Polarizability of ultracold $\textrm{Rb}_2$ molecules in the rovibrational ground state of $\mathrm{a}^3\Sigma_u^+$

We study, both theoretically and experimentally, the dynamical polarizability $\alpha(\omega)$ of $\textrm{Rb}_2$ molecules in the rovibrational ground state of $\mathrm{a}^3\Sigma_u^+$. Taking all relevant excited molecular bound states into account, we compute the complex-valued polarizability $\alpha(\omega)$ for wave numbers up to $20000\:\textrm{cm}^{-1}$. Our calculations are compared to experimental results at $1064.5\:\textrm{nm}$ ($\sim9400\:\textrm{cm}^{-1}$) as well as at $830.4\:\textrm{nm}$ ($\sim12000\:\textrm{cm}^{-1}$). Here, we discuss the measurements at $1064.5\:\textrm{nm}$. The ultracold $\textrm{Rb}_2$ molecules are trapped in the lowest Bloch band of a 3D optical lattice. Their polarizability is determined by lattice modulation spectroscopy which measures the potential depth for a given light intensity. Moreover, we investigate the decay of molecules in the optical lattice, where lifetimes of more than $2\:\textrm{s}$ are observed. In addition, the dynamical polarizability for the $\mathrm{X}^1\Sigma_g^+$ state is calculated. We provide simple analytical expressions that reproduce the numerical results for $\alpha(\omega)$ for all vibrational levels of $\mathrm{a}^3\Sigma_u^+$ as well as $\mathrm{X}^1\Sigma_g^+$. Precise knowledge of the molecular polarizability is essential for designing experiments with ultracold molecules as lifetimes and lattice depths are key parameters. Specifically the wavelength at $\sim1064\:\textrm{nm}$ is of interest, since here, ultrastable high power lasers are available.


I. INTRODUCTION
Owing to the extraordinary control over the internal and external degrees of freedom, ultracold molecules trapped in an optical lattice represent a system with many prospects for studies in ultracold physics and chemistry [1,2], the realization of molecular condensates [3], precision measurements of fundamental constants [4][5][6][7] and quantum computation [8,9] and simulation [10]. In the recent years, several groups have realized the preparation of optically trapped vibrational ground state (v = 0) molecules in either the lowest lying singlet or triplet potential [11][12][13][14][15].
Experiments with these molecules, e.g. ultracold collisions, are typically carried out in optical lattices or optical dipole traps [16][17][18]. In these environments, precise knowledge of the dynamical polarizability of molecules is a prerequisite for well controlled experiments.
The Rb 2 molecule is one of the few ultracold molecular species currently available, with which benchmark experiments for nonpolar molecules can be carried out. Here we investigate the dynamical polarizability α(ω) of a Rb 2 triplet molecule in the lowest rovibrational level of a 3 Σ + u . A similar analysis for Cs 2 regarding the electronic ground state X 1 Σ + g was previously carried out by Vexiau et al. [19]. In addition to calculations of the frequency dependent dynamical polarizability α(ω), we present measurements of the real part Re{α(ω)} at a wavelength of λ = 1064.5 nm.
The experiments are performed with molecules trapped in the lowest Bloch band of a cubic 3D optical lattice which consists of three standing light waves with polarizations orthogonal to each other. By carrying out modulation spectroscopy on one of the standing light waves, we map out the energy band-structure for various light intensities. From these measurements Re{α(ω)} is determined. Our experimental findings at λ = 1064.5 nm and also those at 830.4 nm [12] agree well with the calculations. In addition, we experimentally investigate the decay time of the deeply bound molecules in a 3D optical lattice at 1064.5 nm for various lattice depths. Here, lifetimes of more than 2 s are observed. Furthermore, we present numerical results for the dynamical polarizabilities of the rovibronic ground state, i.e., the lowest rovibrational level of the X 1 Σ + g potential. For convenient application of our results, we provide a simple analytical expression and the corresponding effective parameters which can be used to reproduce the dynamical polarizabilities for all vibrational levels of both, the lowest singlet as well as triplet state outside the resonant wavelength regions.
This article is organized as follows. In section II, we give a brief, general introduction to the dynamical polarizability α(ω) of a homonuclear diatomic molecule. Sections III to VI describe the calculations and measurements related to the polarizability of 87 Rb 2 in the rovibrational ground state of a 3 Σ + u , along with a comparison of our results to reference values from literature. In section VII we present calculations of the polarizability for the rovibrational ground state of X 1 Σ + g . Afterwards, section VIII provides a simple expression which parametrizes the polarizability for all vibrational levels of the lowest singlet as well as triplet state. A table with the corresponding parameters can be found in the Supplemental Material [20].

II. INTERACTION OF A DIATOMIC MOLECULE WITH LIGHT
When a nonpolar molecule is subject to a linearly polarized electric field E =εE 0 cos(ωt) with amplitude E 0 and unit polarization vectorε, a dipole moment p = α ↔ (ω) E is induced. In general, α ↔ (ω) is a tensor (see, e.g., [21]). For the sake of simplicity, we restrict ourselves to molecules and therefore to the real part of the polarizability Re{α(ω)} (see, e.g., [22]). In Eq. (1), the angled brackets . . . indicate time averaging. We note, that the dipole potential is attractive, when the sign of Re{α(ω)} is positive and repulsive otherwise. The imaginary part of the polarizability Im{α(ω)} is related to the power P abs absorbed by the oscillator from the driving field, since We calculate the dynamical polarizability α(ω) following the method described in Ref. [19]. The generic expression of the polarizability for a diatomic molecule in a state |i is Here, the angled brackets refer to the spatial integration over all internal coordinates of the system. The summation covers all the accessible dipole transitions with frequency ω i f and transition electric dipole moment d from the initial state |i to final states | f with line width γ f .

A. Relevant transitions
If the molecule is initially in a vibrational level v a of the a 3 Σ + u state, all rovibrational levels (including the continuum) of the electronic potentials with 3 Σ + g and 3 Π g symmetry need to be accounted for in Eq. (3). As N = 0 in the initial a 3 Σ + u state, only transitions towards final levels with total angular momentum J = 0, 1, 2 must be considered. Here, J = S + N, where S is the total electronic spin. Therefore, when considering a diatomic molecule it is usual to define two contributions to the isotropic polarizability α: the parallel polarizability α along the molecular axisẐ, which is related to d Z , and the perpendicular polarizability α ⊥ , which is related to d X = d Y .
The expression given by Eq. (3) deals only with the transitions involving the two valence electrons of Rb 2 . Following Ref. [23], the contribution to the polarizability of the two Rb + cores, hereafter referred to as α c , must be taken into account, and is added to the results of Eq. (3). More details about this quantity are discussed in Appendix A.
The first step of the calculations is to collect a set of accurate molecular potential energy curves (PECs) and transition electric dipole moments (TEDMs). The a 3 Σ + u PEC is taken from the spectroscopic study of Ref. [24]. For the excited molecular states and the related TEDMs from the a 3 Σ + u state, we use the same data as Refs. [25,26], which we report in the Supplemental Material [20] attached to the present paper, for convenience. The PECs are displayed in Fig. 1, while the TEDMs are drawn in Appendix B. These data are obtained by the quantum chemistry approach described in details in Ref. [27]. Briefly the Rb 2 molecule is considered as two valence electrons moving in the field of the two ionic Rb + cores, which are represented by a large effective core potential (ECP) including a core polarization potential (CPP) [28,29]. A full configuration interaction (FCI) is then performed on the two valence electrons, using a large gaussian basis set [30], with the CIPSI quantum chemistry code developed at Université Paul Sabatier in Toulouse. It is worth mentioning that partial spectroscopic information is available on the 1 3 Σ + g [31] and the 1 3 Π g [32] state, but no complete PEC has been extracted in these studies. As discussed for instance in Refs. [26,32], the computed PECs are suitable to reproduce the observed data provided that they are slightly shifted in frequency (in terms of ω/2πc, by at most 100 cm −1 ). However, as it will be discussed later on, such shifts only have negligible consequences on the results reported in the present paper. Finally, the vibrational wave functions of levels |i and | f for the summation are obtained using the Mapped Fourier Grid Hamiltonian representation [33,34].

B. Results
The real and imaginary parts of the dynamical polarizability α v a =0 (ω) of a molecule in the vibrational ground state of the a 3 Σ + u potential are displayed in Fig. 1(b) and (c) as functions of the trapping laser frequency. The polarizabilities are expressed in atomic units (a.u.), which can be converted into SI units according to 1a.u. = 4πε 0 a 3 0 = 1.649×10 −41 Jm 2 V −2 , where a 0 denotes the Bohr radius and ε 0 is the vacuum permittivity. Note, for some applications, e.g., considerations related to the ac Stark shift, units of HzW −1 cm 2 (1 a.u. corresponds to 4.6883572 × 10 2 HzW −1 cm 2 ) are advantageous. The sum in Eq. (3) has been truncated to include only the vibrational levels of the four lowest 3 Σ + g states and the three lowest 3 Π g states. While it is difficult to provide a strict error bar, we checked that adding a couple of upper electronic states in the sum of Eq. (3) contributes to the polarizability for less than 1%. The associated molecular data are collected in the Supplemental Material [20]. For simplicity, the natural lifetime τ f = (γ f ) −1 has been fixed to 10 ns (γ f ≈ 2π × 15 MHz) for all the excited molecular levels. As discussed below (see experimental section), this is a reasonable approximation. Strongly oscillating patterns in both Re{α v a =0 (ω)} and Im{α v a =0 (ω)} [see Figs. 1(b) and (c), respectively] correspond to frequency ranges of strong absorption which should be disregarded for trapping purpose. The real part smoothly increases from the static polarizability α v a =0 (ω = 0) = 698.5 a.u. up to the bottom of the 1 3 Σ + g potential well, reaching 3147 a.u. at the wavelength of the trapping laser used in the present experiment (1064.5 nm). In the same region the imaginary part increases from about 10 −5 a.u. at ω = 0 to 10 −3 a.u. at the trapping laser frequency which leads to a correspondingly larger photon scattering rate.

A. Experimental setup and measurement scheme
The experiments presented in this work are carried out with a pure sample of about 1.5 × 10 4 87 Rb 2 molecules prepared in the rovibrational ground state of the a 3 Σ + u potential and trapped in a 3D optical lattice. There is no more than a single molecule per lattice site and the temperature of the sample is about 1 µK. As described in detail in Refs. [12,21,35], the molecules are prepared as follows. An ultracold thermal cloud of spin-polarized 87 Rb atoms ( f Rb = 1, m f ,Rb = 1) is adiabatically loaded into the lowest Bloch band of a 3D optical lattice at a wavelength of λ = 1064.5nm. The lattice is formed by a superposition of three linearly polarized standing light waves with polarizations orthogonal to each other, see Fig. 2 on the quantization axis defined by the direction of the magnetic field B [cf. Fig. 2(a)]. The molecule has positive total parity, total electronic spin S = 1, total nuclear spin I = 3 and is further characterized by the quantum number f = 2 ( f = S + I). Moreover, the total angular momentum is given by F = 2 ( F = f + J) and its projection is m F = 2. Henceforth, we simply refer to these molecules as "v a = 0 molecules".
According to Eq. (1), Re{α(ω)} = 4|U|/E 2 0 , i.e., the real part of the dynamical polarizability can be determined by measurements of the potential depth |U| and the electric field amplitude E 0 of an optical trap. For the case of a cubic 3D optical lattice with orthogonal polarizations, the trapping potential is given by represent the contributions of the standing waves of directions β = x, y, z with k = 2π/λ being the wave number of the lattice beams. We now only consider the part of the lattice in the vertical z direction since this axis is the only one relevant for the measurements of the potential depth in the present work. Therefore, we define V z (z) ≡ V (z), |U| z ≡ |U| and φ z ≡ φ . The phase φ is a function of the laser wavelength λ because the standing light wave is created by retroreflecting the laser beam from a fixed mirror at position z m . It is given by φ = 4πz m /λ at z = 0.
For this, we either modulate |U| by periodically changing the intensity of the standing light wave We only find molecules in the lowest Bloch band, not in higher bands. In order to determine the molecule number, we reverse the STIRAP and dissociate the resulting Feshbach dimers by sweeping over the Feshbach resonance. Then, the generated atoms are detected via absorption imaging. By comparing the resonant transition frequencies observed in the modulation spectra to the energy band-structure of the sinusoidal lattice, the lattice depth |U| is deduced. This will be explained in detail further below. 50 kHz which we attribute to a resonant transition from n = 0 to n = 4. Due to the large width of this resonance the uncertainty in the determination of its center frequency is relatively large. Fig. 3 also features a second resonance dip, but here it is located at about half the frequency of the prominent one. This resonance dip can be assigned to a transition from the lowest Bloch band to the second excited band, involving two identical "quanta" with frequency ν.
It is known (see, e.g., [36]) that such subharmonic resonances exist. To be consistent, a similar sub-harmonic resonance dip should be present in Fig. 3(a) at about 15 kHz (as indicated by the vertical, dashed arrow). Indeed, at that position the data points seem to be systematically below the fit curve with respect to the prominent peak. However, the corresponding signal (if at all) is very weak, partially due to its position at the steep flank of the prominent resonance. Now, we turn to Fig. 3(c), which shows an excitation spectrum after phase modulation for v a = 0 molecules. We observe two resonances of similar strength, both of which we attribute to the transition from n = 0 to n = 1. The dip at lower frequency is again a subharmonic resonance.
Surprisingly, it is stronger than the harmonic one at about 25 kHz. However, we have verified the assignment by comparison to the corresponding amplitude modulation spectra.
We calculate the Bloch bands by diagonalizing the Hamilton operator for the lattice in 1D (neglecting gravitation), which is particularly simple in momentum space (see, e.g., [39]). Here, m is the mass of a molecule, i.e., twice the mass of a 87 Rb atom. Figure 4 shows the calculated energy eigenvalues as a function of the lattice depth. The energies are given in terms of the recoil energy , with h being Planck's constant. As we do not specify the quasimomentum, the energy eigenvalues form bands which are broad for low lattice depths. However, the bands n = 0 to n = 2 are quite narrow for lattice depths above ∼ 40 E R . This is the regime where we take most of our measurements. Having measured the resonant excitation frequencies after modulation we could in principle use Fig.4 to read off the corresponding lattice depth |U|. We refine this method and at the same time check for consistency as follows.
In the experiment we control the lattice depth |U| via the laser beam power P that can be measured using photodiodes. The square to the electrical field E 2 0 is proportional to P. Consequently |U| ∝ P, i.e., the precise value of |U| is known up to a calibration factor (which depends linearly on the dynamical polarizability). Thus, given a molecular state, we should be able to adjust the calibration factor such that all data obtained for various powers P match the band structure calculation. The measured data points in Fig. 4 clearly show that this works quite well, both for deeply bound molecules (red) and Feshbach molecules (blue). In this procedure we do not account for the transitions from n = 0 to n = 4 owing to the large uncertainties of the corresponding resonances in the excitation spectra. Nevertheless, these data points are shown in the plot for comparison.
In addition to |U|, the electrical field amplitude E 0 of the optical lattice has to be determined in order to infer the dynamical polarizability α(ω) [see Eq. (1)]. We can circumvent this by referencing the measurements on the lattice depth |U| for the molecules in the rovibrational ground state of the a 3 Σ + u potential to similar measurements with Feshbach molecules, of which the polarizability α Fesh (ω) is known to be twice the one of a Rb atom α Rb (ω) in the electronic ground state [19].
According to Eq. (1) the lattice depths |U v a =0 | and |U Fesh | for the v a = 0 and Feshbach molecules for a given lattice beam intensity, i.e., a given E 0 .

V. COMPARISON OF RESULTS
We have again studied this issue in this work and find that population of higher lattice bands can be suppressed even in the absence of a magic wavelength when working with deep lattices.
At λ = 1064.5 nm there is still a factor of 2.5 difference in polarizability between Feshbach and  4]. This strongly increases the selectivity of the transition (see, e.g., [47]). Indeed, in our experiments we do not observe any significant population of higher bands.
As can be seen in Fig. 5, the absolute dynamical polarizability of the triplet rovibrational ground state molecules at 1064.5 nm is about four times larger than at 830.4 nm. This is convenient since it results in a four times deeper interaction potential at the same laser intensity. For  Table I with the corresponding wavelengths indicated by green dashed vertical lines.
longer wavelengths than 1064.5 nm, Fig. 5 reveals, that the dynamical polarizabilities of v a = 0 molecules and Feshbach molecules approach each other. Hence working at even longer wavelengths than 1064.5 nm might be advantageous for some applications.

VI. LIFETIME OF THE MOLECULES
According to Eq. (2), the imaginary part of the dynamical polarizability, Im{α}, is linked to the light power absorbed by a molecule, P abs , which in turn can be expressed in terms of the photon scattering rate Γ sc =h −1 ω −1 P abs [22]. Using this and Eq. (1), Γ sc can be written as For a 3D optical lattice with equal lattice depths |U| in each direction the scattering rate is given by Γ 3D sc = 3Γ sc . As an example, we consider the case of |U| = 50 E R at λ = 1064.5 nm. Then, the corresponding values for the polarizability obtained in the present work, Im (α) = 0.96 × 10 −3 a.u and Re{α} = 3430 a.u., yield Γ sc = 0.18 s −1 . Note, this calculation only accounts for an ideal optical lattice. As there is always background light that does not contribute to the lattice, the estimated value for the scattering rate represents just a lower bound. Once a photon is absorbed, the molecule is excited and typically decays to a nonobservable state. Assuming excitation to be the only loss-mechanism a lifetime τ = 1.9 s of the v a = 0 molecules is expected in a 50 E R deep 3D optical lattice at 1064.5 nm.
We experimentally investigate the lifetimes of the molecules in the rovibrational ground state of a 3 Σ + u by varying the holding time t h in the lattice. Figure 6 shows lifetime measurements of v a = 0 molecules for various potential depths |U|, which are adjusted to be equal in each direction. Applying an exponential fit, we obtain a 1/e decay time τ of more than 2 s for both our measurements at |U| = 33 E R and |U| = 58 E R .
In order to estimate possible loss induced by inelastic molecular collisions, we calculate the tunneling rates Γ tu between adjacent lattice sites within the lowest Bloch band (n = 0). When considering a lattice depth of 33 E R (58 E R ) one obtains Γ tu = 2.37 s −1 (Γ tu = 0.09 s −1 ). In our setup at most 20% of the lattice sites are occupied in the region of highest molecule density. Thus, for |U| = 33 E R decay due to collisions cannot be neglected, whereas for |U| = 58 E R and beyond only loss owing to photon scattering is relevant. For such deep lattices, the lifetime τ scales directly inversely with the lattice depth |U|. We confirm this for the measurements at |U 1 | = 58 E R and As the agreement between calculations and measurements for the triplet molecules is in general good, we also provide calculations for the singlet ground state of 87 Rb 2 . Using the same approach as above (see also [19]), we compute the dynamical polarizability α v X =0 (ω), i.e., with respect to the v X = 0, J = N = 0 level. The X 1 Σ + g PEC has been derived from spectroscopic data of Ref. [48]. The A 1 Σ + u and the b 3 Π u PECs and the related spin-orbit coupling between those two states are taken from Ref. [49]. The PECs for all the other states and for the TEDMs are taken from the computations reported in Refs. [25,26]. Again, the sum in Eq. (3) has been truncated to include only the levels of the four lowest 1 Σ + u states and the three lowest 1 Π u states. The natural lifetime of the excited levels has been fixed at 10 ns. Results are presented in Fig. 7, showing that two magic wavelengths can be identified at 990.1 nm and 1047.2 nm. The latter is located close to the region of strong absorption resonances and consequently, from the imaginary part [cf. Fig. 7(b)], the photon scattering rate at 1047.2 nm is expected to be by a factor of four larger than at 990.1 nm.

VIII. PARAMETRIZATION OF THE POLARIZABILITY
with ω Σ (resp. ω Π ) the effective transition frequencies and d Σ (resp. d Π ) the corresponding effective dipole moment. We have isolated in this expression the core polarizability α c as its frequency dependance is much weaker than the one of the terms coming from valence electron excitation (see Appendix A). The imaginary part of the polarizability is neglected since the model is designed for the ranges outside the resonant regions.
We extract the effective parameters from a fit to the full numerical results using Eq. (8). The results with respect to the a 3 Σ + u and X 1 Σ + g rovibrational ground state are shown in Fig. 8. For the triplet case, data points with frequencies close to resonances, i.e., from ω/(2πc) = 9527 cm −1 to 11018 cm −1 and above 13173 cm −1 , are excluded from the fit. We obtain ω Σ /(2πc) =  B. Rovibrational ground state of X 1 Σ + A similar fit can be performed for the dynamical polarizability of X 1 Σ + g molecules in the (v X = 0, N = 0) level [ Fig. 8(b)]. Frequency domains from ω/(2πc) = 9432 cm −1 to 9960 cm −1 , from 10613 cm −1 to 12890 cm −1 and above 14736 cm −1 , corresponding to resonances towards the b 3 Π u , A 1 Σ + u and B 1 Π u excited states, respectively, were excluded from the fit. We omitted to account for levels of the b 3 Π u state as they have a very small singlet character. Then, the fit to the numerically calculated dynamical polarizability yields ω Σ /(2πc) = 11450.31 cm −1 , d Σ = 2.647731 a.u., ω Π /(2πc) = 15019.57 cm −1 and d Π = 2.965774 a.u. Again, we point out that the effective transition dipole moments obtained are very close to the electronic transition dipole moments taken at equilibrium distance multiplied by the Hönl-London factors, i.e., 2.613 a.u. and 2.959 a.u., respectively. Here, the rRMS of the fit is 0.5%.

C. Excited vibrational states
Both the frequency domain with good Franck-Condon factors and transition dipole moments depend significantly on the initial vibrational level. This is reflected in the variation of the effective parameters when we perform an individual fit for each vibrational level (with N = 0) of the a 3 Σ + u (v a = 0 to 40) and X 1 Σ + g (v X = 0 to 124) states. The corresponding parameters, which can be used to reproduce the dynamical polarizabilities in the nonresonant frequency domains, are reported in the Supplemental Material [20]. Figure 9 shows the resulting rRMS values as a function of the vibrational level with respect to the lowest triplet and singlet state. The shape of the a 3 Σ + u potential is quite different compared to the relevant excited states. Consequently, the inner turning point region of a given vibrational level v a will induce couplings to levels of excited molecular states with energies strongly different from those related to couplings induced by the outer turning point region. This effect mainly occurs for the excited 3 Σ + g potential wells as they are deep. Instead, the depth of the 3 Π g potential is not sufficient to create such a variation.
Thus we added one more transition term to the ansatz of Eq. (8) in order to account for both the inner and outer part of the a 3 Σ + u -3 Σ + g transitions in the model. This reduced significantly the rRMS of the effective polarizabilities of the v a levels (see Fig. 9). We want to emphasize that such an interpretation gives a reasonable physical picture for most levels. However, for some levels like the deeply bound ones or those close to the dissociation limits, the three-effective-transition model is somewhat artificial and the effective parameters should be taken only as numerical parameters needed to easily obtain the corresponding polarizability. We have studied the dynamical polarizability of the 87 Rb 2 molecule in the rovibrational ground state of the a 3 Σ + u potential. Calculations of both, the real and imaginary part are provided and we measured Re{α(ω)} at λ = 1064.5 nm. Our experimental and theoretical findings show good agreement. From our computed value of Im{α(ω)} at this wavelength, we expect trapping times of the molecules on the order of seconds for lattice depths around 50 E R , which was confirmed by our observations. We also have investigated theoretically the dynamical polarizability of the singlet ground state X 1 Σ + g . These results are interesting for future STIRAP transfer of Rb 2 to the corresponding rovibronic ground state. Furthermore, we have introduced a simple analytical expression to parametrize the dynamical polarizabilities for all levels of both, a 3 Σ + u and X 1 Σ + g states. By fitting this expression to the numerical results, we have extracted effective parameters, which can be used to reproduce Re{α(ω)} of a given vibrational state with high fidelity.

IX. CONCLUSION
The precise knowledge of the dynamical polarizability enables accurate control of optical dipole potentials and therefore is of importance for future experiments with deeply bound Rb 2 molecules.

X. Appendix
A. Polarizability of the Rb + + + core.
As the two ionic Rb + cores are only weakly perturbing each other, we consider the molecular core polarizability as twice the atomic Rb + polarizability. First, we calculate the atomic polarizabilities at imaginary frequencies α Rb (iω) [50], which includes only the contribution of the valence electron. We subtract them from the values of Ref. [23], where the resulting differences represent the contribution of the core electrons, and thus the Rb + polarizability. Following Ref. [23] this estimate assumes that the influence of the valence electrons on the core polarizability is negligible.
We use an ansatz similar to the one of Eq.  [20]. Some of the data with respect to the lowest transitions have already been reported in Refs. [25,26].
The largest TEDMs in the range of the PECs concern the first excited Σ + and Π potentials [see Fig. 10(a)]. At large distances the molecular excited electronic wave functions become close to the form [φ 5s (1)φ nl (2) ± φ 5s (2)φ nl (1)]/ √ 2 and therefore the TEDMs converge towards d 5s−nl , which is equal to the atomic TEDMs multiplied by √ 2. We find d 5s−5p = 4.23 a.u. and d 5s−6p = 0.347 a.u., in excellent agreement with the values extracted from the NIST database [40] (4.226 a.u. and 0.3531 a.u.), obtained by averaging the TEDMs corresponding to 5s − np 1/2 and 5s − np 3/2 for n = 5 and 6, respectively. This confirms the good quality of the present representations of the atomic electronic wave functions. Transition electric dipole moment (in a.u.) starting from the Rb_2 ground state (singlet Sigma_g^+) towards the 7 lowest singlet Pi_u states (labelled as (1) to (7)), as functions of the internuclear distance R (in a.u. Tran !"on electric dipole moment (in a.u.) st#$"ng from the Rb_2 lowest triplet Sigma_u^+ state towards the 9 lowest triplet Pi_g states (labelled as (1) to (9)), as fun%"ons of the internuclear distance R (in a.u.). Transition electric dipole moment (in a.u.) starting from the Rb_2 ground state (singlet Sigma_g^+) towards the 6 lowest singlet Sigma_u^+ states (labelled as (1) to (6)), as functions of the internuclear distance R (in a.u.).

#R
(1) (2) (3) (4) (5) 5.2 3.92460 -0.93280 -0.24210 -0.61350 0.08000 Table: Parameters for the fit of the dynamic polarizability of the vibrational levels v (with N=0) of the lowest triplet Sigma_u^+ state of Rb_2, using the ansatz of the paper with two effective transitions. omega_eff: effective frequency; d_eff: effective dipole moment. The frequency ranges taken in account for the fit are indicated as intervals.  Table: Parameters for the t of the dynamic polarizability of the vibra!onal levels v (with N=0) of the lowest singlet Sigma_g^+ state of Rb_2, using the ansatz of the paper with two e"ec!ve transi!ons. omega_e": e"ec!ve frequency; d_e": e"ec!ve dipole moment. The frequency ranges taken in account for the t are indicated as intervals.