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Theoretical Study of The Feasibility of Laser Cooling The Mg Cl Molecule Including Hyperfine Structure and Branching Ratios Quan-Shun Yang, Shichang Li, You Yu, and Tao Gao J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b11047 • Publication Date (Web): 02 Mar 2018 Downloaded from http://pubs.acs.org on March 4, 2018

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The Journal of Physical Chemistry

24

Theoretical Study of The Feasibility of Laser Cooling The Mg35Cl Molecule Including Hyperfine Structure and Branching Ratios

Quan-Shun Yang, Shi-Chang Li, You Yu, Tao Gao∗ Institute of Atomic and Molecular Physics, Sichuan University, 610065, Chengdu, China College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu 610225, China Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Sichuan University, Chengdu, 610064, China

Abstract The possibility of laser cooling the 24Mg35Cl molecule is investigated using the electronic, rovibrational and hyperfine structure. Twelve low-lying Λ-S electronic states of the 24

Mg35Cl molecule have been calculated at the multi-reference configuration interaction

level of theory. The spin-orbit coupling effects are taken into account in the electronic structure calculations. Spectroscopic constants agree well with previously obtained theoretical and experimental values. Based on the potential energy curves and transition dipole moments, the highly diagonally distributed Franck-Condon factors for the A2Π → X2Σ+ transition and short radiative lifetime of the A2Π state are determined. Then, employing a quantum effective Hamiltonian approach, we investigate the hyperfine manifolds of the X2Σ+ state and obtain the zero-field hyperfine spectrum with the errors relative to the experimental data not exceeding 8 kHz ∼ 20 kHz. Finally, we design a laser cooling scheme with one cooling main laser beam and two repumping laser beams with modulated sidebands, which is sufficient for the implementation of efficient laser slowing and cooling of the 24Mg35Cl molecule. Moreover, it is important to note that the dissociation energy (2.2593 eV) of the B2Σ+ state is obtained for the first time at the multi-

∗

Corresponding authors. Tel.:+86 028 85405234. E-mail: [email protected] (T. Gao) 1 ACS Paragon Plus Environment

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reference configuration interaction level. We hope that this can provide a helpful reference for experimental observation.

1. Introduction The combination of laser cooling and trapping has produced ultracold temperature atoms, resulting in revolutionary advances in atomic clocks1, quantum information processing2 and simulation of condensed-matter systems3,4. With their additional types of internal motion (such as vibration and rotation), ultracold polar molecules yield even richer phenomena and applications in quantum simulations of condensed-matter system5, quantum computation6 and chemical dynamics7,8. However, cooling of molecules to cold and ultracold temperature has not been realized before 2010. One of the main difficulties in cooling molecules is the lack of a closed two-level transition cycle. This is due to the absence of rigorous selection rules controlling the branching ratios for the decay of an excited electronic state into different vibrational levels of the ground state. Fortunately, the direct laser cooling of SrF9 has paved the way to a new approach to this problem. In this case, the branching ratios are governed by the molecule’s Franck–Condon factors (FCFs). The completion realization of a full three-dimensional MOT for SrF was employed10, and a further improvement based on a MOT beam polarization scheme was derived from a detailed calculation by Tarbutt11. Then, one- and two-dimensional transverse laser cooling and MOT of YO12 and the longitudinal laser cooling of CaF13 molecule was successfully demonstrated experimentally. Finally, the YO molecule was cooled to 2 mK via Doppler cooling with the addition of magneto-optical trapping forces. Recently, the molecules had been cooled to below 1 mk by some group14-18. As early as 2004, Di Rosa presented a brief survey19 of candidate molecules for laser cooling. Therefore, theoretically, a large number of polar molecules20-23 have also been investigated for the feasibility of their use in laser cooling. As with atomic systems19, a potential laser cooling candidate molecule should meet certain conditions： a) highly diagonal Franck-Condon factors (FCFs) that limit the vibrational branching and the number of lasers, b) the excited state should exhibit a short radiative lifetime for a high photon scattering rate, and c) the absence of an intermediate electronic state between the

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ground state and the first excited state or the intermediate state having no effect on the laser-cooling cycle. The properties of 24Mg35Cl have been studied theoretically24-27 and experimentally2835

by many researchers. For instance, the electronic structures for low-lying states of

24

Mg35Cl have been elucidated through the first-order configuration interaction (FOCI)

and second-order configuration interaction (SOCI) method by Yarkony et al.26 Then, the rotational spectra for the X2Σ+ state of the 24Mg35Cl molecule in various isotropic forms and vibrational states were observed in the 130-290 GHz frequency range by Bogey et al.29 Hirao et al.30 studied the A2Π–X2Σ+ emission band system of the 24Mg35Cl molecule by high-resolution Fourier transform spectroscopy in 2002. The emission bands systems of the 24Mg35Cl and 24Mg37Cl molecules were further studied by high resolution Fourier Transform Spectroscopy (FTS) by Gutterres et al.31 We exploit

24

Mg35Cl for the

following reasons: (i) Recently, laser cooling of 24Mg35Cl has been studied theoretically20. However, some problems are still not solved well, such as the relevant hyperfine structure (HFS) of

24

Mg35Cl, the modulating frequency covering all of the hyperfine

energy levels by the proposed sideband frequency distributions. (ii) Cold molecules have found great utility in precision measurements36. Thus, in the present work, we seek to perform a systematic theoretical investigation of the feasibility of laser cooling the 24

Mg35Cl molecule including hyperfine structure and branching ratios. The paper is organized as follows. In Sec. 2, the ab initio methods and basis set for

twelve low-lying Λ-S electronic states calculations of

24

Mg35Cl molecule are briefly

described. Sec. 3 shows the computational results. The hyperfine structure and hyperfine structure branching ratio calculation procedures are presented in Sec. 4 and Sec. 5. A brief summary is given in Sec. 6.

2. METHOD 3 ACS Paragon Plus Environment


For

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Mg35Cl, we calculate the potential energy curves (PECs) of twelve low-lying

Λ-S states. The complete active space self-consistent field (CASSCF)37, 38 mothed is used for the calculations. Afterward, the multi-reference configuration interaction (MRCI)39, 40 plus Davidson correction (+Q)41 calculations with the CASSCF wave functions as a zeroorder reference function are carried out. The calculations are performed in the C2v symmetry, where all molecular orbitals are labeled by (a1, b1, b2, a2) irreducible representations, and the electronic states are labeled by (A1, B1, B2, A2) irreducible representations. In the CASSCF calculations, the nine active (valence) electrons representing the Mg 3s2 and Cl 3s23p5 shells are placed into the active space. The twenty electrons in the 1s22s22p6 shell of the Mg and Cl atoms are placed into the closed spaces but are still optimized. The closed spaces are denoted (6,2,2,0) and the total orbital space is denoted (10,4,4,0). In this case, the active space is referred to as CAS (9, 8). Additionally, state averaged were computed using the CASSCF method which contained all possible electronic states (7A1, 5B1, 5B2, 1A2). We choose the same orbital space and closed spaces in the subsequent MRCI calculations. The correlation consistent polarized valence quadruple zeta cc-PVQZ-DK (=VQZDK)42 and augmented correlation consistent polarized valence quadruple zeta aug-ccPVQZ-DK (=AVQZ-DK)43 all-electron basis sets are used for the Mg and Cl atoms in the calculations of the Λ-S states, respectively. The scalar relativistic effects are accounted for using the Douglas–Kroll–Hess (DKH)44,

45

transformation of the relativistic

Hamiltonian. The spin-orbit coupling (SOC) effects are also considered using the SOC operator Hso defined within the Breit-Pauli approximation46. The same active space and basis sets are used in the SOC calculations at the MRCI theory level. All calculations are implemented with the MOLPRO47 software. The spectroscopic constants (Re, Te, ωe, ωeχe, Be, De) are obtained by using Le Roy’s LEVEL 8.048 program. Using the PECs and transition dipole moments (TDMs), the FCFs and radiative lifetimes of the various vibrational levels can also be determined from LEVEL

calculations48.

8.0


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FIG. 1. PECs of the low-lying double electronic states of the Σ+, Π and ∆ of 24Mg35Cl molecule at the MRCI level of theory.

FIG. 2. PECs of the

low-lying

quadruple

electronic states of the

Σ+ and Π of

molecule at the MRCI

level of theory.

24

Mg35Cl

3. Results and discussion 3.1. PECs and spectroscopic constants In Figs. 1 and 2, we present the PECs of twelve low-lying Λ-S states at the MRCI level of theory. The corresponding spectroscopic constants of these bound states calculated with the LEVEL 8.048 program are listed in Table 1. Previous theoretical and experimental values for these bound states are also tabulated in Table 1. Table 1. Spectroscopic constants of bound states of 24Mg35Cl calculated at the MRCI level of theory. state X2Σ+

Te(cm-1) 0 0 0

Re(Å) 2.19 2.20

ωe(cm-1) 483.20 462.12 467.53

ωeχe(cm-1) 2.24 2.10


Be(cm-1) 0.25 0.26 0.24

De(eV) 3.42 3.30

This work Expa Ref20


A2Π

22Π B2Σ+(int) B2Σ+(ext) C2Σ+ a Ref. 35. b c

0 0 0 26958.71 26062.04 26310.80 26741.33 26739.91 32363.35 31945.56 38613.09 30867.66 42918.81

2.17 2.20 2.17 2.18 2.16 2.17 2.52 2.55 2.15 3.66 2.37

469.90 466.08 462.10 515.92 492.33 487.60 490.80

1.95 2.00

0.25 0.25 0.25 0.25

0.55 0.54

22.98

0.25 0.25 0.19 0.18 0.26

2.07 2.01 0.10

0.21 16.22

0.09 0.21

2.26 0.77

2.20 1.27 2.18

681.20 622.72 540.39 552.00 179.32 705.16

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14.29

3.37

Ref 27 Expb Expc This work Ref 20 Ref 27 Expb Expc This work Ref 20 This work Expa This work This work

Ref. 34.

Ref. 31.

For the X2Σ+ state, Wan et al.20 found a slightly larger equilibrium distance Re of 2.2025 Å compared to the experimental data. Wan et al.20 did not report their result for the anharmonicity constant ωeχe for the X2Σ+ state. However, our calculations yield a shorter equilibrium distance (2.1940 Å) and a slightly larger anharmonicity constant (2.2378 cm-1) for the X2Σ+ state compared to the observed values (2.1960 Å and 2.1000 cm-1). The percentage errors for our calculated equilibrium distance and the anharmonicity constant are 0.09% and 6.56%, respectively. The Be values for the X2Σ+ and A2Π states are 0.2462 cm-1 and 0.2520 cm-1 in our calculation, in excellent agreement with the experimental values of 0.2456 cm-1 and 0.2517 cm-1. As shown in Table 1, the spectroscopic parameters for the A2Π state are also close to the experimental data, for example, the anharmonicity constant ωeχe of 2.1989 cm-1 is in good agreement with the experimental value of 2.1800 cm-1. Moreover, it is important to note that the dissociation energy of the B2Σ+ state is obtained here for the first time at the multi-reference configuration interaction level. The calculated harmonic constant ωe (540.3859 cm-1) for the B2Σ+(int) state is in accordance with the experimental data (552.0000 cm-1). These values will be very helpful as reference data for experimental observation. Overall, the spectroscopic constants derived from our present work are in good agreement with previous theoretical and experimental values.


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Table 2. Spectroscopic constants of the first six Ω states of 24Mg35Cl at MRCI level.

A2Π3/2

Te(cm-1) 0 0 26888.87

A2Π1/2

26947.78

state X2Σ+1/2

2

2 Π3/2

32255.48

22Π1/2

32396.04

B2Σ+1/2(int) B2Σ+1/2(ext) a Ref. 35.

38616.28 30923.27

Re(Å) 2.19 2.20 2.17 2.18 2.17 2.18 2.17 2.52 2.54 2.54 2.57 2.16 3.66

ωe(cm-1) 483.55 467.52 518.27 492.79 519.18 495.04 491.60 681.20 616.72 720.03 635.54 525.75 179.96

ωeχe(cm-1) 2.17 2.44 2.65 2.54 14.29 17.12 22.46 0.21

Be(cm-1) 0.25 0.24 0.25 0.25 0.25 0.25 0.25 0.19 0.18 0.18 0.18 0.26 0.09

De(eV) 3.38 3.26 0.52 0.51 0.57 0.56 2.07 2.03 2.07 2.01 0.11 2.22

FIG. 3. PECs of six Ω states of 24Mg35Cl molecule at MRCI+Q level.


This work Ref 20 This work Ref 20 This work Ref 20 Expa This work Ref 20 This work Ref 20 This work This work


FIG. 4. The PDMs (a) of the double electronic states of the Σ+, Π, and ∆, (b) of the quadruple electronic states of the Σ+, and Π of 24Mg35Cl molecule.

The SOC effects are also taken into account for the 24Mg35Cl molecule. The PECs of six Ω states are shown in Fig. 3. The spectroscopic parameters together with previous theoretical values for the six Ω states are collected in Table 2. Comparison of the spectroscopic constants presented in Table 1 (with the SOC) and in Table 2 (without the SOC) shows that the equilibrium distance Re, anharmonicity constant ωexe and rotation constant Be of the X2Σ+1/2, B2Σ+1/2(int), B2Σ+1/2(ext), and 22Π3/2 states are almost identical to those of their X2Σ+, B2Σ+(int), B2Σ+(ext), and 22Π parents. For the harmonic constant ωe, the differences of X2Σ+, B2Σ+(ext), A2Π, and 22Π with X2Σ+1/2 B2Σ+1/2(ext), A2Π1/2, A2Π3/2, and 22Π3/2 are 0.3494 cm-1, 0.6448 cm-1, 3.2655 cm-1, 2.3559 cm-1 and 0.0033 cm-1, respectively. Therefore, the SOC did not show a significant impact for the spectroscopic parameters. The permanent dipole moments (PDMs) of the twelve low-lying electronic states and the TDMs of the 2Π → 2Σ+ transitions are presented in Figs. 4 and 5. As shown in Fig. 4, the calculated PDMs are decreasing as the internuclear distance R increases for X2Σ+, reaching a minimum (2.74 a.u.) at R > Re and finally tending to zero. The evaluated PDM data (3.167 Debye) at Re are in accordance with the previous theoretical value (3.381 Debye)28. For the A2Π state, the magnitude of the PDMs gradually decreases with increasing R and reaches a minimum (−1.82 a.u.) at R=2.30 Å. Subsequently, it starts to rise, reaches a maximum (0.95 a.u.) at R=2.65 Å, and drops thereafter, in reasonable agreement with the results obtained by Wan et al.20 8 ACS Paragon Plus Environment

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FIG. 5. TDMs for the transition from A2Π, 22Π and 32Π states (a) to the ground state X2Σ+, (b) to the B2Σ+ state and (c) to the C2Σ+ state.

3.2. FCFs and radiative lifetime We obtain the FCFs of the A2Π → X2Σ+ transition and spontaneous radiative lifetimes of the A2Π state using the LEVEL 8.0 program48. The FCFs describe the overlap of the vibrational wave functions, and diagonal FCFs can restrain the vibrational branching of a state. The corresponding FCF data along with the previous theoretical values for the 24

Mg35Cl molecule are listed in Table 3 for the transition discussed here. The FCF of

~0.932 for the A2Π → X2Σ+ transition is slightly larger than the previously calculated value of ~0.928 obtained by Wan et al.20 and is in extremely close agreement with the value of ~0.932 predicted by Kumaran et al.49 using the closed-form approximation method. However, the FCFs are not the only important parameters for laser cooling molecules. A potential candidate molecule should also exhibit a rate of optical cycling that produces a large spontaneous scattering force for laser cooling. Examination of the 9 ACS Paragon Plus Environment


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data presented in Table 4 shows that the A2Π (v′=0) state has a sufficiently short radiative lifetime of γ-1 = 12.28 ns, allowing for fast optical cycling. In addition, there is no intermediate state that can affect the laser cooling cycle of the

24

Mg35Cl molecule.

Therefore, the 24Mg35Cl molecule meets the three conditions of a potential candidate for laser cooling.

FIG. 6. Proposed laser-cooling scheme for 24Mg35Cl using the X2Σ+ → A2Π transition (solid lines) and spontaneous decay (dotted lines) with calculated fv'v″ for 24Mg35Cl. Solid short line indicates relevant upper and lower vibrational structure in 24Mg35Cl, respectively.


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Table 3. The calculated FCFs fv'v″ along with theoretical values and wavelength (unit is nm) λν'ν″ of the A2Π → X2Σ+ transition. The diagonal FCFs are in bold. Numbers in parentheses indicate the power of 10.

Present work

The closed-form approximationa

Referenceb a

f00 f01 f02 f03 9.32(10-1) 6.50(10-2) 2.62(10-3) 9.51(10-5) 9.32(10-1) 6.53(10-2) 2.28(10-3) 6.43(10-5) 9.28(10-1) 6.90(10-2) 3.00(10-3) 1.40(10-4)

f10 f11 f12 f13 6.55(10-2) 8.00(10-1) 1.26(10-1) 8.00(10-3)

f20 f21 f22 f23 2.16(10-3) 1.28(10-1) 6.69(10-1) 1.84(10-1)

f30 f31 f32 f33 6.01(10-5) 7.09(10-3) 1.86(10-1) 5.42(10-1)

λ00

λ10

λ21

371.30

378.02

377.52

8.10(10-1) 1.21(10-1) 6.54(10-3) 7.00(10-2) 7.92(10-1) 1.27(10-1) 1.40(10-4)

Ref. 49.

b

Ref. 20.

A three-laser cooling scheme for the laser cooling of 24Mg35Cl is presented in Fig. 6. The vibrational and rotational branching loss must be addressed for the laser cooling cycle. Therefore, the FCF (0.932) of the X2Σ+ (v″=0) → A2Π (v ′ =0) transition is suitable as the main pump. The first vibrational repump (X2Σ+ (v″=0) → A2Π (v′=1)) and second vibrational repump (X2Σ+ (v″=1) → A2Π (v ′ =2)) are also required to obtain more than 104 scattered photons. Table 4. Spontaneous radiative lifetimes (ns) of A2Π → X2Σ+ transition of 24Mg35Cl molecule Radiative lifetime transition

A 2Π

v′=0

v′=1

v′=2

v′=3

v′=4

12.28

12.35

12.43

12.51

12.65



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4. HYPERFINE STRUCTURE OF THE 24Mg35Cl RADICAL In laser cooling of molecules, it is important to consider the hyperfine structure (HF) of the ground electronic state for the elimination of dark states. Thus, all hyperfine levels should be pumped simultaneously in the case of the molecules accumulating into one state, and then, nearly closed optical transitions are implement. In this work, we have estimated the HFs of the ground state X2Σ+ of the

24

Mg35Cl molecule using the matrix

diagonalization technique. In a diatomic molecular system, the effective Hamiltonian is a sum of terms representing the various inter-coupling of angular momenta within the molecule50. Therefore, the effective Hamiltonian for the 24Mg35Cl ground state is given by H hf = γvT 1 ( Sˆ )T 1 ( Nˆ ) + bF T 1 ( Iˆ)T 1 ( Sˆ ) + cvTq1= 0 ( Iˆ)Tq1= 0 ( Sˆ ) − eT 2 (Q)T 2 (∇E )

(1)

where γv is the spin-rotational constant, bF is the Fermi contact constant, and cv is the dipole-dipole constant. All parameters36 are listed in Table 5. Table 5. Rotational constants and hyperfine constants for 24Mg35Cl (in MHz) v=0 B 7339.1014 D 8.1640×10-3 γ 66.547 bF 36.520 cv 36.463 eQq -11.622

A typical Hund’s case (b) case molecule first S (electron spin) couples with N (the total angular momentum excluding electron spin) to form J (total angular momentum). Then, J couples with I (total nuclear spin) to form the total angular momentum F (total angular momentum of molecule). The angular momentum states are written as | ((S, N) J, I) F)〉, and this is known as the bβJ basis. Based on the bβJ basis, the corresponding matrix elements for each term of the hyperfine structure are tabulated as follows: η′, Λ′, N ′, S , J ′, I , F | γvT 1 ( Sˆ )T 1 ( Nˆ ) | η, Λ, N , S , J , I , F S N J  = δJJ ′δFF ′ γv ( −1) N + J + S [ S ( S + 1)(2 S + 1)]1 / 2 [ N ( N + 1)(2 N + 1)]1 / 2    N S 1 12 ACS Paragon Plus Environment

(2)

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η′, Λ′, N ′, S , J ′, I , F | bF T 1 ( Iˆ)T 1 ( Sˆ ) | η, Λ, N , S , J , I , F = δFF ′bF ( −1) J ′ + F + I + N + J +1+ S [( 2 J ′ + 1)( 2 J + 1)]1 / 2 [ S ( S + 1)(2 S + 1)]1 / 2 [ N ( N + 1)( 2 N + 1)]1 / 2 (3)

I J ′ F  J S N  × ×    J I 1   S J′ 1

η′, Λ′, N ′, S , J ′, I , F | cvTq1= 0 ( Iˆ)Tq1= 0 ( Sˆ ) | η, Λ, N , S , J , I , F = δFF ′C (−1) J ′ + F + I + N [(2 J ′ + 1)(2 J + 1)]1 / 2 [ S ( S + 1)(2 S + 1)]1 / 2 [ N ( N + 1)(2 N + 1)]1 / 2 (2 N + 1) J N 2 N   ×  N ×  0 0 0   S 

J ′ 1  I J ′ F  N 2 ×   J I 1   S 1

(4)

η′, Λ′, N ′, S , J ′, I , F | −eT 2 (Q )T 2 (∇E ) | η, Λ, N , S , J , I , F  I J ′ F  ( I + 1)( 2 I + 1)(2 I + 3) eQq ( −1) J ′ + F + I  (−1) J ′ + N + S [( 2 J ′ + 1)( 2 J + 1)]1 / 2  2 J I 4 ( 2 − 1 ) I I   (5) J′ S N 2 N   N (2 N + 1) ( −1) ×  N 2  0 0 0 

= δ JJ ′ δ F F ′ N × J

where δii' is defined as δii' =0 for i ≠ i' and δii' =1 for i = i', and eQq is the nuclear quadrupole coupling constant of

35

Cl (ICl = 3/2). By numerically diagonalizing the

constructed matrix, we obtain the state eigenvectors and the energy eigenvalues. Then, using the energy eigenvalues of different rotational hyperfine energy levels, we obtain the transition frequencies among the rotational hyperfine levels. The experimental observational spectral data and our theoretically calculated results for the X2Σ+ (v″=0) state of the 24Mg35Cl molecule are presented in Table 6. The fourth column presents the theoretical calculation spectrum data, with the errors relative to the experimental data not exceeding 8 kHz ∼ 20 kHz. Therefore, the theoretical methods and results used in this work are reliable. However, the A2Π state is best described by Hund’s case (a); the parity of the rotational ladder in the A2Π states is given by half-integral J, levels with parities (1) J-1/2 or (-1) J+1/2 are designated + or -, respectively. The hyperfine splitting between F = 0 and F = 1 for the Hund’s case (a) state A2Π is still unresolved.



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Table 6. The experimental spectral data and our theoretically computed results and their comparisons for 24Mg35Cl. The first three columns are the permissible transitions between the rotational hyperfine levels of the X2Σ+ state. The forth column is observational spectrum data in Ref. 36. Our theoretical results are included in the fifth column and the differences between experimental data and theoretically calculated results are given in the last column. N″→N′

J″→J′

0→1

1/2→1/2

0→1

1/2→3/2

F″→F′ 2→2 1→2 2→3 2→2 1→2 1→0

vexp (MHz) 14570.719 14643.751 14708.397 14690.968 14764.000 14720.436

vcal (MHz) 14570.701 14643.741 14708.378 14690.952 14763.992 14720.426

vexp-vcal (MHz) 0.018 0.010 0.019 0.019 0.008 0.010

Based on the calculated data, we designed the closure of the rotational structures for 24

Mg35Cl, as presented in Fig. 7. Clearly, we sought to cool on the N=0 rotational state

for the ground state (X2Σ+). However, due to parity and angular momentum selection rules, molecules can decay back to the N=0 and N=2 levels, and rotational branching opens up. For

24

Mg35Cl, this loss of N = 2 and N = 0 still exist. Therefore, we should

eliminate this loss by using the microwaves resonant with both |N = 0; J = 1/2; F = 0,1〉 ↔ |N = 1; J = 1/2; F =1,0〉 transitions. The rotational state (N=1) is excited to the |A, J=1/2, +〉 state with the wavelength of 371.30 nm. Moreover, six hyperfine energy levels (|J=3/2, F=3〉, |J=3/2, F =2〉, |J=3/2, F=1〉, |J=3/2, F=0〉, |J=1/2, F=2〉, and |J=1/2, F=1〉) for N=1 are obtained due to the I=3/2 nuclear spin of Cl, which splits J into F=J±1/2 levels through magnetic and electric hyperfine interaction. Meanwhile, each of the population levels requires a repump laser. However, experimentally, it is challenging to use so many lasers. Fortunately, we can use an electro-optical modulator (EOM) with the laser detuning of δ = −17.5 MHz and the modulation frequency of fMod = 35 MHz to cover all hyperfine energy levels by the proposed sideband frequency distributions. So far, we have discussed all of the conditions for the laser cooling of the

24

Mg35Cl molecule.

A2Π shows a short lifetime and highly diagonal FCFs in the vibrational level. We eliminated the rotational branching by parity selection rules and microwave remixing. Additionally, the sideband modulation will cover all six hyperfine levels of |X, N=1〉


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FIG. 7. The relevant hyperfine energy-level structure, laser frequencies, and spontaneous radiation (dashed lines) for 24Mg35Cl. Due to the angular momentum and parity selection rules a spontaneous decay back to the odd parity rotational state N=1 of X2Σ+ is allowed.

Table 7. The J-mixing phenomena in N=1 rotational states of the ground X state. N

1

Nominal label |J=3/2, F=3〉 |J=3/2, F=2〉 |J=1/2, F=2〉 |J=1/2, F=1〉 |J=3/2, F=1〉 |J=3/2, F=0〉

Actual label |J=3/2, F=3〉 0.3711|J=1/2, F=2〉+0.9286|J=3/2, F=2〉 0.9286|J=1/2, F=2〉-0.3711|J=3/2, F=2〉 0.9354|J=1/2, F=1〉+0.3535|J=3/2, F=1〉 -0.3535|J=1/2, F=1〉+0.9354|J=3/2, F=1〉 |J=3/2, F=0〉

5. BRANCHING RATIOS We calculate the branching ratios, which reflect the distributions of the transition strengths for all possible hyperfine decays from |A, J=1/2, +〉 to |X, N=1, −〉. These calculations are done in the Hund’s case (a) basis. Thus, we must transform the basis sets |A, J, +〉 and |X, N, J, F, −〉 to the Hund’s case (a) basis (|Λ, S, Σ, J, Ω, I, F, mF〉). We also should consider the mixing among the J states for the same N before deriving the matrix elements for the transition in A-X. The mixing is summarized in Table 7. The conversion for the pure J state between the Hund’s case (b) and the Hund’s case (a) is given by



N, J = N +

1 1  1 1 1 1  Σ = ,Ω = = − Σ = − ,Ω = −  2 2 2 2 2 2

N, J = N −

1 1  1 1 1 1  Σ = ,Ω = = + Σ = − ,Ω = −  2 2 2 2 2 2

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   

(6)

   

(7)

while for the A state, it is given by Π, J =

1 1  1 1 1 1 1 1 1 1  Λ = 1, S = , Σ = − , J = , Ω = ,+ = − Λ = −1, S = , Σ = , J = , Ω = −  2 2 2 2 2 2 2 2 2 2

   

(8)

Here, we calculate the transition dipole moments d = Ψe T p1 (dˆ ) Ψg

(9) ^

where |Ψe〉 and |Ψg〉 are Zeeman sublevels under the Hund’s case (a) basis. T1(d) is the first rank tensor denoting the electric dipole moment, and p denotes the polarization of light. Employing the Wigner-Eckart theorem, we obtain d = Λ′, S , Σ′, Ω′, J ′, I , F ′, m′F T p1 ( dˆ ) Λ, S , Σ, Ω, J , I , F , mF  F′ 1 F  J F I  (−1) F + J ′+ I +1 (2 F ′ + 1)( 2 F + 1)  = (−1) F ′−m    F ′ J ′ 1  − m′F p mF  F

(10)

× Λ′, S , Σ′, Ω′, J ′ T 1 (dˆ ) Λ, S , Σ, Ω, J

We can further reduce the final term Λ′, S , Σ ′, Ω′, J ′ T 1 ( dˆ ) Λ, S , Σ, Ω, J 1

=

∑（ −1）

J ′− Ω′

q = −1

 J′ 1 J   × Λ′, S , Σ ′ T 1 ( dˆ ) Λ, S , Σ ( 2 J ′ + 1)(2 J + 1)   − Ω′ q Ω  ^

where the term 〈Λ′, S, Σ′ |T1(d)|Λ, S, Σ〉 is common to all branches.


(11)

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Table 8. Branching ratios for the A2Π1/2, J′=1/2 → X2Σ+, N=1, i.e. the main optical cycling transition. Columns are denoted by the excited state’s hyperfine numbers (F′, m′F) and the rows are denoted by the ground states numbers (J, F, mF). J 3/2 3/2

1/2

3/2

1/2

3/2

F mF 0 0 -1 1 0 1 -1 1 0 1 -2 -1 2 0 1 2 -2 -1 2 0 1 2 -3 -2 -1 3 0 1 2 3

m′F=-1 1/18 0.1088 0.1088 0 0.0162 0.0162 0 0.0029 0.0014 0.0005 0 0 0.4138 0.2069 0.0690 0 0 0 0 0 0 0 0 0

F′=1 m′F=0 1/18 0.1088 0 0.1088 0.0162 0 0.0162 0 0.0014 0.0020 0.0014 0 0 0.2069 0.2759 0.2069 0 0 0 0 0 0 0 0

m′F=1 1/18 0 0.1088 0.1088 0 0.0162 0.0162 0 0 0.0005 0.0014 0.0029 0 0 0.0690 0.2069 0.4138 0 0 0 0 0 0 0

m′F=-2 0 0.0069 0 0 0.3431 0 0 0.1551 0.0775 0 0 0 0.1227 0.0614 0 0 0 1/6 1/18 1/90 0 0 0 0

m′F=-1 0 0.0035 0.0035 0 0.1715 0.1715 0 0.0775 0.0388 0.1163 0 0 0.0614 0.0307 0.0920 0 0 0 1/9 4/45 1/30 0 0 0

F′=2 m′F=0 0 0.0012 0.0046 0.0012 0.0572 0.2287 0.0572 0 0.1163 0 0.1163 0 0 0.0920 0 0.0920 0 0 0 1/15 1/10 1/15 0 0

m′F=1 0 0 0.0035 0.0035 0 0.1715 0.1715 0 0 0.1163 0.0388 0.0775 0 0 0.0920 0.0307 0.0614 0 0 0 1/30 4/45 1/9 0

m′F=2 0 0 0 0.0069 0 0 0.3431 0 0 0 0.0775 0.1551 0 0 0 0.0614 0.1227 0 0 0 0 1/90 1/18 1/6

Due to the angular momentum selection rules for the electric dipole transition, only the decays of |A, J=1/2, +〉 to the N = 1 level in the ground state are allowed. The calculated branching ratios for A2Π1/2, J′=1/2 → X2Σ+, and N=1 are listed in Table 8. As a consistency check, we sum the branching ratios in each column and obtain 1, as expected. The theoretically calculated spectra are then plotted with the branching ratio as the strength of

each peak (see

Fig. 8).



FIG. 8. The proposed sideband modulation scheme to the (re)pumping lasers to simultaneously cover all four hyperfine levels of |X, N=1〉. The theoretically calculated spectra (black line) are plotted with the branching ratio from the |A, J=1/2, +〉 to each hyperfine level as the strength of each peak, the calculated energy value in Fig. 7 as the center frequency. The red solid line indicates the sidebands of an EOM with the laser detuning δ = −17.5MHz and the modulated frequency fMod = 35MHz. The units of the x axis are MHz.

6. Conclusion In this work, the PECs and PDMs for the twelve low-lying Λ-S electronic states and TDMs for the 2Π →2Σ+ transition of the

24

Mg35Cl molecule have been assessed at the

CASSCF/MRCI (+Q) level of theory. The cc-PVQZ-DK (=VQZ-DK) and aug-cc-PVQZDK (=AVQZ-DK) all-electron basis sets are used for the Mg and Cl atoms, respectively, in the calculations of the Λ-S states. The SOC effects are also taken into account at the MRCI+Q level by using the same basis sets and active space. The SOC calculation results show that the influence of SOC on the spectroscopic properties is weak. Moreover, the hyperfine structure and branching ratio are also computed for the X2Σ+ ground state of the 24Mg35Cl molecule. The spectroscopic constants and transition frequencies derived from the present work are in good agreement with experimental values and available theoretical data. Using the potential energy curves and transition dipole moments, the highly diagonally distributed Franck-Condon factors for the A2Π → X2Σ+ transition and radiative lifetimes for the first six vibrational levels (v'=0-5) of the A2Π state are 18 ACS Paragon Plus Environment

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determined. Our results reveal that the radiative lifetime of A2Π (v'=0) is 12.28 ns, which is sufficiently short for rapid laser cooling. There is no intermediate state affecting the laser cooling cycle of the

24

Mg35Cl molecule. Therefore, the three criteria for efficient

and rapid laser cooling are satisfied well. Finally, a three-laser cooling scheme for the laser cooling of 24Mg35Cl is presented. The X2Σ+ (v″=0) → A2Π (v′=0) transition can serve as the main pump due to its suitable FCF (0.932). The first vibrational repump (X2Σ+ (v″=0) → A2Π (v′=1)) and second vibrational repump (X2Σ+ (v″=1) → A2Π (v′ =2)) are also required to obtain more than 104 scattered photons. Each of the hyperfine population levels also requires a pump laser. However, the use of so many lasers will be challenging experimentally. Fortunately, we can use an EOM with the laser detuning of δ = −17.5 MHz and the modulation frequency of fMod = 35 MHz to cover all hyperfine energy levels by the proposed sideband frequency distributions. Overall, our results indicate that the 24Mg35Cl molecule is a promising candidate laser cooling molecule.

Acknowledgments The authors wish to thank the anonymous referee for valuable comments.

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Mg37Cl A²pi-X²sigma+ band system. Brazilian Journal of Physics. 2003, 4, 886-

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TOC graphic 23 ACS Paragon Plus Environment



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FIG. 1. PECs of the low-lying double electronic states of the Σ+,  and Δ of 24Mg35Cl molecule at the MRCI level of theory.

FIG. 2. PECs of the low-lying quadruple electronic states of the Σ+ and  of 24Mg35Cl molecule at the MRCI level of theory.

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FIG. 3. PECs of six Ω states of 24Mg35Cl molecule at MRCI+Q level.

FIG. 4. The PDMs (a) of the double electronic states of the Σ+, , and Δ, (b) of the quadruple electronic states of the Σ+, and  of 24Mg35Cl molecule.


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FIG. 5. TDMs for the transition from A2Π, 22Π and 32Π states (a) to the ground state X2Σ+, (b) to the B2Σ+ state and (c) to the C2Σ+ state.

FIG. 6. Proposed laser-cooling scheme for 24Mg35Cl using the X2Σ+ → A2Π transition (solid lines) and spontaneous decay (dotted lines) with calculated fv'v″ for 24Mg35Cl molecule. Solid short line indicates relevant upper and lower vibrational structure in 24Mg35Cl, respectively.



FIG. 7. The relevant hyperfine energy-level structure, laser frequencies, and spontaneous radiations for 24Mg35Cl

molecule. Due to the angular momentum and parity selection rules allow a spontaneous decay

only back to the odd parity rotational state N=1 of X2Σ+.

FIG. 8. The proposed sideband modulation scheme to the (re)pumping lasers to simultaneously cover all four hyperfine levels of |X, N=1. The theoretically calculated spectra (black line) are plotted with the branching ratio from the |A, J=1/2, + to each hyperfine level as the strength of each peak, the calculated energy value in Fig. 7 as the center frequency. The vertical solid color line indicates the sidebands of an EOM with the laser detuning δ = −17.5MHz and the modulated frequency fMod = 35MHz. The units of the x axis are MHz.


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Theoretical Study of the Feasibility of Laser Cooling the - American

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