Laser cooling with intermediate state of spin–orbit coupling of LuF molecule

Abstract

This work presents a theoretical study of the laser cooling feasibility of the molecule LuF, in the fine structure level of approximation. An ab-initio complete active space self-consistent field (CASSCF)/MRCI with Davidson correction calculation has been done in the Λ^(±) and Ω^(±) representations. The corresponding adiabatic potential energy curves and spectroscopic parameters have been investigated for the low-lying electronic states. The calculated values of the internuclear distances of the X³Σ₀₊ and (1)³Π₀₊ states show the candidacy of the molecule LuF for direct laser cooling. Since the existence of the intermediate (1)³Δ₁ state cannot be ignored, the investigation has been done by taking into consideration the two transitions (1)³Π₀₊−(1)³Δ₁ and (1)³Π₀₊ −X³Σ₀₊. The calculation of the Franck–Condon factors, the radiative lifetimes, the total branching ratio, the slowing distance, and the laser cooling scheme study prove that the molecule LuF is a good candidate for Doppler laser cooling.

Introduction

Cold and ultracold molecules offer new insights into many-body physics, revolutionize physical chemistry, and provide techniques for probing new states of quantum matter. They represent an exciting new frontier that enables the investigation of measurements at an unprecedented level of detail. Besides, polar ultracold molecules present new platforms for quantum information, quantum computing^1,2,3,4, and quantum simulation of many-body interactions^5,6. Dipole–dipole interactions among ultracold polar molecules, such as LuF can lead to discoveries beyond traditional molecular science⁷.

The group of lutetium mono-halides LuX (X = F, Cl, Br, I) has been the subject of numerous experimental and theoretical studies and has an increasing interest in various types of research^8,9,10,11. LuF is an element of the lutetium mono-halides LuX group, which is significant in astrophysics due to their presence in many stars enriched by the r-nucleosynthesis process^12,13,14,15, in the interstellar medium¹⁶, and in the cool stellar atmospheres¹⁷. This molecule has been studied experimentally in literature^18,19,20,21, while the theoretical studies in Λ^(±) representation are given in Refs.^{22,23,24,25,26,27,28}. The Ω^(±) states of LuF molecules have been investigated by Assaf et al.²⁹, where 36 electronic states have been investigated.

Computation and results

We initially performed preliminary investigations of the electronic structure of the LuF molecule in the spin-free approximation. The Quantum Chemistry Package Molpro 2010³⁰ was used by applying the ab-initio Complete Active Space Self-consistent field (CASSCF) method. The adiabatic potential energy curves have been calculated by employing the internally contracted Multi Reference Configuration interaction plus Davidson correction (MRCI + Q) techniques within the Born–Oppenheimer approximation. In the Λ^(±) representation, the adopted basis sets for the Lu and F atoms are respectively ECP60MWB (using s, p, d atomic orbitals) and ECP2SDF (using s, p atomic orbitals). The investigated potential energy curves in this representation for the lower electronic states are shown in Fig. 1. We can note that the first low-lying states of the LuF molecule are of triplet multiplicity as in all electronic structures reported previously^27,28,29. The spectroscopic constants T_e, R_e, ω_e, ${\omega }_{e}{x}_{e},$ B_e, α_e, and D_e have been calculated by fitting the potential energy curve values around the minimum of the internuclear distance R_e to a polynomial in terms of R, the significant degree of which is determined from the evaluation of the statistical error for the coefficients, for the ground and the low-lying states of the LuF molecule, as presented in Table 1. In this Table, the comparison of our calculated value of R_e for the ground and the investigated excited electronic states with those given in the literature proved a suitable accuracy with relative difference values that are less than 3%.

Figure 1

The potential energy curves of the low-lying free spin singlet and triplet electronic states of LuF molecule.

Table 1 The spectroscopic constants of the spin-free ^2s+1Ʌ^(+/−) electronic states of LuF molecule.

Similarly, the values of T_e and ω_e also present a good agreement for the six studied states with relative differences close to 12% compared with the literature. Comparing our values with the experimental data^18,19,20,21 for the ground state proves a high accuracy with average percentage errors of 4.6%, 4.5%, 0.6%, and 11.7% for ${\omega }_{e}{x}_{e, }$ B_e, α_e, and D_e respectively. Furthermore, the electronic spectrum of the LuF molecule has been recorded experimentally using a hollow cathode lamp by D’Incan et al.¹⁸ and Effantin et al.¹⁹; they assigned the following notation for the excited states : A¹Σ⁺, B¹Π, D¹Π, E¹Π, and F¹Σ⁺. However, similarly to Assaf et al.²⁹ and Hamadeh et al.²⁷, in our work, the first excited state was found to be a ¹Π state, with a transition energy T_e = 24 973 cm⁻¹. This state doesn’t correspond to the observed A, B, or D systems, but to the reported E¹Π state, where the spectroscopic constants average relative differences with our calculations are 2.0% for Te, 1.5% for Re, 0.3% for ω_e and 2.7% for Be.

We perform our calculations with Spin–Orbit Coupling (S.O.C) effects for the LuF molecule in the Ω^(±) representation for a more accurate description of experimentally observed systems. We use the same basis sets ECP28-MWB with ANO-SO³¹ for the Lu atom and an all-electrons scheme for the F atom³² as presented by Assaf et al.²⁹. The potential energy curves and the splitting energies for the three lowest electronic states X¹Σ⁺ (X¹Σ₀₊), (1)³Π ((1)³Π₀₊, ³Π₀₋), and (1)³Δ ((1)³Δ₁, (1)³Δ₂) are respectively given in Figs. 2 and 3. Also, the accurate spectroscopic constants of the bound Ω states of LuF molecule have been calculated and listed in Table 2 with the same method used for the spin-free states. As shown in Figs. 2 and 3, we find relatively large values of the splitting energies for ³Π (about 375 cm⁻¹) and ³Δ states (about 930 cm⁻¹) indicating the significant effects of the spin–orbit coupling on the electronic states of the LuF molecule. Effantin et al. observed five bands for singlet transition systems: A¹Σ⁺$\to$ X¹Σ⁺, B¹Π $\to$ X¹Σ⁺, D¹Π $\to$ X¹Σ⁺, E¹Π $\to$ X¹Σ⁺, and F¹Σ⁺$\to$ X¹Σ⁺. As discussed by Hamade et al.²⁷ a comparison of the observed levels with those obtained throught our calculations, shows that the upper states A and B are not equivalent to any ¹Σ⁺ and ¹Π. Rather, the transition A¹Σ⁺$\to$ X¹Σ⁺ predicted by Effantin et al. shows a band of $\Delta\Omega =0$, which is attributed to the spin–orbit transition ³Π₀₊$\to$ X¹Σ₀₊. On other hand, the observed transition B¹Σ⁺$\to$ X¹Σ⁺ transition of $\Delta\Omega =1$, is equivalent to that of ³Π₁ $\to$ X¹Σ₀₊. These results led us to confirm that the upper states A¹Σ⁺ and B¹Π are the components ³Π₀₊ and ³Π₁ of (1)³Π state respectively. Consequently, our calculations show a good agreement with the experiment conducted by Effantin et al. for the (1)³Π₀₊ state with a relative discrepancy $\Delta {R}_{e}/{R}_{e}=0.2 \%$, $\Delta {T}_{e}/{T}_{e}=7.2 \%$, $\Delta {\upomega }_{e}/{\omega }_{e}=0.5 \%$, $\Delta {{\omega }_{e}x}_{e}/{{\omega }_{e}x}_{e}=4.6 \%$, and $\Delta {\mathrm{B}}_{e}/{B}_{e}=0.4 \%$.

Figure 2

The potential energy curves of the low-lying spin orbit states of LuF molecule.

Figure 3

Values of the splitting energy of ³Π and ³Δ of the molecule LuF.

Table 2 The spectroscopic constants of the spin–orbit ${\Omega }^{(\pm )}$ electronic states of LuF molecule.

We also performed a rovibrational study of the investigated states using the canonical function approach^33,34,35 and cubic spline interpolation between every two consecutive points of the potential energy curves. Table 3 shows the vibrational energy Ev, the rotational constant ${B}_{v}$, and the centrifugal distortion constant ${D}_{v}$ for the investigated spin-free (Λ^(±) representation) and spin–orbit (Ω^(±) representation) curves, with a comparison with previously published data. Our values for the different vibrational levels of the (X)¹Σ⁺, (1)¹Π, X¹Σ₀₊, and (1)³Π₀₊ states of LuF molecule agree well with the experimental ones^19,21,22, with an relative difference of $0.3 \%\le \Delta {E}_{v}/{E}_{v}\le 2.6 \%$, $0.3 \%\le \Delta {B}_{v}/{B}_{v}\le 5.1 \%$, and $0 \%\le \Delta {\mathrm{D}}_{v}/{D}_{v}\le 21.5 \%$. We compared in Table 3 the ro-vibrational constants values for X¹Σ⁺ that were reported by Effantin et al.¹⁹ with our calculated X¹Σ⁺ and X¹Σ₀₊ values to confirm that this state is of X¹Σ₀₊ nature. In fact, the values of ${B}_{v}$ and ${D}_{v}$ of X¹Σ₀₊ are closer to experimental data than those for X¹Σ⁺, as previously discussed. Generally, our present calculations agree with the available experimental values, which confirms the credibility of our work. In addition, Table 4 shows the ro-vibrational constant values for the remaining low-lying excited states of the LuF molecule. No comparison has been reported for these levels since they are given here for the first time.

Table 3 Values of the eigenvalues, and the rotational constants for the different vibrational levels of the (X)¹Σ⁺, (1)¹Π, X¹Σ₀₊, and (1)³Π₀₊ states of LuF molecule comparison with the experimental values.

Table 4 Values of the eigenvalues, and the rotational constants for the different vibrational levels of the low-lying states (1)¹Δ, (2)¹Σ⁺, (2)¹Δ, (1)³Δ, (1)³Σ⁺, (2)³Δ, and (1)³Π of LuF molecule.

To further verify the truthfulness of our data, we calculated the wavenumbers of the rotational components for P-branch and R-branch for (1)³Π₀₊$-$ X¹Σ₀₊ system as listed in Table 5 by applying the concept of Loomis-Wood diagrams for linear molecules³⁶. This method is based on expressing the rovibrational transitions for the P-branch and R-branch as a polynomial of fourth degree in $m$ with $m=- J$ for the P-branch and $m=J+1$ for the R-branch using the following relation:

$$ \tilde{v} = \tilde{v}_{0} + (B_{{v^{\prime}}} + B_{{v^{\prime\prime}}} )m + (B_{{v^{\prime}}} - B_{{v^{\prime\prime}}} - D_{{v^{\prime}}} + D_{{v^{\prime\prime}}} )m^{2} - 2\left( {D_{{v^{\prime}}} + D_{{v^{\prime\prime}}} } \right)m^{3} + \left( {D_{{v^{\prime\prime}}} - D_{{v^{\prime}}} } \right)m^{4} $$

(1)

where ${\widetilde{v}}_{0}$ is the vibrational transition band center, ${B}_{{v}^{^{\prime}}}$ and ${B}_{{v}^{{^{\prime}}{^{\prime}}}}$ are the rotational constants for the (1)³Π₀₊ (upper vibrational state) and X¹Σ₀₊ (lower vibrational state) respectively, ${D}_{{v}^{^{\prime}}}$ is the centrifugal distortion constant for the upper vibrational state, and ${D}_{{v}^{{^{\prime}}{^{\prime}}}}$ is the centrifugal distortion constant for the lower vibrational state. Comparing these values with those reported by Effantin et al.¹⁹ yields a good agreement, where the percentage relative difference is 7.2% for the P and R branches. At the same time, the constant shift ($\sim 1162 {\mathrm{cm}}^{-1}$), which corresponds to a relative difference of approximately 7.2% among all presented ro-vibrational energy levels shows that there may have been an experimental setting in¹⁹ (possible calibration issues) that would have led to a discrepancy in the vibrational transition band center value ${\widetilde{v}}_{0}$, that propagated to all investigated rotational levels.

Table 5 The theoretical wavenumbers (in cm⁻¹) of the rotation lines of the electronic spectrum of LuF molecule compared with the available experimental values.

The fine structure selection rules state that the transitions Σ−Δ, ΔΩ > 1, and 0⁺−0⁻ are forbidden²⁰. Consequently, we analyzed only the transitions ¹Π − ¹Σ⁺, ¹Π − ¹Δ among the lowest states shown in Fig. 3. More precisely, we present their Transition Dipole Moment (TDM) curves in the considered region $1.5~{\text{{\AA}}} \le R \le 2.12~{\text{{\AA}}}$ in Fig. 4. We then deduced the electronic emission coefficients proposed by Hilborn et al.³⁷, based on the values of the transitions’ dipole moments at the equilibrium positions of the upper states for each electronic transition $\left|{\mu }_{21}\right|$:

$$ {\upomega }_{{{\text{ij}}}} = 2{{ \pi \nu }}_{{{\text{ij}}}} $$

(2)

$$ {\text{A}}_{{{\text{ij}}}} = \frac{{2{\upomega }_{{{\text{ij}}}}^{3} {\upmu }_{{{\text{ij}}}}^{2} }}{{3{\upvarepsilon }_{0} {\text{hc}}^{3} }} $$

(3)

$$ {\upgamma }_{{{\text{cl}}}} = \frac{{{\text{e}}^{2} {\upomega }_{{{\text{ij}}}}^{2} }}{{6{{ \pi \varepsilon }}_{0} {\text{m}}_{{\text{e}}} {\text{c}}^{3} }} $$

(4)

$$ {\text{f}}_{{{\text{ij}}}} = \frac{{ - {\text{ A}}_{{{\text{ij}}}} }}{{3{{ \gamma }}_{{{\text{cl}}}} }} $$

(5)

${\omega }_{21}$ is the emission angular frequency and ${A}_{21}$ is the Einstein coefficients for spontaneous emissions. For the perpendicular transitions with ΔΛ =  ± 1 (or ΔΩ =  ± 1) such as ¹Π − ¹Σ⁺, ¹Π − ¹Δ, the Einstein coefficient A_ij must be divided by an additional factor of two³⁸ depending on the exact definition of ${\mu }_{ij}$. The constants ${\upvarepsilon }_{0}$ and m_e are respectively the vacuum permittivity and the mass of the electron. $\left|{f}_{21}\right|, {\gamma }_{cl}, \mathrm{and} {\upnu }_{\mathrm{ij}}$ are respectively the oscillator strength constant, the classical radiative decay rate of the single-electron oscillator, and the transition frequency between the two states. h and c represent the Planck constant and the speed of light, respectively. The calculated values of these constants for the two transitions (1)³Π₀₊ −X³Σ₀₊ and (1)³Π₀₊−(1)³Δ₁ are given in Table 6. No comparison of these results with literature is available since they are given here for the first time. However, the value of the radiative lifetime τ will be discussed in the next section.

Figure 4

The transition dipole moment curves for the X¹Σ₀₊−(1)³Π₀₊ and (1)³Δ₁ − (1)³Π₀₊ transitions of LuF molecule.

Table 6 The transition dipole moment values at the upper state equilibrium position $\left|{\mu }_{21}\right|$, the emission angular frequency ${\omega }_{21}$, the Einstein spontaneous coefficients ${A}_{21}$, the spontaneous radiative lifetime ${\tau }_{21}$, the classical radiative decay rate of the single-electron oscillator ${\gamma }_{cl}$, and the emission oscillator strength ${f}_{21}$ of some transitions among the doublet states of LuF molecule.

Laser cooling study of LuF molecule

The difference in the values of the equilibrium positions ΔR_e between the ground X¹Σ₀₊ and (1)³Π₀₊ and (1)³Δ₁ states of LuF molecule is minimal, which is an encouraging factor in verifying the laser cooling feasibility for this molecule. The main criteria to keep a closed-loop cycle in a laser cooling process are:

• 1. A highly diagonal Franck–Condon factor (FCF) among the lowest vibrational levels of the ground and considered excited states.

• 2. The absence of an intervening intermediate state, unless it was found possible to include it within the laser cooling scheme.

• 3. A short radiative lifetime of a given transition (in the range of ns to μs) ensures a rapid spontaneous deexcitation of the molecules to provide a high number of cycles per second.

We consequently calculated the FCF values among specific states using LEVEL 11 program³⁹. Our results for the allowed transitions (1)³Π₀₊ −X³Σ₀₊ and (1)³Π₀₊ − (1)³Δ₁ show a diagonal FCF among the first three vibrational levels for the two transitions, as shown in Fig. 5.

Figure 5

The Franck-Condon factor for X¹Σ₀₊ − (1)³Π₀₊ and (1)³Δ₁ − (1)³Π₀₊ transitions of LuF molecule.

To probe how substantially an intermediate state influences a given cooling cycle, one can rely upon the vibrational branching ratio loss $\left( {\gamma = \left( {{\text{A}}_{{\nu^{\prime\prime}\nu^{\prime} - {\text{ Excited}}/{\text{Intermediate}}}} } \right)/({\text{A}}_{{\nu^{\prime\prime}\nu - {\text{ Excited}}/{\text{Ground}}}} )} \right)$ between the considered intermediate state and the excited state involved in the cooling process. Here, A_{ν′′ν′−Excited/Intermediate)} is the Einstein coefficient for transitions between the excited and Intermediate states, and A_{ν′′ν−excited/ground} is that for transitions between the excited and ground-state. If the value of γ is less than 10^–4, then the intermediate state should have a minimal effect on the cooling cycle⁴⁰.

In our case, we follow a similar procedure to understand the implication of the intermediate state (1)³Δ₁, in a cooling loop cycle consisting of the ground state X¹Σ₀₊, and the excited state (1)³Π₀₊. In general, Einstein's coefficient ${A}_{\nu {^{\prime}}\nu }$ among vibrational levels can be written as the following⁴¹:

$$ A_{{\nu^{\prime}\nu }} = (\left( {3.1361891} \right)\left( {10^{ - 7} } \right)(\Delta E)^{3} \left( {\left. {\psi_{{\nu^{\prime}}} } \right|M\left( r \right)\left| {\psi_{\nu } } \right.)^{2} } \right) $$

(6)

where ΔE is the emission frequency (in cm⁻¹), and M(r) is the electronic transition dipole moment between the two electronic states that are considered (in Debye).

Our calculated value of the transition dipole moment, obtained with the Molpro software³⁰, is vertical (given as μ_x, μ_y, and μ_z)). In our calculations, we choose the highest value of these transition matrix elements (μ_x in this case₎. Consequently, we considered the Einstein coefficient ${A}_{\nu {^{\prime}}\nu }$ (Eq. (2)) to be:

$$ A_{{\nu^{\prime}\nu }} = (\left( {3.1361891} \right)\left( {10^{ - 7} } \right)(\Delta E)^{3} \left( {\left. {\psi_{{\nu^{\prime}}} } \right|\mu_{x} \left| {\psi_{\nu } } \right.)^{2} } \right) $$

(7)

The values of the vibrational branching ratio for the first five vibrational levels, which represents the percentage of transition probability between two vibrational levels, are given in Tables 7, 8 and are obtained by using the formula⁴²:

$$ R_{{\nu^{\prime}\nu }} = \frac{{A_{{\nu^{\prime}\nu }} { }}}{{\mathop \sum \nolimits_{\nu } A_{{\nu^{\prime}\nu }} }} $$

(8)

Table 7 The radiative lifetimes τ, and the vibrational branching ratio of the vibrational transitions between the electronic states (1)³Π₀₊−X¹Σ₀₊ of the molecule LuF.

Table 8 The radiative lifetimes τ, and the vibrational branching ratio of the vibrational transitions between the electronic states (1)³Π₀₊−(1)³Δ₁ of the molecule LuF.

Finally, each transition's radiative lifetimes are calculated using τ(s) = $1/{\sum }_{\nu }{A}_{\nu {^{\prime}}\nu }$_, and presented in Table 7 and Table 8. Up to our knowledge, the radiative lifetimes of LuF molecule spin–orbit states are presented here for the first time in the literature. One can notice that the spontaneous emission of the transition (1)³Π₀₊ − X³Σ₀₊ is dominant over that of (1)³Π₀₊ − (1)³Δ₁, where the FCF and the radiative lifetime of the former are f₀₀ = 0.930636 and τ = 3.45 μs while those of the later are f₀₀ = 0.828539 and τ = 0.259 ms. The variation in the radiative lifetime for the two transitions is due to the difference in energy ΔE, which is much more important between the ground state X¹Σ₀₊ and the excited state (1)³Π₀₊, compared to that between the excited state (1)³Π₀₊ and the intermediate state (1)³Δ_1. The comparison of these values for the radiative lifetime with those calculated by using Hilborn emission coefficients³⁷ that are given in Table 6 for the two transitions (1)³Π₀₊ − X³Σ₀₊ and (1)³Π₀₊ − (1)³Δ₁ shows an excellent agreement with the relative differences 6.2% and 1.3% respectively.

Our calculated value for the vibrational branching loss ratio γ  = γ_Δ/γ_Σ = 0.02812 where, γ_Δ and γ_Σ represent the total emission rate of the (1)³Π₀₊ − (1)³Δ₁, and (1)³Π₀₊ − X¹Σ₀₊ transitions, respectively. The order of this ratio is two times higher than the minimum required value of 10^–440. Consequently, the intermediate state (1)³Δ₁ must be considered while setting a convenient laser cooling scheme. At the same time, the forbidden transitions (1)³Π₀₊ − (1)³Π₀₋, (1)³Π₀₊ − (1)³Δ₂, and X¹Σ₀₊ − (1)³Δ₂²⁰ do not disturb the transition X¹Σ₀₊ − (1)³Π₀₊.

Laser cooling schemes with an intermediate state have already been proposed in the literature^43,44. We use the technique proposed by Yuan et al.⁴¹ to include the intermediate state in the laser cooling cycle. To this end, one must calculate the Einstein coefficients for transitions among the three involved electronic states, i.e., X¹Σ₀₊, (1)³Π_0, and (1)³Δ_1. For the two transitions (1)³Π₀₊ − X³Σ₀₊ and (1)³Π₀₊ − (1)³Δ₁, the values of the vibrational branching ratio for the first five vibrational levels are given in Table 9 by using the formulas:

$$ \left( {\text{1}} \right)^{{\text{3}}} \Pi _{{0 + }} - {\text{X}}^{{\text{1}}} \Sigma _{{0 + }} \to R_{{\nu ^{\prime\prime}\nu }} ~ = \frac{{A_{{\nu ^{\prime\prime}\nu }} {\text{~}}}}{{\mathop \sum \nolimits_{\nu } A_{{\nu ^{\prime\prime}\nu }} + \mathop \sum \nolimits_{{\nu ^{\prime}}} A_{{\nu ^{\prime\prime}\nu ^{\prime}}} }} $$

(9)

$$ \left( {1} \right)^{{3}} \Pi_{0 + } - \left( {1} \right)^{{3}} \Sigma_{{1}} \to R^{\prime}_{{\nu^{\prime\prime}v^{\prime}}} = \frac{{A_{{\nu^{\prime\prime}v^{\prime}}} }}{{\mathop \sum \nolimits_{\nu } A_{{\nu^{\prime\prime}\nu }} + \mathop \sum \nolimits_{{\nu^{\prime}}} A_{{\nu^{\prime\prime}\nu^{\prime}}} }} $$

(10)

Table 9 The radiative lifetimes τ, and the vibrational branching ratio of the vibrational transitions between the electronic states (1)³Π₀₊−X¹Σ₀₊ and (1)³Π₀₊−(1)³Δ₁ of the molecule LuF.

${A}_{\nu {^{\prime}}{^{\prime}}\nu }$ and ${A}_{v{^{\prime}}{^{\prime}}v{^{\prime}}}$ are the Einstein coefficients for the transitions (1)³Π₀₊−X¹Σ₀₊ and (1)³Π₀₊−(1)³Δ₁, respectively. For the main optical cycle of the transition (1)³Π₀₊ − X¹Σ₀₊, the number of cycles (N) for photon absorption/emission among vibrational levels (denoted as a, b, c, etc.…) is reciprocal to the total loss:

$$ N = \frac{1}{{1 - \left[ {(R_{00} + R^{\prime}_{00} + R_{0a} + R^{\prime}_{0a } \ldots ) + (R_{0b} + R^{\prime}_{0b} + R_{0c} + R^{\prime}_{0c} \ldots .)\left( {R_{a0} + R^{\prime}_{a0} + R_{aa} + R^{\prime}_{ab} + \ldots .} \right)} \right]}} $$

(11)

Having the value of N, the experimental parameters needed to realize the cooling of a molecule can be obtained. If k_b and h are the Boltzmann and Planck constants, and m is the mass of the molecule, the mathematical expressions of these parameters are⁴⁵:

$$ V = \frac{hN}{{m\lambda_{00} }} $$

(12)

$$ {\text{T}}_{{{\text{ini}}}} = \frac{{mV^{2} }}{{2k_{B} }} $$

(13)

$$ {\text{a}}_{{{\text{max}}}} = \frac{{hN_{e} }}{{N_{tot} m\lambda_{00} \tau }} $$

(14)

$$ L = \frac{{k_{B} T_{ini} }}{{ma_{max} }} $$

(15)

where V and T_ini are the initial velocity and temperature of the molecule, respectively. The maximum acceleration is a_max, and the slowing distance is L. N_e is the number of excited states in the main cycling transition and N_tot is the number of the excited states connected to the ground state plus N_e.

We considered the cooling scheme presented in Fig. 6 to obtain suitable experimental values. The driving lasers are given in solid lines for the two transitions along with their corresponding wavelengths. Dotted lines represent the spontaneous decays with the values of their FCF (f_ν′′ν and f′_{ν′′ν′}) and the vibrational branching ratios R_ν′′ν and R’_{ν′′ν′}. The suggested scheme includes the two transitions X¹Σ₀₊ − (1)³Π₀₊ and (1)³Π₀₊ − (1)³Δ₁. The wavelength of the main cycling laser for the transition X¹Σ₀₊ − (1)³Π₀₊ is λ₀₀ = 336.8 nm, while that of the repump laser is λ₀₁ = 343.8 nm. Since the influence of the intermediate state (1)³Δ₁ cannot be ignored, there is a need for two additional lasers to handle the loss to the vibrational levels for the transition (1)³Π₀₊ − (1)³Δ₁, at λ′₀₀ = 840.2 nm and λ′₀₁ = 802.4 nm.

Figure 6

The laser cooling scheme of X¹Σ₀₊ − (1)³Π₀₊ and (1)³Δ₁ − (1)³Π₀₊ transitions of LuF molecule.

The value of N for this scheme is calculated as the following:

$$ {\text{N}} = \frac{1}{{1 - [\left( {R_{00} + R^{\prime}_{00} + R_{01} + R^{\prime}_{01 } } \right)}} $$

(16)

The corresponding experimental parameters for this scheme are N = 442, L = 1.04 cm,V = 2.73 m/s, T_ini = 86.7 mK, a_max = 358 m/s², and N_e/N_tot = 1/3. The temperatures that can be reached during the cooling process can be obtained by calculating the Doppler limit temperature T_D and the recoil temperature T_r⁴²:

${T}_{D}=\frac{h}{4{k}_{B}\pi \tau }=1108.6 nK$ and ${T}_{r}=\frac{{h}^{2}}{m{k}_{B}{\uplambda }_{00}^{2}}=888 mK$

Suppose T_i is the temperature of the LuF molecules obtained by using laser ablation to produce the atoms of the LuF molecule⁴⁶, typically in the order of T_i = 7000 K. In that case, there is a need for an intermediate process for the molecules to reach the mK regime. This regime can be obtained by collisions between LuF hot molecules of mass M, and cold buffer helium gas of mass m and temperature T_B. After N collision, the temperature T_N of the molecule is given by⁴⁷:

$$ T_{N} = \, T_{B} + \, \left( {T_{i} - \, T_{B} } \right){\text{exp}}\left( { - 2Mm/\left( {M \, + \, m} \right)^{2} } \right)N $$

(17)

We suggest a pre-cooling temperature for LuF molecule T_N = 86.7 mK, corresponding to T_ini of the laser cooling process, and a helium gas temperature T_B = 2 K. From Eq. (6), the number of collisions in the buffer cell equals N = 285. At low temperatures in the buffer gas cell, the collision between the molecules can be ignored. If the density of helium n_He = 5 × 10¹⁴ cm³ and the collision cross-section σ_X−He = 10^–14 cm² the average distance (mean free path) λ between two collisions is given by⁴⁸

$$ \lambda \, = { 1}/\left( {\left( {\sigma_{{{\text{X}} - {\text{He}}}} \cdot {\text{ n}}_{{{\text{He}}}} } \right)\left( {{1} + {\text{M}}/{\text{m}}} \right)^{{{1}/{2}}} } \right) $$

(18)

The corresponding value of λ = 0.0287 cm. Based on the rules of the kinetic theory of ideal gases, the molecules in the buffer gas cell will be thermalized during the time⁴⁸:

$$ t = \mathop \sum \limits_{N = 0}^{98} \frac{\lambda }{{\sqrt {\frac{{3k_{b} }}{M}\left( {T_{i} - T_{B} } \right)e^{ - N/\kappa } + T_{i} } }} $$

(19)

where $\kappa =\frac{{(M+m)}^{2}}{2mM}$. During this short time, t = 0.444 ms in the buffer gas, the LuF molecules will reach a suitable before being sent to the Doppler laser cooling setup.

Conclusion

The adiabatic potential energy curves for the singlet and triplet electronic states of the LuF molecule have been investigated with spin–orbit calculation upon employing the MRCI + Q technique with Davidson correction. The calculation of the spectroscopic constants and the FCF show the candidacy of the LuF molecule for a direct laser cooling between the two states X¹Σ₀₊ and (1)³Π₀₊ with the intermediate state (1)³Δ₁. Since the influence of this state cannot be ignored, the study of the laser cooling of this molecule has been done by taking into consideration the two transitions (1)³Δ₁−(1)³Π₀₊ and X¹Σ₀₊ − (1)³Π₀₊. Correspondingly, a total branching ratio is investigated with a short radiative time (τ = 3.40 μs) along with the slowing distance, the number of cycles (N) for photon absorption/emission, and the Doppler and recoil temperatures. The time needed to thermalize the molecules in the buffer gas cell is calculated, with the number of collisions in this cell between the molecules with the helium atoms and the mean free path between two collisions. This study of the laser cooling of the molecule LuF paves the way to an experimental laser cooling of this molecule.