Polarizability of ultracold ${\mathrm{Rb}}_{2}$ molecules in the rovibrational ground state of ${a}^{3}{{\boldsymbol{\Sigma }}}_{{\bf{u}}}^{+}$

If the molecule is initially in a vibrational level v_a of the ${a}^{3}{\Sigma }_{u}^{+}$ state, all rovibrational levels (including the continuum) of the electronic potentials with ${}^{3}{\Sigma }_{g}^{+}$ and ${}^{3}{\Pi }_{g}$ symmetry need to be accounted for in equation (3). As N = 0 in the initial ${a}^{3}{\Sigma }_{u}^{+}$ state, only transitions toward final levels with total angular momentum $J=0,1,2$ must be considered. Here, $\vec{J}=\vec{S}+\vec{N}$, where $\vec{S}$ 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 ${\alpha }_{\parallel }$ along the molecular axis $\hat{Z}$, which is related to d_Z, and the perpendicular polarizability ${\alpha }_{\perp }$, which is related to d_X = d_Y. In general, ${\alpha }_{\parallel }$ involves ${}^{3}{\Sigma }^{+}\;\to$ ${}^{3}{\Sigma }^{+}$ transitions and ${\alpha }_{\perp }$ is related to ${}^{3}{\Sigma }^{+}\;\to$ ${}^{3}\Pi$ transitions. One can show that $\alpha =({\alpha }_{\parallel }+2{\alpha }_{\perp })/3$ (see, e.g., [18, 19]).

The expression given by equation (3) deals only with the transitions involving the two valence electrons of Rb₂. Following [21], the contribution to the polarizability of the two Rb⁺ cores, hereafter referred to as ${\alpha }_{c}$, must be taken into account, and is added to the results of equation (3). More details about this quantity are discussed in appendix A.1.

The first step of the calculations is to collect a set of accurate molecular potential energy curves (PECs) and TEDMs. The ${a}^{3}{\Sigma }_{u}^{+}$ PEC is taken from the spectroscopic study of [22]. For the excited molecular states and the related TEDMs from the ${a}^{3}{\Sigma }_{u}^{+}$ state, we use the same data as [23, 24], which we report in the supplemental material (see footnote 3) attached to the present paper, for convenience. The PECs are displayed in figure 1(a), while the TEDMs are drawn in figure A1 (see appendix A.2). These data are obtained by the quantum chemistry approach described in details in [25]. Briefly the Rb₂ 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 including a core polarization potential [26, 27]. A full configuration interaction is then performed on the two valence electrons, using a large Gaussian basis set [28], 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 ${1}^{3}{\Sigma }_{g}^{+}$ state [29], the ${1}^{3}{\Pi }_{g}$ state [30], and on the ${2}^{3}{\Pi }_{g}$ state [22], but no complete PEC has been extracted in these studies. As discussed for instance in [23, 30], the computed PECs are suitable to reproduce the observed data provided that they are slightly shifted in frequency (in terms of $\omega /(2\pi c)$, by at most $100\;{{\rm{cm}}}^{-1}$). We will estimate in section 5 the limited influence of such shifts on the results reported in the present work. Finally, the vibrational wave functions of levels $| i\rangle$ and $| f\rangle$ for the summation are obtained using the mapped Fourier grid Hamiltonian representation [31, 32].

The real and imaginary parts of the dynamical polarizability ${\alpha }_{{v}_{a}=0}(\omega )$ of a molecule in the vibrational ground state of the ${a}^{3}{\Sigma }_{u}^{+}$ potential are displayed in figures 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 $1\;{\rm{a.u.}}=4\pi {\epsilon }_{0}{a}_{0}^{3}=1.649\;\times \;{10}^{-41}\;{\text{J m}}^{2}\;{{\rm{V}}}^{-2}$, where a₀ denotes the Bohr radius and ${\epsilon }_{0}$ is the vacuum permittivity. Note, for some applications, e.g., considerations related to the ac Stark shift, units of ${\text{Hz W}}^{-1}\;{{\rm{cm}}}^{2}$ ($1\;{\rm{a.u.}}$ corresponds to $4.6883572\;\times \;{10}^{2}\;{\text{Hz W}}^{-1}\;{{\rm{cm}}}^{2}$) are advantageous. The sum in equation (3) has been truncated to include only the vibrational levels of the four lowest ${}^{3}{\Sigma }_{g}^{+}$ states and the three lowest ${}^{3}{\Pi }_{g}$ states. Furthermore, electric-dipole-forbidden transitions are not considered in the sum, as they would appear as very weak and narrow resonances in the polarizability. The associated molecular data are collected in the supplemental material (see footnote 3). For simplicity, the natural lifetime ${\tau }_{f}={({\gamma }_{f})}^{-1}$ has been fixed to $10{\rm{ns}}$ (${\gamma }_{f}\approx 2\pi \;\times \;15$ MHz) for all the excited molecular levels.

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Strongly oscillating patterns in both ${\rm{Re}}\{{\alpha }_{{v}_{a}=0}(\omega )\}$ and ${\rm{Im}}\{{\alpha }_{{v}_{a}=0}(\omega )\}$ (see figures 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 ${\alpha }_{{v}_{a}=0}(\omega =0)=698.5\;{\rm{a.u.}}$ up to the bottom of the ${1}^{3}{\Sigma }_{g}^{+}$ potential well, reaching $3147\;{\rm{a.u.}}$ at the wavelength of the trapping laser used in the present experiment ($1064.5\;\mathrm{nm}$). In the same region the imaginary part increases from about ${10}^{-5}\;{\rm{a.u.}}$ at $\omega =0$ to ${10}^{-3}\;{\rm{a.u.}}$ at the trapping laser frequency which leads to a correspondingly larger photon scattering rate.

It is difficult to provide a well-defined error bar on the theoretical values of the dynamical polarizabilities as their accuracy depends on the considered wavelength. Various causes of global inaccuracies have been analyzed in depth in [33]. First, the choice of a constant radiative lifetime for all excited levels in equation (3) influences only the strongly oscillating regions of the polarizability, changing the amplitude of the resonances. We checked that this approximation has no effect in the smoothly varying regions which are relevant for trapping experiments. We verified also that adding a couple of upper electronic states in the sum of equation (3) contributes to the polarizability for less than 1%. Moreover, the first excited Σ and Π states contribute together for more than 90% to the polarizability. Usually, these are the most well known states either because accurate spectroscopic results are available, or because they are well-determined by quantum chemistry calculations. The accuracy of TEDMs is tedious to analyze as their experimental determination relies on line intensities which are difficult to measure accurately. However, one argument in favor of the accuracy of the TEDMs results when comparing the values obtained from different methods. For instance, in [34, 35], with respect to various alkali-metal dimers, the TEDMs computed by two different methods are found to agree within 2%. Finally, an indication for the accuracy of the present work is provided by the measurement of the dynamical polarizability for the v = 0, J = 0 level of the ${{\rm{Cs}}}_{2}$ electronic ground state at $1064.5\;\mathrm{nm}$, which is quite far away from the lowest resonant region [18]. The experimentally determined value with respect to $\lambda =1064.5$ nm is $2.42(15)\times {\alpha }_{\mathrm{Cs}}$ [37], where ${\alpha }_{\mathrm{Cs}}$ is the dynamical polarizability of the ${\rm{Cs}}$ atom. This is in remarkable agreement with the computed value of $2.48\times {\alpha }_{\mathrm{Cs}}$ [18].

The experiments presented in this work are carried out with a pure sample of about $1.5\times {10}^{4}$ ⁸⁷Rb₂ molecules prepared in the rovibrational ground state of the ${a}^{3}{\Sigma }_{u}^{+}$ 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 $1\;\mu {\rm{K}}$. As described in detail in [13, 19, 36], the molecules are prepared as follows. An ultracold thermal cloud of spin-polarized ${}^{87}\mathrm{Rb}$ atoms (${f}_{\mathrm{Rb}}=1$, ${m}_{f,\mathrm{Rb}}=1$) is adiabatically loaded into the lowest Bloch band of a 3D optical lattice at a wavelength of $\lambda =1064.5$ nm. The lattice is formed by a superposition of three linearly polarized standing light waves with polarizations orthogonal to each other, see figure 2(a). The three lattice beams are derived from the same laser source with a linewidth of a few kHz and have relative intensity fluctuations of less than 10⁻³. In order to avoid interference effects, the frequencies of the standing waves are offset by about 100 MHz relative to each other. At the location of the atomic sample the beam waists ($1/{e}^{2}$ radii) are about 130 μm and the maximum available power per beam is about 3.5 W. By slowly crossing the magnetic Feshbach resonance at 1007.4 G we produce weakly bound diatomic molecules. After a purification step which removes remaining atoms, a stimulated Raman adiabatic passage (STIRAP) is performed at $1000\;{\rm{G}}$, transferring the dimers into the rovibrational ground state (v_a = 0, N = 0, m_N = 0) of the ${a}^{3}{\Sigma }_{u}^{+}$ potential. Here, m_N is the projection of N on the quantization axis defined by the direction of the magnetic field $\vec{B}$ (cf figure 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($\vec{f}=\vec{S}+\vec{I}$). Moreover, the total angular momentum is given by F = 2 ($\vec{F}=\vec{f}+\vec{J}$) and its projection is m_F = 2. Henceforth, we simply refer to these molecules as 'v_a = 0 molecules'.

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According to equation (1), ${\rm{Re}}\{\alpha (\omega )\}=4| U| /{E}_{0}^{2}$, i.e., the real part of the dynamical polarizability can be determined by measurements of the potential depth $| U|$ and the electric field amplitude E₀ of an optical trap. For the case of a cubic 3D optical lattice with orthogonal polarizations, the trapping potential is given by $V(x,y,z)={\displaystyle \sum }_{\beta =x,y,z}{V}_{\beta }(\beta )$ where the

$$\begin{eqnarray}&&{V}_{\beta }(\beta )=-{| U| }_{\beta }{\mathrm{cos}}^{2}\left(k\beta +{\phi }_{\beta }\right)\end{eqnarray} \tag{ 4 }$$

represent the contributions of the standing waves of directions $\beta =x,y,z$ with $k=2\pi /\lambda$ 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 ${V}_{z}(z)\equiv V(z)$, $| U{| }_{z}\equiv | U|$ and ${\phi }_{z}\equiv \phi$. 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 $\phi =4\pi {z}_{m}/\lambda$ at z = 0.

In order to obtain $| U|$, we carry out lattice modulation spectroscopy (see, e.g., [12, 38–40]). For this, we either modulate $| U|$ by periodically changing the intensity of the standing light wave (amplitude modulation) or we modulate the phase ϕ by periodically changing the laser wavelength λ (phase modulation). Resonant amplitude (phase) modulation drives transitions from the lowest Bloch band (n = 0), in which the molecules have been initially prepared, to even (odd)-numbered excited lattice bands (see figure 2(b)). This can cause either direct loss from the trap owing to heating (see, e.g., [38]) or molecules in higher lattice bands collide with each other and those of the lowest Bloch band, respectively, resulting in decay to nonobservable states. In consequence, resonant excitation leads to a decreased molecular signal in our measurements.

For amplitude modulation spectroscopy, we modulate the intensity of the lattice laser beam sinusoidally by a few percent. When performing phase modulation spectroscopy, we modulate the laser frequency by a few MHz corresponding to a phase difference on the order of a few ${10}^{-2}\;{\rm{rad}}$ as ${z}_{m}\sim 0.4\;{\rm{m}}$. The modulation duration is typically on the order of $1\;{\rm{ms}}$. At the end of each experimental cycle (which takes about $40\;{\rm{s}}$), the remaining number N of molecules is measured. 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.

Figure 3 shows measured excitation spectra of Feshbach (a) and v_a = 0 (b), (c) molecules, obtained via amplitude or phase modulation spectroscopy. A single data point typically consists of 5–30 repetitions of the experiment (for a given spectrum the number of repetitions is constant). Figure 3(a) as well as (b) exhibit a prominent resonance after amplitude modulation. This resonance is related to a transition from the lowest Bloch band (n = 0) to the second excited lattice band (n = 2). Spectrum (a) for Feshbach molecules in addition shows a broad shoulder at around $50\;{\rm{kHz}}$ 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. Spectrum (b) in figure 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., [38]) that such subharmonic resonances exist. To be consistent, a similar subharmonic resonance dip should be present in figure 3(a) at about $15\;{\rm{kHz}}$ (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.

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Now, we turn to figure 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 assign 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. We attribute this to a purely technical issue, as the strength of the phase modulation varied with the frequency in our setup. However, we have verified the assignment of the resonances 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),

$$\begin{eqnarray}&&H=-\displaystyle \frac{{{\hslash }}^{2}}{2m}\displaystyle \frac{{\partial }^{2}}{\partial {z}^{2}}+V(z),\end{eqnarray} \tag{ 5 }$$

which is particularly simple in momentum space (see, e.g., [41]). Here, m is the mass of a molecule, i.e., twice the mass of a ${}^{87}\mathrm{Rb}$ 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 ${E}_{R}={h}^{2}/(2m{\lambda }^{2})$, 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 $\sim 40\;{E}_{R}$. This is the regime where we take most of our measurements. Having measured the resonant excitation frequencies after modulation we could in principle use figure 4 to read off the corresponding lattice depth $| U|$. We refine this method and at the same time check for consistency as follows.

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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 ${E}_{0}^{2}$ is proportional to P. Consequently $| U| \propto P$, 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 figure 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 E₀ of the optical lattice has to be determined in order to infer the dynamical polarizability $\alpha (\omega )$ (see equation (1)). We can circumvent this by referencing the measurements on the lattice depth $| U|$ for the molecules in the rovibrational ground state of the ${a}^{3}{\Sigma }_{u}^{+}$ potential to similar measurements with Feshbach molecules, of which the polarizability ${\alpha }_{\mathrm{Fesh}}(\omega )$ is known to be twice the one of a Rb atom ${\alpha }_{\mathrm{Rb}}(\omega )$ in the electronic ground state [18]. According to equation (1) the lattice depths $| {U}_{{v}_{a}=0}|$ and $| {U}_{\mathrm{Fesh}}|$ for the v_a = 0 and Feshbach molecules are related by

$$\begin{eqnarray}&&\displaystyle \frac{| {U}_{{v}_{a}=0}| }{| {U}_{\mathrm{Fesh}}| }=\displaystyle \frac{{\rm{Re}}\{{\alpha }_{{v}_{a}=0}\}}{{\rm{Re}}\{{\alpha }_{\mathrm{Fesh}}\}}\end{eqnarray} \tag{ 6 }$$

for a given lattice beam intensity, i.e., a given E₀.

From our experiments at $\lambda =1064.5$ nm we obtain ${\rm{Re}}\{{\alpha }_{{v}_{a}=0}\}=(2.5\pm 0.1)\times {\rm{Re}}\{{\alpha }_{\mathrm{Fesh}}\}$, whereas ${\rm{Re}}\{{\alpha }_{{v}_{a}=0}\}=(0.1\pm 0.02)\times {\rm{Re}}\{{\alpha }_{\mathrm{Fesh}}\}$ was found at $\lambda =830.4\;\mathrm{nm}$ [13]. Using our calculated atomic values of 685.8 (1064.5 nm) and 2995.9 a.u. (830.4 nm) yields molecular polarizabilities ${\rm{Re}}\{{\alpha }_{{v}_{a}=0}\}$ of 3430 ± 140 (1064.5 nm) and 600 ± 120 a.u. (830.4 nm), respectively.

In general, using figures (e.g., figures 1 and 7) to read off the dynamical polarizabilities at specific wavelengths is cumbersome. Therefore, we provide here a simple analytical fitfunction and parameters that allow for reproducing the numerical results with respect to nonresonant wavelength regimes. In the infrared and optical domain we are studying, the main contributions to the polarizability of the X (a) state outside of resonances come from the transitions toward the first excited ${}^{1(3)}{\Sigma }^{+}$ state and the first ${}^{1(3)}\Pi$ state. Thus we attempt to model the polarizability by reducing those transitions to a single effective transition toward each of the two different symmetries. The approximate real part of the polarizability is expressed as

$$\begin{eqnarray}&&{\alpha }_{\mathrm{eff}}(\omega )=\displaystyle \frac{2{\omega }_{\Sigma }}{{\hslash }({\omega }_{\Sigma }^{2}-{\omega }^{2})}{d}_{\Sigma }^{2}+\displaystyle \frac{2{\omega }_{\Pi }}{{\hslash }({\omega }_{\Pi }^{2}-{\omega }^{2})}{d}_{\Pi }^{2}+{\alpha }_{c}\end{eqnarray} \tag{ 8 }$$

with ${\omega }_{\Sigma }$ (resp. ${\omega }_{\Pi }$) the effective transition frequencies and ${d}_{\Sigma }$ (resp. ${d}_{\Pi }$) the corresponding effective dipole moments. We have isolated in this expression the core polarizability ${\alpha }_{c}$ as its frequency dependence is much weaker than the one of the terms coming from valence electron excitation (see appendix A.1). 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 equation (8). The results with respect to the ${a}^{3}{\Sigma }_{u}^{+}$ and ${X}^{1}{\Sigma }_{g}^{+}$ rovibrational ground state are shown in figure 8. For the triplet case, data points with frequencies close to resonances, i.e., from $\omega /(2\pi c)=9527\;{{\rm{cm}}}^{-1}$ to $11018\;{{\rm{cm}}}^{-1}$ and above $13173\;{{\rm{cm}}}^{-1}$, are excluded from the fit. We obtain ${\omega }_{\Sigma }/(2\pi c)=10112.33\;{{\rm{cm}}}^{-1}$, ${d}_{\Sigma }=2.792881\;{\rm{a.u.}}$, ${\omega }_{\Pi }/(2\pi c)=13481.32\;{{\rm{cm}}}^{-1}$ and ${d}_{\Pi }=3.211023\;{\rm{a.u.}}$ Note, in terms of dipole moments, a.u. can be converted into SI units according to $1\;{\rm{a.u.}}={{ea}}_{0}=8.478\;\times \;{10}^{-30}\;{\rm{Ams}}$. Using these parameters, the effective polarizability reproduces the numerical results in the fitted region to within a relative root mean square value (rRMS) of around 1% (see figure 8(a)). The rRMS value is defined by

$$\begin{eqnarray}&&{\rm{rRMS}}=\sqrt{\underset{i=1}{\overset{M}{\displaystyle \sum }}\displaystyle \frac{1}{M}{\left(\displaystyle \frac{{\alpha }_{\mathrm{eff}}({\omega }_{i})-\alpha ({\omega }_{i})}{\alpha ({\omega }_{i})}\right)}^{2}}\end{eqnarray} \tag{ 9 }$$

with $\alpha ({\omega }_{i})$ being the numerical values, and ${\alpha }_{\mathrm{eff}}({\omega }_{i})$ the fitted ones. Here, M is the number of considered values ${\omega }_{i}$, which depends on the vibrational level as the resonant frequency regions excluded from the fit vary. We point out that if we take the transition dipole moment d_Z (resp. d_X = d_Y) at the equilibrium distance of the ${a}^{3}{\Sigma }_{u}^{+}$ state and multiply it by the appropriate Hönl–London factor ($1/\sqrt{3}$ for ${\Sigma }^{+}-{\Sigma }^{+}$ transition and $\sqrt{2/3}$ for ${\Sigma }^{+}-\Pi$ transition), we get the value $2.767\;{\rm{a.u.}}$ (resp. 3.142 a.u.), very close to the effective dipole moment found above. Moreover, the effective transition frequencies correspond roughly to the average frequencies of transitions with favorable Franck–Condon factor.

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A similar fit can be performed for the dynamical polarizability of ${X}^{1}{\Sigma }_{g}^{+}$ molecules in the (v_X = 0, N = 0) level (figure 8(b)). Frequency domains from $\omega /(2\pi c)=9432\;{{\rm{cm}}}^{-1}$ to $9960\;{{\rm{cm}}}^{-1}$, from $10613\;{{\rm{cm}}}^{-1}$ to $12890\;{{\rm{cm}}}^{-1}$ and above $14736\;{{\rm{cm}}}^{-1}$, corresponding to resonances toward the ${b}^{3}{\Pi }_{u}$, ${A}^{1}{\Sigma }_{u}^{+}$ and ${B}^{1}{\Pi }_{u}$ excited states, respectively, were excluded from the fit. We omitted to account for levels of the ${b}^{3}{\Pi }_{u}$ state as they have a very small singlet character. Then, the fit to the numerically calculated dynamical polarizability yields ${\omega }_{\Sigma }/(2\pi c)=11450.31\;{{\rm{cm}}}^{-1}$, ${d}_{\Sigma }=2.647731\;{\rm{a.u.}}$, ${\omega }_{\Pi }/(2\pi c)=15019.57\;{{\rm{cm}}}^{-1}$ and ${d}_{\Pi }=2.965774\;{\rm{a.u.}}$ 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 and $2.959\;{\rm{a.u.}}$, respectively. Here, the rRMS of the fit is 0.5%.

Both, the frequency domains 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 ${a}^{3}{\Sigma }_{u}^{+}$ (v_a = 0–40) and ${X}^{1}{\Sigma }_{g}^{+}$ (v_X = 0–124) states. The corresponding parameters, which can be used to reproduce the dynamical polarizabilities in the nonresonant frequency domains, are reported in the supplemental material (see footnote 3). Figure 9 shows the resulting rRMS values as a function of the vibrational level with respect to the lowest triplet and singlet state.

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Using the ansatz of equation (8) gives a poor result for most excited vibrational levels of the ${a}^{3}{\Sigma }_{u}^{+}$ potential with an rRMS exceeding 4% in the range of v_a = 2–27. This behavior is related to the fact that Franck–Condon factors mainly depend on the amplitude of the excited wave function around both the inner and the outer classical turning points of the initial vibrational level studied. The shape of the ${a}^{3}{\Sigma }_{u}^{+}$ 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 ${}^{3}{\Sigma }_{g}^{+}$ potential wells as they are deep. Instead, the depth of the ${}^{3}{\Pi }_{g}$ potential is not sufficient to create such a variation.

Thus we added one more transition term to the ansatz of equation (8) in order to account for both the inner and outer part of the ${a}^{3}{\Sigma }_{u}^{+}$–${}^{3}{\Sigma }_{g}^{+}$ transitions in the model. This reduced significantly the rRMS of the effective polarizabilities of the v_a levels (see figure 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.

As the two ionic ${\mathrm{Rb}}^{+}$ cores are only weakly perturbing each other, we consider the molecular core polarizability as twice the atomic ${\mathrm{Rb}}^{+}$ polarizability. First, we calculate the atomic polarizabilities at imaginary frequencies ${\alpha }_{\mathrm{Rb}}({\rm{i}}\omega )$ [53], which includes only the contribution of the valence electron. We subtract them from the values of [21], where the resulting differences represent the contribution of the core electrons, and thus the ${\mathrm{Rb}}^{+}$ polarizability. Following [21] 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 equation (8) to model the core polarizability with two effective transitions. This yields the effective frequencies $\omega /(2\pi c)=165912$ and $362918\;{{\rm{cm}}}^{-1}$, and the effective dipole moments 2.72799 and 1.40119 a.u. We note that these two transition frequencies are on the order of magnitude of the main transition in ${\mathrm{Rb}}^{+}$ and ${\mathrm{Rb}}^{2+}$, respectively. The core polarizability is a small contribution slowly varying from $17.9\;{\rm{a.u.}}$ at vanishing frequency up to $18.1\;{\rm{a.u.}}$ in the optical domain relevant here. For the static case, we can compare the result of our calculations to the value of $18.2\;{\rm{a.u.}}$ reported in [45] and find good agreement.

For the sake of completeness we show in figure A1 the TEDMs included in the calculations of the dynamical polarizabilities as functions of the internuclear distances. The corresponding numerical values are given in the supplemental material (see footnote 3). Some of the data with respect to the lowest transitions have already been reported in [23, 24]. The largest TEDMs in the range of the PECs concern the first excited ${\Sigma }^{+}$ and Π potentials (see figure A1(a)). At large distances the molecular excited electronic wave functions become close to the form $[{\phi }_{5{\rm{s}}}(1){\phi }_{{nl}}(2)\pm {\phi }_{5{\rm{s}}}(2){\phi }_{{nl}}(1)]/\sqrt{2}$ and therefore the TEDMs converge toward ${d}_{5{\rm{s}}-{nl}}$, which is equal to the atomic TEDMs multiplied by $\sqrt{2}$. We find ${d}_{5{\rm{s}}-5{\rm{p}}}=4.23$ a.u. and ${d}_{5{\rm{s}}-6{\rm{p}}}=0.347$ a.u., in excellent agreement with the values extracted from the NIST database [42] (4.226 and 0.3531 a.u.), obtained by averaging the TEDMs corresponding to $5{\rm{s}}-n{{\rm{p}}}_{1/2}$ and $5{\rm{s}}-n{{\rm{p}}}_{3/2}$ for n = 5 and 6, respectively. This confirms the good quality of the present representations of the atomic electronic wave functions.