Laser cooling of InF, InCl and InH with an ab initio study

Abstract

The feasibility of laser cooling InF, InCl and InH is investigated based on ab initio quantum chemistry. To determine their suitability for laser cooling molecules, we have calculated the electronic structures, spectroscopic parameters, transition dipole moments (TDMs), radiative lifetimes, Franck–Condon factors (FCFs) and diode laser excitation wavelengths of InF, InCl and InH. Calculated spectroscopic constants of the first three electronic states for InF, InCl and InH show good agreement with available theoretical and experimental results. InF has highly diagonally distributed FCFs (f₀₀ = 0.961, f₁₁ = 0.909) for the C¹Π → X¹Σ⁺ transition, and the rather short lifetime of the state C¹Π is computed to be 2.77 ns at the lowest vibrational level. Notable is that the ³Π → X¹Σ⁺ transition of InF also has large diagonal FCFs and short lifetimes. Therefore, InCl and InH are not potential laser-cooling candidates because the FCFs of the ¹Π → X¹Σ⁺ transition are off-diagonal. We further propose laser cooling schemes for InF. The present results could provide a promising theoretical reference for further theoretical and experimental research on InF, InCl and InH.

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1. Introduction

The production of ultracold diatomic molecules has drawn considerable interest for several years because of their wide range of prospective applications, for example, quantum computations,¹ controlling chemistry² and precision measurement.^3–5 Recently, laser cooling of diatomic molecules SrF,^6,7 YO,⁸ CaF⁹ and KRb¹⁰ have been experimentally performed. These successful laser cooling experiments have initiated a search for more molecules since more than 90 elements in the periodic table possibly form more than 4000 different diatomic molecules. The potential laser cooling candidates must have highly diagonal Franck–Condon factors (FCFs) and shorter lifetimes (i.e. high spontaneous-emission rates). Highly diagonal FCFs limit the number of lasers required to keep the molecule in a quasi-closed-loop cooling cycle and rapid laser cooling demands a shorter lifetime.

The A²Π → X²Σ⁺ transition suggested by the experiments^6–9 for SrF, YO and CaF has been theoretically used to establish the laser cooling schemes for RaF,⁴ BeF,¹¹ MgF¹² and MH¹³ (M = Be, Mg, Ca, Sr, and Ba). In 2015, laser cooling of LiBe molecule based on the B²Π → X²Σ⁺ transition has been reported by You et al.¹⁴ Interestingly, AlX (X = H,¹⁵ F,¹⁵ Cl¹⁶ and Br¹⁶) and BBr¹⁷/BCl¹⁷ molecules with an intervening state a³Π existed between the A¹Π and X¹Σ⁺ states have also been identified as potential laser-cooling candidates. The laser cooling scheme using the A¹Π → X¹Σ⁺ and a³Π → X¹Σ⁺ transitions was theoretically suggested. In our previous works, theoretical optical schemes have been predicted for AlCl¹⁶/AlBr¹⁶ and BBr¹⁷/BCl.¹⁷ Li et al.¹⁸ propose to utilize the ¹Σ–¹Σ electronic transition system for direct laser cooling of AgH and AgD molecules.

The boron group molecules are of persistent interest in the investigation of cold molecules. BH¹⁹/BCl¹⁷/BBr,¹⁷ AlH¹⁵/AlF¹⁵/AlCl¹⁶/AlBr¹⁶ and GaF¹⁹ were identified as the promising laser-cooling targets theoretically. However, the study on laser cooling of indium diatomics is very limited. On the basis of previous studies, we believe InF, InCl and InH molecules are possible laser-cooling candidates. In this paper, we focus on the theoretical studies of the laser cooling of InF, InCl and InH. The electronic structure and spectroscopic properties of the low-lying electronic states of InF were given by Banerjee et al.²⁰ from the relativistic configuration interaction calculations. They also determined the lifetime of the state C¹Π was about 2.3 ns at the lowest vibrational level. Zou et al.²¹ obtained the potential energy curves and the spectroscopic constants on the ground and low-lying excited states of InCl using all-electron relativistic calculations. The theoretical calculations have been performed on the InH molecule based on multi-reference configuration interaction plus Davidson corrections method.²² Leininger et al.²³ performed the calculations for spectroscopic constants of InH, InF, and InCl at the self-consistent-field and correlated levels. Several experimental results on the spectroscopic parameters were available in the literature for InF, InCl and InH.

Though InF, InCl and InH have been extensively investigated in previous works, as mentioned above, a systematic study of laser cooling of InF, InCl and InH is not available in the literature to the best of our knowledge. The main objective of this paper is to identify whether InF, InCl and InH are potential laser-cooling candidates and briefly design laser-cooling schemes. To determine their suitability for laser cooling molecules, we have calculated the electronic structures, spectroscopic parameters, TDMs, radiative lifetimes, FCFs and diode laser excitation wavelengths of InF, InCl and InH. Since in this work we mean to focus on the vibrational cooling, the rotational degrees of freedom can be neglected in the first approximation. Here, the C¹Π → X¹Σ⁺ transition for InF/InCl and A¹Π → X¹Σ⁺ transition for InH are the main transitions of laser cooling and the transition from intervening state ³Π to the ground state X¹Σ⁺ is spin forbidden.

The paper is organized as follows. Section 2 details the theoretical methods and basis sets used in the calculations. Section 3 presents the results and discussions of the data, outlining the schemes for direct laser cooling. Section 4 provides a conclusion for this work.

2. Computational details

All the ab initio calculations are performed with the MOLPRO package.²⁴ It should be noted that only the subgroup C_2v point group symmetry can be adopted for InF, InCl and InH in the package. The electronic structures for the X¹Σ⁺, C¹Π and ³Π states of InF/InCl and the X¹Σ⁺, A¹Π and a³Π states of InH are calculated using the complete active space self-consistent field method (CASSCF),^25,26 which is followed by MRCI plus Davidson corrections (MRCI+Q)^27–29 calculations. The scalar relativistic effect is taken into account using the Douglas–Kroll–Hess (DHK)^30,31 transformation of the relativistic Hamiltonian. The spectroscopic constants, including equilibrium bond distance (R_e), dissociation energy (D_e), harmonic frequency (w_e), anharmonic vibrational frequency (w_eχ_e), rotational constant (B_e) and electronic transition energy (T_e) are evaluated by LEVEL 8.0 program.³² The TDMs of InF, InCl and InH are computed by taking the expectation and transition values using the MRCI wave functions. With the potential energy curves (PECs) and TDMs of different electronic states, we have obtained the FCFs and radiative lifetimes of the various vibrational levels from LEVEL 8.0 program.

For the indium atom, being considerably heavy, we use the small-core scalar relativistic effective core potential ECP28MDF³³ together with the corresponding valence basis sets. The 4s²4p⁶4d¹⁰5s²5p¹ electrons of the In atom are kept in the valence space, while the remaining core electrons are replaced by the above mentioned pseudopotentials. The correlation consistent polarized valence quintuple zeta aug-cc-pV5Z (=AV5Z)³⁴ is chosen for H. Meanwhile, the AVQZ³⁵ all-electron basis sets are employed for F and Cl. That is to say, for InF, nineteen molecule orbitals (MOs) are put into active space, including 9a₁, 4b₁, 4b₂, 1a₂ symmetry (9,4,4,1), which correspond to the 1s2s2p shells of the F atom and 4s4p4d5s5p shells of the In atom; for InCl, twenty two MOs are selected as the active space, including 11a₁, 5b₁, 5b₂, 1a₂ symmetry (11,5,5,1), which correspond to the Cl 1s2s2p3s3p and In 4s4p4d5s5p shells; for InH, the active space is (7,3,3,1), which correspond with the H 1s and In 4s4p4d5s5p.

The main transitions of laser cooling for InF, InCl and InH are the ¹Π → X¹Σ⁺ transitions which are singlet states. Moreover, the excited state decays exclusively on the ¹Π → X¹Σ⁺ transition because of the selection rules for the change in parity and angular momentum in an electric dipole transition. Besides, the transition from intervening state ³Π to the ground state X¹Σ⁺ is spin forbidden. Therefore, spin–orbit coupling effects are not considered in the calculations. Previous works on laser cooling of MH¹³ (M = Be, Mg, Ca, Sr, and Ba) and GaF¹⁹ also indicate that the influence of spin–orbit coupling on the spectroscopic properties is generally weak.

3. Results and discussions

3.1 PECs and spectroscopic constants

The calculated PECs of X¹Σ⁺, ¹Π and ³Π states of InF, InCl and InH acquired at the MRCI level are shown in Fig. 1. The PECs of X¹Σ⁺ and ³Π states look surprisingly similar, implying that highly diagonal FCFs for ³Π → X¹Σ⁺ transitions are possible. In Table 1, we report the corresponding spectroscopic constants of InF/InCl for the X¹Σ⁺, C¹Π and ³Π states and InH for the X¹Σ⁺, A¹Π and a³Π states. The experimental³⁶ and theoretical^20–23 values for these states are also listed in Table 1 for comparison.

Fig. 1 Potential energy curves of the first three electronic states of InF (a), InCl (b) and InH (c) at the MRCI level of theory.

Table 1 Spectroscopic constants of X¹Σ⁺, C¹Π and ³Π states for InF/InCl and X¹Σ⁺, A¹Π and a³Π states for InH calculated at the MRCI level of theory

Molecule	States	T e	R e (Å)	w e (cm^−1)	w e χ e (cm^−1)	B e	D e	Ref.
InF	X^1Σ^+	0	1.987	550.04	2.36	0.2619	5.45	This work
		0	1.985	535.35	—	—	5.48	Exp.^36
		0	2.010	564.00	—	—	5.37	Ref. 20
	C^1Π	43 050	1.964	467.05	16.51	0.2673	0.44	This work
		42 809	1.966	463.90	—	—	—	Exp.^36
		42 255	2.000	460.00	—	—	—	Ref. 20
	^3Π	30 463	1.953	588.93	3.13	0.2727	1.94	This work
		—	1.946	575.25	—	—	—	Exp.^36
		28 145	1.980	590.00	—	—	—	Ref. 20
InCl	X^1Σ^+	0	2.398	325.51	1.03	0.1094	4.70	This work
		—	2.401	317.40	—	—	4.68	Exp.^36
		0	2.431	313.70	1.38	—	—	Ref. 21
	C^1Π	37 478	2.455	210.06	14.07	0.1334	0.06	This work
		37 484	2.473	177.30	12.60	—	—	Exp.^36
		37 757	2.577	106.90	2.33	—	—	Ref. 21
	^3Π	27 802	2.335	350.00	1.90	0.1153	1.23	This work
		—	2.333	340.30	—	—	—	Exp.^36
		27 871	2.340	325.20	1.16	—	—	Ref. 21
InH	X^1Σ^+	0	1.821	1537.00	24.83	4.8703	2.88	This work
		0	1.838	1476.00	25.61	4.9945	2.75	Exp.^36
		—	1.867	1434.00	—	—	2.65	Ref. 23
	A^1Π	22 500	1.862	284.30	—	—	—	This work
		22 570	1.954	—	—	—	—	Ref. 22
	a^3Π	16 766	1.752	1523.74	61.55	5.389	—	This work
		16 303	1.793	1495.85	63.62	5.270	—	Ref. 22

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For the ground state X¹Σ⁺ of InF, the calculated R_e and w_e results are only 0.002 Å and 14.69 cm⁻¹ larger, while the D_e result is only 0.03 cm⁻¹ smaller than the experimental data.³⁶ The corresponding percentage errors in R_e, w_e and D_e are 0.10%, 3.08% and 0.55%, respectively. Banerjee et al.²⁰ provided even larger R_e (1.26%) and w_e (5.35%), an even smaller D_e (2.00%) for the X¹Σ⁺ of InF compared to the experimental data.³⁶ The first excited state of InF is ³Π which lies well above the ground state. Our calculated R_e and w_e of the ³Π state are in remarkably good agreement with experiment,³⁶ and the relative errors are only 0.36% and 2.38%, respectively. While the percentage errors in R_e and w_e given by Banerjee et al.²⁰ are 1.74% and 2.56%, respectively. The present calculation yields a T_e value for the ³Π of 30 463 cm⁻¹, and the observed A³Π₀₊, B³Π₁–X¹Σ⁺ transitions³⁶ take place in the range 30 400–31 300 cm⁻¹. The relativistic configuration interaction calculations by Banerjee et al.²⁰ yielded a T_e value for the ³Π of 28 145 cm⁻¹ which was not within the observed range. It is also encouraging to see that the present values of R_e, w_e and T_e for the C¹Π state are 1.964 Å, 467.05 cm⁻¹ and 43 050 cm⁻¹, which are in very good agreement with the experimental data 1.966 Å, 463.90 cm⁻¹ and 42 809 cm⁻¹ and the theoretical data 2000 Å, 460.00 cm⁻¹ and 42 255 cm⁻¹.

For InCl, good agreement is found between our results and experimental data. The percentage errors of R_e, w_e and D_e for the ground state X¹Σ⁺ are 0.12%, 2.56% and 0.43%, respectively. The equilibrium bond distance R_e by the available all-electron relativistic calculations²¹ is 2.431 Å, and the corresponding percentage error is 1.25%. The present w_eχ_e of the X¹Σ⁺ state is 1.03 cm⁻¹, which is slightly smaller than the previous theoretical value²¹ 1.38 cm⁻¹. For the C¹Π, our present results compare very well with the experimental data³⁶ (T_e ∼ 6 cm⁻¹, R_e ∼ 0.018 Å, w_e ∼ 32.76 cm⁻¹, w_eχ_e ∼ 1.47 cm⁻¹). While the differences between the previous theoretical value²¹ and experimental data are bigger (T_e ∼ 273 cm⁻¹, R_e ∼ 0.104 Å, w_e ∼ 70.40 cm⁻¹, w_eχ_e ∼ 9.27 cm⁻¹). Whereas for the ³Π, our spectroscopic constants calculated at MRCI level are in reasonable agreement with the available all-electron relativistic calculations²¹ (T_e ∼ 69 cm⁻¹, R_e ∼ 0.005 Å, w_e ∼ 24.8 cm⁻¹, w_eχ_e ∼ 0.74 cm⁻¹). In the case of R_e and w_e, our results (2.335 Å, 350.0 cm⁻¹) are in good agreement with experiment³⁶ (2.333 Å, 340.3 cm⁻¹). The complete active space self-consistent field method (CASSCF) is used in both the previous work²¹ and our work. So the effects of the different active spaces (this work: 11,5,5,1) and (ref. 20: 6,3,3,1) on the spectroscopic constants of InCl can be seen from Table 1. It is obvious that the calculated spectroscopic constants of InCl for active space (11,5,5,1) are more close to the experimental values.

Concerning the X¹Σ⁺ of InH, our calculated parameters R_e, w_e, w_eχ_e, B_e and D_e are also in very good agreement with experiment,³⁶ and the relative errors are 0.92%, 4.13%, 3.05%, 2.49% and 4.73%, respectively. While the percentage errors in R_e, w_e and D_e given by Leininger et al.²³ are 1.58%, 2.85% and 3.64%, respectively. For the A¹Π and a³Π states of InH, Table 1 shows that our present results accord with existing theoretical calculations.^22,23 For example, the A¹Π and a³Π states are located at 22 500 cm⁻¹ and 16 766 cm⁻¹ in our calculations, 22 570 cm⁻¹ and 16 303 cm⁻¹ were reported by Zhang et al.²²

In summary, our present work yield spectroscopic parameters of InF, InCl and InH for the first three electronic states that agree well with previous experimental and theoretical results.

3.2 TDM curves

Because the transition ³Π → X¹Σ⁺ is spin forbidden at spin-free level, we only obtain the TDMs of the ¹Π → X¹Σ⁺ transition. Fig. 2 shows the computed TDMs of the dipole-allowed transitions C¹Π → X¹Σ⁺ for InF/InCl and A¹Π → X¹Σ⁺ for InH as functions of the internuclear distance. We note that the ¹Π → X¹Σ⁺ transitions for InF, InCl and InH are strong. As seen in Fig. 2, the TDM curves of InF, InCl and InH demonstrate similar behavior. The magnitude gradually increases as internuclear distance R increases, reaches a maximum (InF: 1.80 a.u., InCl: 1.40 a.u., InH: 0.92 a.u.), and drops thereafter. The TDMs of InF, InCl and InH all trend to zero at around 4.0 Å, 3.5 Å and 3.0 Å, respectively.

Fig. 2 TDMs for the C¹Π → X¹Σ⁺ transitions of InF/InCl and A¹Π → X¹Σ⁺ transitions of InH at MRCI level.

3.3 FCFs, spontaneous radiative lifetime and radiative width

3.3.1 InF molecule. The potential laser cooling candidates must have highly diagonal Franck–Condon factors (FCFs). The calculated FCF data of the C¹Π → X¹Σ⁺ transition have been tabulated in Table 2. It is clear that InF has highly diagonal FCFs f_ν′ν (f₀₀ = 0.961 and f₁₁ = 0.909) for the C¹Π(ν′) → X¹Σ⁺(ν) transition. Most FCFs of experimental cooling molecules are greater than 0.9, as listed in the previous literature.^6–10 The predicted f₀₀ of 0.961 for InF is greater than 0.9, thus it is sufficiently large to be potentially viable for cooling. Of course, a larger FCF has significant benefits for limiting the number of lasers required to keep the molecule in a quasi-closed-loop cooling cycle. To visually demonstrate the distributions of FCFs f_ν′ν for the different vibrational states of the C¹Π(ν′) → X¹Σ⁺(ν) transition, we have plotted the cooling transition C¹Π(ν′) → X¹Σ⁺(ν) between 0 ≦ ν′ ≦ 6 and 0 ≦ ν ≦ 6 in Fig. 3. The FCFs for the ³Π → X¹Σ⁺ transition are also presented in Table 2. The ³Π → X¹Σ⁺ transition also have large diagonal FCFs, although the calculated f₀₀ of 0.807 for the ³Π → X¹Σ⁺ transition is smaller than that predicted for the C¹Π → X¹Σ⁺ transition (f₀₀ = 0.961).

Table 2 The calculated FCFs f_ν′ν for C¹Π(ν′) → X¹Σ⁺(ν) and ³Π(ν′) → X¹Σ⁺(ν) transitions of InF

Molecule	Transition	f 00, f10	f 01, f11	f 02, f12	f 03, f13
InF	C^1Π → X^1Σ^+	0.9606, 0.0377	0.0326, 0.9085	0.0062, 0.0369	0.0004, 0.0155
	^3Π → X^1Σ^+	0.8068, 0.1788	0.1670, 0.4997	0.0231, 0.2542	0.0027, 0.0565

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Fig. 3 The calculated FCFs of InF for the lowest vibrational levels of the cooling transition C¹Π → X¹Σ⁺.

Except for large FCFs, the potential laser cooling candidates must also have sufficiently short lifetimes. In Table 3, we have reported the computed lifetimes for the C¹Π → X¹Σ⁺ transition. Our computed lifetimes can be compared with those of the results predicted by Banerjee et al.²⁰ The radiative lifetimes of the C¹Π → X¹Σ⁺ transition are computed to be 2.77–3.41 ns for the first five vibrational levels of InF, which shows that the C¹Π of InF is a rather short lifetime state. At the same time, the radiative width for the C¹Π → X¹Σ⁺ transition are computed to be 19.15–15.53 cm⁻¹ for the first five vibrational levels of InF. The radiative lifetimes (7.58–8.34 ns) of the ³Π are a little longer lived than these of the electronic state C¹Π, which shows that the ³Π of InF is also a rather short lifetime state. As shown in Table 3, the radiative lifetimes show a slight increase with increasing vibrational level, and the radiative widths show a slight decrease with increasing vibrational level.

Table 3 Estimated radiative lifetimes (ns) and radiative width (cm⁻¹) (in italics) (theoretical values obtained in brackets)

Molecule	Transition	ν′ = 0	1	2	3	4
a Characteristic base 10 given parenthetically.
InF	C^1Π → X^1Σ^+	2.77	2.88	3.02	3.20	3.41
		(2.32)^a	(2.44)^a	(2.61)^a	—	—
		19.15(−4)	18.38(−4)	17.53(−4)	16.55(−4)	15.53(−4)
	^3Π → X^1Σ^+	7.58	7.74	7.94	8.12	8.34
		6.99(−4)	6.85(−4)	6.68(−4)	6.52(−4)	6.35(−4)

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3.3.2 InCl molecule. For the C¹Π(ν′) → X¹Σ⁺(ν) transition of InCl, the diagonal term f₁₁(0.07) is obtained, and the off-diagonal term f₁₀(0.29), f₁₂(0.16), f₁₃(0.18) are also calculated (see Table 4). For the ³Π → X¹Σ⁺ transition of InCl, off-diagonal term f₁₀(0.29), f₁₂(0.39), f₁₃(0.15) are larger than the diagonal term f₁₁(0.14) (see Table 4). It is clear that InCl does not have highly diagonal FCFs for the C¹Π → X¹Σ⁺ and ³Π → X¹Σ⁺ transitions. Besides, the present FCFs f₀₀(0.61) for the C¹Π → X¹Σ⁺ transition and f₀₀(0.60) for the ³Π → X¹Σ⁺ transition are not large enough to be potentially viable for cooling. The relative probabilities from C¹Π(ν′ = 0) to X¹Σ⁺(ν) are governed by the FCFs which are approximately 61% for ν = 0, 23% for ν = 1, 10% for ν = 2, 4% for ν = 3, 1.5% for ν = 4, 0.5% for ν = 5, 0.2% for ν = 6, 0.07% for ν = 7, and negligibly small for all ν > 7. To limit the inefficiency, or loss of InCl molecules in the cooling cycle, eight cooling lasers are required. Experimentally, cycling transitions requiring one or two repump lasers are common in atomic systems. Here the number of lasers required is not practical. There are similarities to the laser cooling of InCl on the ³Π → X¹Σ⁺ transition. So we identify that InCl is not a potential laser-cooling candidate. Thus the radiative lifetimes and diode laser excitation wavelengths of InCl do not need to be discussed in the following section.

Table 4 The calculated FCFs f_ν′ν for C¹Π(ν′) → X¹Σ⁺(ν) and ³Π(ν′) → X¹Σ⁺(ν) transitions of InCl

Molecule	Transition	f 00, f10	f 01, f11	f 02, f12	f 03, f13
InCl	C^1Π → X^1Σ^+	0.6079, 0.2912	0.2293, 0.0716	0.1010, 0.1620	0.0385, 0.1766
	^3Π → X^1Σ^+	0.5957, 0.2907	0.3282, 0.1437	0.0691, 0.3913	0.0067, 0.1521

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3.3.3 InH molecule. As listed in Table 5, the A¹Π state is not suitable for laser cooling because of the off-diagonal FCFs and the small diagonal FCF f₀₀(0.790) for the A¹Π → X¹Σ⁺ transition. Fig. 4 shows the FCFs of the a³Π → X¹Σ⁺ transition are highly diagonal. The present FCF of 0.918 for the a³Π → X¹Σ⁺ transition of InH is still large enough for laser cooling. However, this a³Π → X¹Σ⁺ transition is very weak because it is spin-forbidden. Because of the poor linestrength of the a³Π → X¹Σ⁺ transition, the InH molecule must have been precooled using the A¹Π → X¹Σ⁺ transition before the a³Π → X¹Σ⁺ transition could be adopted for laser cooling of InH. It is similar to the laser cooling of Ca, where cooling on the weak ¹S → ¹P transition must followed by the strong ¹S → ³P transition. The problem with the laser cooling scheme using the a³Π → X¹Σ⁺ transition is that the InH molecule could not be precooled using the A¹Π → X¹Σ⁺ transition. So InH is not a promising laser-cooling candidate. Thus the radiative lifetimes and diode laser excitation wavelengths of InH do not need to be discussed in the following section.

Table 5 The calculated FCFs f_ν′ν for A¹Π(ν′) → X¹Σ⁺(ν) and a³Π(ν′) → X¹Σ⁺(ν) transitions of InCl

Molecule	Transition	f 00, f10	f 01, f11	f 02, f12	f 03, f13
InH	A^1Π → X^1Σ^+	0.7897, 0.0573	0.1172, 0.0169	0.0604, 0.0372	0.0181, 0.0715
	a^3Π → X^1Σ^+	0.9176, 0.0672	0.0809, 0.8217	0.0012, 0.0969	0.0001, 0.0014

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Fig. 4 The calculated FCFs of InH for the lowest vibrational levels of the cooling transition a³Π → X¹Σ⁺.

3.4 Laser cooling scheme

We have established the laser cooling schemes for InF, and Fig. 5 shows the proposed scheme. The corresponding computed radiative wavelengths are collected in Table 6. For InF, the main cycle is the X¹Σ⁺(ν = 0) → C¹Π(ν′ = 0) transition at wavelength λ₀₀ = 232.4 nm. As shown in Fig. 5(a), there is a small probability of decay to the X¹Σ⁺(ν = 1) state (≈3%), an even smaller probability of decay to the X¹Σ⁺(ν = 2) state (≈0.6%), and a negligible probability of decay to the X¹Σ⁺(ν ≥ 3) state (<10⁻⁴). To enhance the cooling effect, we add the X¹Σ⁺(ν = 1) → C¹Π(ν′ = 0) transition as the first vibrational pump and the X¹Σ⁺(ν = 2) → C¹Π(ν′ = 1) transition for the second pump. With this scheme, two repumping lasers λ₁₀ = 229.8 nm and λ₂₁ = 230.2 nm are required. The required cooling wavelengths of λ₀₀ = 232.4 nm, λ₁₀ = 229.8 nm and λ₂₁ = 230.2 nm can be generated by a frequency quadrupled Ti:sapphire laser (189–235 nm). One problem with the above laser cooling scheme, is the presence of the intervening ³Π state between the two states (X¹Σ⁺ and C¹Π). Notable, the ³Π → X¹Σ⁺ transition also have large diagonal FCFs and short lifetimes. Thus, this state ³Π could also be another option for laser cooling. We propose a four laser cyclic system involving the ν = 0–3 of the X¹Σ⁺ state and ν′ = 0, 1 of the ³Π state. The main cycling laser at the wavelength of 328.1 nm can drive the X¹Σ⁺(ν = 0) → ³Π(ν′ = 0) transition. Similar to the case in Fig. 5(a), a pair of repumping lasers with λ₁₀ = 321.9 nm and λ₂₁ = 321.6 nm are required to enhance the cooling effect. A third repumping laser with λ₃₁ = 315.8 nm may be required on this scheme due to the non-negligible ν′ = 1 → ν = 3 transition (FCF = 0.0565). The required cooling wavelengths on this scheme are located in the ultraviolet light (UVA) range (400–320 (or 315) nm). There are similarities here to the laser cooling of AlF. Wells and Lane¹⁵ have studied on laser cooling of AlF molecules, and the X¹Σ⁺ → C¹Π and X¹Σ⁺ → ³Π transitions of AlF had strongly FCFs. So for InF, the cooling on a strong C¹Π → X¹Σ⁺ transition may be followed by further cooling on the X¹Σ⁺ → ³Π transition, which is similar to the laser cooling of AlF. The relatively long lifetimes of the ³Π state can be exploited to reach a much lower Doppler temperature than possible on the X¹Σ⁺ → C¹Π transition.

Fig. 5 Proposed laser cooling schemes for InF using (a) the C¹Π(ν′) → X¹Σ⁺(ν) (solid red) and (b) the ³Π(ν′) → X¹Σ⁺(ν) (solid red) transition. The decay pathways with calculated f_ν′ν are shown as dotted line.

Table 6 The calculated wavelength λ_ν′ν

Molecule	λ 00 (nm)	λ 10 (nm)	λ 21 (nm)	λ 31 (nm)	Transition
InF	232.4	229.8	230.2	227.8	C^1Π → X^1Σ^+
	328.1	321.9	321.6	315.8	^3Π → X^1Σ^+
InH	596.9	549.5	556.3	525.0	a^3Π → X^1Σ^+

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Compared with other potential laser-cooling candidates, diagonal FCFs f₀₀ of InF (f₀₀ = 0.961) is slightly larger than that predicted for BeF (f₀₀ = 0.897), LiRb (f₀₀ = 0.872), InH (f₀₀ = 0.918) and KRb (f₀₀ = 0.947). Experimentally, KRb is a typical laser-cooled molecule. Radiative lifetime τ(ν′ = 0) = 2.8 ns of InF is shorter than the one of BeF (7.9 ns), LiRb (102.6 ns), InH (205.7 ns) and KRb (229.3 ns). A shorter τ of the transition could produce a strong Doppler force, which will produce rapid laser cooling. We have identified InF is a more promising laser-cooling candidate. Not only it has bigger FCFs and shorter radiative lifetime, it also appears to attain a lower Doppler temperature on the X¹Σ⁺ → C¹Π transition followed by the weak X¹Σ⁺ → ³Π transition. Generally, it is true that the Doppler-limit temperature is inversely proportional to the excited state lifetime. As shown in Table 3, the lifetime (7.58 ns) of the ³Π state is about three times larger than that of the C¹Π state. However, the lifetime (7.58 ns) of InF is still shorter than that of KRb (229.3 ns). The short lifetime is sufficient to produce large spontaneous scattering forces.

4. Conclusions

The PECs of X¹Σ⁺, ¹Π and ³Π states of InF, InCl and InH are acquired at the MRCI level. Our present work yield spectroscopic constants (R_e, D_e, w_e, w_eχ_e, B_e, T_e) of InF, InCl and InH for the first three electronic states that agree well with previous experimental and theoretical results. The TDMs of the dipole-allowed transitions C¹Π → X¹Σ⁺ for InF/InCl and A¹Π → X¹Σ⁺ for InH are obtained. Using the FCFs and radiative lifetimes obtained, we have identified InF is a promising candidate for laser cooling. Because of its off-diagonal FCFs, InCl is not a potential laser-cooling candidate. For InH, the A¹Π state has off-diagonal FCFs and the FCFs of the a³Π → X¹Σ⁺ transition are highly diagonal. However, this a³Π → X¹Σ⁺ transition is very weak because it is spin-forbidden. The problem with the laser cooling scheme using the a³Π → X¹Σ⁺ transition is that the InH molecule could not be precooled using the A¹Π → X¹Σ⁺ transition. So InH is not a promising laser-cooling candidate. The proposed laser cooling schemes of InF that drive the X¹Σ⁺ → C¹Π transition use three laser wavelength. Furthermore, the ³Π → X¹Σ⁺ transition of InF is also strongly diagonal and the lifetimes of the excited ³Π state of InF is relatively long. So for InF, the X¹Σ⁺ → C¹Π transition may be followed by the weak X¹Σ⁺ → ³Π transition to obtain a lower Doppler temperature. All of these three required laser-cooling wavelengths are in the visible range.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11704052).

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