van-der-Waals constant and its components as a function of the principal quantum number of a pair of identical Rydberg atoms is nearly one and the same, which may be presented in a general asymptotic form, as follows: However, the coefficients A(lJ) and a lJ i ( ) ( ) in the right-hand side of this equation depend on the magnetic quantum numbers M and on the angle θ of the interatomic axis orientation. Therefore, it is more convenient to use the asymptotic series of the form (34) for the M-and θ-independent irreducible components (20)-(23) of the van-der-Waals constant (19).
The asymptotic series in parentheses of equation (34) may be truncated to a polynomial in the inverse powers of the principal quantum number. Numerical calculations demonstrate, that for n 30  , the third-order polynomial P n 1 3 ( ) in powers of n 1 evaluates the components of C 6 with uncertainties below 1%. Therefore, we confine ourselves to the asymptotic equation R nlJ A lJ n P n 1 , 35 11 3 = ab ab where the subscripts s, a, T should substitute for α and β to denote corresponding components of C 6 (R ss , R aa , R sT and R TT ). The third-order polynomial P x a lJ x a lJ x a lJ x 1 , i 1, 2, 3 = may be determined from a standard curve polynomial fitting procedure (see for example [37,38]). Thus, the most important part of dependence on the principal quantum number of the C 6 components is determined by the factor n 11 with the amplitudes A ss (lJ), A aa (lJ), A sT (lJ) and A TT (lJ). Separate terms of the polynomial P n 1 3 ( ) account for the lower powers (n 10 , n 9 , n 8 ), the fractional contribution from which gradually disappears, when n  ¥.
In the case of 'resonances', where the energy defect (29) of the most contributing states changes its sign, the asymptotic equation (35) may be generalised, as follows:

R nlJ
A lJ n n lJ n n lJ n P n 1 , of the asymptotic presentation (36) for irreducible components of the van-der-Waals constants of Rb atoms in identical Rydberg states nlJMñ | are listed in table 1. Quite different from (36) asymptotic approximations to the long-range interaction of atoms were presented in [39] where the inter-atomic potentials in two-atomic molecules were discussed. In particular, the dependence on the principal quantum numbers n for the van-der-Waals constant, together with the basic term of c n 0 11 for ns-ns and np-np asymptotic states, involved also the terms c n 1 12 and c n 2 13 , which begin dominate already at n 50 > for all symmetries of bi-atomic molecules, thus giving a wrong n-dependence for C 6 of highly excited Rydberg atoms. For the np-np asymptotic state of the molecular symmetries g 1 S + and u 3 S + an additional 'resonance'-type term c n n 29.5 determined from currently most reliable data on quantum defects [10,[12][13][14] and energy levels [36].
Evidently, the polynomial approximations to C 6 of [39] may work only in a narrow region of n between 30 and 100 and become wrong for n 100 > . However, the basic target of studies in [39] was the long-range molecular potentials. So the individual atomic states were not addressed there. In addition, the data for the quantum defects were not yet so detailed as in our days. Therefore the polynomial approximations of [39] could not account for neither the current data on the Förster-type resonances, nor the fine structure of Rydberg states.
The van-der-Waals coefficients (19) may be written also as the resolution in powers of the geometric factor cos 2 q ( ), as follows where the notations are used Finally, the coefficients B nlJM k ( ) may be written as combinations of the two-atomic radial matrix elements (24), for which the asymptotic presentations of the form (36) also hold.
Numerical values for coefficients C 6 determined from equations (19), (35)- (37) and from the data of table 1 demonstrate a good agreement with the approximation for n A B S 1 2 ñ = ñ = ñ | | | -states presented in [39] for 30n95. It is impossible to compare the results for P-and D-states, because the data of table 1 accounts for the fine-structure splitting and for the relative orientation of the quantisation and inter-atomic axes, which were hidden in the molecular symmetries of the data [39]. Also, our numerical results demonstrate a satisfactory agreement with the data of [19] for the dependence of the van-der-Waals constant on orientation of the interatomic axis (on the angle θ) for the interaction of Rb atoms in their 60D 5/2 Zeeman substates. The origin of small discrepancies may be caused by some difference of the newest data for the quantum defects of P-, D-and F-states, used in our calculations, from those used in [19].
The negative asymptotic value of B n 72.3 . The repulsive interaction remains up to infinite principal quantum numbers of nS 1 2 ñ | -states. For states nP 3 2 , nD 5 2 and nF 5 2,7 2 the singularities appear in the vicinity of n P 38 3 ) and n F 9 1 5 2,7 2 » ab ( ) , correspondingly. The component R ss provides the basic contribution to the constant C 6 , as follows from equation (19). Since the values of R ss are positive for all nlJñ | -series presented in table 1 for J 3 2  , the van-der-Waals interaction of atoms in states of n n <˜is repulsive (C 0 6 < ), while for n n >˜the interaction becomes attractive (C 0 6 > ). This property seems rather general, but the dependence on the magnetic quantum number and on the angle θ seen explicitly in the right-hand side of equation (19) may cause some deviations from this regularity. In particular, the series of states nP 3  r ----from the right-hand side of equation (19) at 0 q = . So the repulsive interaction of Rb atoms remains at small angles θ for all n M P 32 3 2 = states (n 39 > ). The following regularity holds here: the greater n the greater is the region of the 'repulsive angles' 0 q q < . In particular, 12.6 0 q » , 28.8 and 32.5 for n=39, n=100 and for all n 3000 > , respectively. The singularities correspond to vanishing of the difference between the two-atomic energies. In particular, , the contribution of the radial matrix element with this singularity disappears at 0 q = together with the singularity in C 6 dependence on the principal quantum number n. This effect is confirmed by the asymptotic values of coefficients of the resolution (37) following from the data of ( ) for Rb atoms may result in transition from the van-der-Waals interaction energy of the form C R 6 6 to a specific form of the dipole-dipole energy described by a function C R 3 3 q ( ) . However, this dependence on R holds only in a region of interatomic distances confined from below and from above to the region R R R min max < < , where R min max ( ) depend on the energy defect (29) and on the matrix elements of the dipole transitions to the resonance states. Besides that, the angular dependence of C 3 q ( ) does not obey the law of the Legendre polynomial P cos 3 cos 1 2 ) , characteristic of the interaction between constant dipoles, and depends on the total angular momentum J and magnetic M quantum numbers of the Rydberg state nlJMñ | , as is discussed in the next section. For nD 3 2 states the singularities appear near n=39 and n=58. In both cases an evidence appears of anomalous dependence of the C 6 constant on the principal quantum number in the vicinity of these numbers. However, the attractive nature of the van-der-Waals interaction for states of the principal quantum number n n 39.6 1 < » ab ( ) remains also attractive for n 40  and becomes repulsive only for states of n 55  , up to n=58. For 39D 3 2 state of the magnetic quantum number M 3 2 =  the sign of the van-der-Waals constant depends on the angle θ, changing from repulsion (C 0 6 < ) at 0.24 q p < to attraction (C 0 6 > ) at 0.24 q p > . The numerical data of table 1 provides rather accurate values of the van-der-Waals constant, which agree with the most reliable data of the literature. In particular, the fractional departure of the tabulated data from the experimental data for C 6 values of Rb atoms in their nD M 3 2 -states [40], does not exceed 6% for n=53 and falls down to about 3% for n=62 and n=82. which is equivalent to the diagonalization of the 3×3 matrix (here ij d is the Kronecker symbol, for brevity the argument R of the matrix elements is omitted) [8,41]:

Close-state perturbation theory for asymptotic interaction between Rydberg atoms
corresponding to the close states, which are withdrawn from the two-fold spectral summation of the Green's function (8). The interaction-induced shift E E E D =refers to the Green's function energy, which may be taken as the mean energy of the close states E E E E 3 ) [8]. As follows from the resolution (3), each term of the matrix element series (41) and the terms of the series for energy (42) may be also resolved in powers of R 1 . So, in the first order (k = 1) we have a sum of finite number of terms, which includes only allowed multipole transitions between initial and final states of atoms A and B, just similar to the first-order shift (6)  ). The starting index q 1 also depends on the angular momenta of the two-atomic states iñ | and jñ | . If the states iñ | and jñ | are dipole-coupled for both atoms ( l l i l j 1 ), then the summation in the non-diagonal matrix element (43) starts from q 3 1 = . Otherwise, if the states are dipole-coupled for only one of the atoms and for another one the quadrupole transition is allowed, then q 4; 1 = if the dipole transitions are forbidden and quadrupole transitions are allowed for both atoms, then the sum for the first-order matrix element (43) starts from the quadrupole-quadrupole term, as in the resolution for the first-order energy (6), determined by the diagonal element W ii 1 ( ) , where the minimal power is q 5 1 = .
The second-order term of the resolution (41) may be presented as series of infinite number of terms where the sum for diagonal matrix elements W ii 2 ( ) performs over even powers of R 1 , starting from q 6 2 = . The parity and starting value q 2 of the power q for non-diagonal matrix elements W W | and final 2ñ | , 3ñ | two-atomic states. In the case of opposite parities for one of the two atoms, q 7 2 = and the sum (44) involves only odd indices q. In the case of opposite parities of states for both atoms, the sum starts from q 8 2 = and includes only even powers q. However, we confine ourselves to the most interesting case of dipole-coupled close states 1ñ | and 2ñ | ( 3ñ | ), therefore the account of only first-order dipole term W R 1 is sufficient. With account of the lowest-order matrix elements (43) and (44)  ). It should be noted, that this kind of degeneracy of the two-atomic states, related with the transposition of states, should be taken into account in calculating the van-der-Waals interaction between identical atoms in different states [5]. . So, for large distances R n 3 > the main contribution to R D( ) comes from the R-independent two-atomic difference of energies δ. In particular, for n 50 » states and 100 d > | | MHz, the fractional difference between R D( ) and δ is below 10% at R 10 m m > . The non-diagonal matrix element W n R 12 is also a rapidly vanishing function of the distance R, so in the indicated region of R n 3 > the inequality holds and Y X X J J 3 1 8 1   In states of the magnetic quantum number M 3 2 = the θ-dependence of the R-region, where the inequality R, 1 c q  ( ) holds, is yet stronger since the first-order matrix element W R, So, the region of the dipole-dipole interaction between Rydberg atoms is restricted by the finite value of the energy defect δ between dipole-coupled two-atomic energy levels: the smaller is d | |, the wider is the region of interatomic distances, where the first-order dipole-dipole law of interaction between atoms (49) holds. Therefore different methods of extending the region of the Förster dipole-dipole interaction were developed and confirmed experimentally with the use of static or radio-frequency electric fields for reducing the magnitude of the energy defect δ [7,11,25,28,42]. However, consecutive theoretical considerations of the external-field-induced Förster resonance with account of simultaneous actions of the field and the long-range interaction of Rydberg atoms, are still missing in the literature. The solution to this problem may be based on the use of the higher-order perturbation theory for the atom-field and atom-atom interactions on the straightforward analogy to the case of ground-state atoms in a field [22]. Corresponding susceptibilities of the two-atomic system would describe the energy shift dependence on the amplitude of external field, on the distance between atoms and on the relative orientation of the interatomic R and external field vectors.
Outside the Förster region, the energy of interaction between Rydberg atoms follows the van-der-Waals law E C R vdW 6 6 D = -. Numerical evaluations demonstrate, that the contribution to the long-range interactioninduced shift of 38P 3/2 -state energy levels of Rb atoms from all the remaining 'non-resonant' terms of the twoatomic basis for the Green's function (8) within the 'Förster region of distances' 1 m m R 6 m m < < is more than 3 orders of magnitude smaller than the contribution of the resonance terms. Outside this region (for R 6 m m > ) this shift follows the law R 6 of the usual van-der-Waals equation with the constant C 6 determined by equation (19), by the asymptotic equation (36) for irreducible components (20)-(23) and by the numerical data of table 1.

Conclusion
For Rydberg levels of very high principal quantum numbers n the absolute value of the energy splitting δ between the bi-atomic dipole-coupled states gradually decreases with the increase of the principal quantum numbers n and may become smaller than the magnitude of of the van-der-Waals interaction energy E C R vdW 6 6 D =at the interatomic distances outside the Le Roy sphere R R LR > . In this case the region of applicability for the perturbation theory for isolated levels extends to R R R F L R > > , where R F is the 'Förster radius'. In the region of the distances R between R LR and R F the perturbation theory for close states should be applied, as discussed in section 5. ) tends to infinity due to the node of the R-dependent Förster energy defect R, , which is practically independent of θ.