Investigating the Selective Control of Photoassociation of Yb2

*e selective control of photoassociation of Yb2 is investigated in theory. Based on ab initio to rationalize Franck–Condon filtering, the optimal target states of photoassociation have been obtained. *e corresponding vibrational transitions from XΣg to the excited state (AΣu, BΠu, CΣu, and DΠu) are v′ � 23, 50, 55, and 0, respectively. By using quantumwave packet dynamic methods, we calculated the yields with time evaluation for the selected target states. *e projections of time-dependent wave functions of initial states on the target vibrational eigenstates reflected the synthetic yields of Yb2. For target AΣu, we used Gaussian pulse to make the yield of v′ � 23 up to 97% at 725 fs. After a laser pulse, the positive chirp promoted the yield of vibrational states to increase, but the negative chirp inhibited its decrease. For the DΠu state, when laser intensity is 1.0×10W/ cm, the purity and yield of target state v′ � 0 reached the maximum at 1350 fs. *at is to say, changing the laser parameters and pulse shapes could control the photochemical reaction along our desired direction. *ese conditions will provide an important reference and suggest a scheme for a feasible photoassociation of further experimental and theoretical research studies. Current study may promote an important step toward the realization of highly accurate quantum manipulation and material synthesis.


Introduction
Optical atomic clocks [1,2] have made it possible to test the fundamentals of physics [3] and place limits on temporal variation of fundamental constants [4][5][6], or to explore quantum many-body systems [7,8], even to search for topological dark matter through its impact on the finestructure constant [9,10]. Meanwhile, molecular clocks promise increased sensitivity to the variation of the electronto-proton mass ratio [11][12][13][14][15]. Laser cooling of spin-singlet atoms, such as calcium, strontium, or ytterbium, has gained widespread attention due to their possible applications in making nonmagnetic Bose-Einstein condensate, as well as atomic reference for optical clocks [16]. e effect of cold collisions is especially important for optical clocks with atoms trapped in one-dimensional optical lattices [17,18]. Information about cold-collisional shifts may be obtained from molecular potentials [19,20]. Directly photoassociative approach to measure 1 S 0 + 3 P 0 and 3 P 0 + 3 P 0 potentials may be challenging since nothing is known about bound state positions [21]. Additionally, Franck-Condon factors may be unfavorable in case of ytterbium [15] which will make direct measurement even more challenging. Bober et al. reported an indirect photoassociative approach that may allow for determining the scattering lengths for 1 S 0 + 3 P 0 and 3 P 0 + 3 P 0 cold collisions using 679 nm laser [22]. Photoassociation (PA) is a process in which two colliding atoms form an excited molecule by absorbing a photon.
is technique had been successfully applied to create ultracold molecules from ultracold atomic gases. Besides, ultracold atoms have also generated new avenues of research related to quantum physics, like Bose-Einstein condensation and atom optics. PA spectroscopy provides a versatile tool for probing the physics of rovibrational molecular states and precisely determining the collisional properties of atoms, such as scattering length and interatomic potential coefficients [23]. Furthermore, the PA process can be actively used to control the strength of atomic interaction via coupling to an excited molecular state. Schemes for internal rovibrational cooling in Yb 2 and Cd 2 based on photodissociation of ( 171 Yb) 2 are presented, which are based on exploration of the rotational and vibrational energy structures using both theoretical and experimental approaches [24].
In addition, Borkowski et al. reported observations of photoassociative spectra near the intercombination line in isotopic mixtures of ultracold ytterbium gases. Several heteronuclear bound states have been found for the excited 170 Yb 174 Yb and 174 Yb 176 Yb molecules [16]. Roy et al. demonstrated interspecies collisions in a dual-species Yb-Li magnetooptical trap (MOT) and carried out PA spectroscopy of molecular potentials below the Yb( 1 S 0 ) + Li( 2 P 1/2 ) asymptote [25]. In theory, the potential energy curves of the ground and twelve low-lying excited electronic states of RbYb molecule have been calculated using the multireference perturbation theory method at the CASSCF/ XMCQDPT2 level [26]. Some key theoretical predictions of the electronic structure and dipole moment of the ground and low-lying excited states of LiYb and the ground state of LiSr have recently been obtained [27].
In this work, we investigated the selective control of photoassociation of Yb 2 at short range that is organized as follows. In Section 2, the methods of ab initio calculations and wave packet dynamics for the allowed transitional states (X 1 Σ + g , A 1 Σ u + , B 1 Π u , C 1 Σ u + , D 1 Π u ) of photoassociation are described. Section 3 includes the results and discussions of the data. Firstly, highly correlated ab initio calculations were performed for accurate determination of the potential energy curves (PECs) and transition dipole moment curves (TDMCs). en, by Franck-Condon filtering, the optimal target states have provided a reliably photoassociative route for laser cooling. Finally, the yield and purity of photoassociation of Yb 2 molecule have been calculated by changing laser parameters and pulse shapes. Finally, the conclusions are made in Section 4.

Ab Initio Calculation.
e ab initio calculations of electronic structure for the allowed transitional electronic states (X 1 Σ g , D 1 Π u ) of Yb 2 have been performed with MOLPRO [28] in the D 2h point group, where the symmetries of A g , B 3u , B 2u , B 1g , B 1u , B 2g , B 3g , and A u representations were used. e methods of complete active space self-consistent field (CASSCF) [29,30] and internally contracted multireference configuration interaction plus Davidson corrections (icMRCI + Q) [31] have been adopted. For Yb atom, the basis set of effective core potential (ECP) ECP60MDF has been used [32]. We performed CASSCF and MRCI calculations with reference spaces consisting of 5s5p6s5d6p. e eight electrons in the 5s5p shells were put in the closed spaces, where it was doubly occupied in all reference configuration state but still optimized. For Yb 2 molecule, the active space of 4 electrons in 15 orbitals (6s, 6p, 5d) generates the appropriate reference wave functions.
us, the corresponding active spaces are (32213220).

Wave Packet Dynamic Method.
Combining with highly correlated ab initio calculations for accurate determination of the potential energy curves (PECs) and transition dipole moment curves (TDMCs), one could obtain the Franck-Condon factors and transitional probabilities. Using Franck-Condon filtering, vibrational coherence could be generated according to the maximum overlap [33]. e selected optimal target states have provided a feasible route for photoassociative process. Furthermore, the quantitative understanding of the yield and purity of target states after association has been studied by using quantum wave packet dynamic methods [34]. e projections of time-dependent wave functions of initial states on the vibrational eigenstates of excited states, reflecting the photoassociation yields, were investigate. e detailed calculation process can be found in our previous published article [35]. By changing the laser parameters and pulse shape, we studied the effects on yield and purity of photoassociation of Yb 2 molecule.

Potential Energies and Transition Dipole Moments of
States. In 1, the potential energy curves (PECs) of Yb 2 molecule with a valid range 2.0 ≤ R ≤ 20Å have been presented, which included X 1 Σ g + and allowed transitions from ground to excited states A 1 Σ u + , B 1 Π u , C 1 Σ u + , and D 1 Π u . In order to estimate the accuracy of these various PECs over the range of R, we evaluated equilibrium internuclear distance (R e ), electronic transition energy (T e ), dissociation energy (D e ), harmonic frequency (ω e ), anharmonic vibrational frequency (ω e x e ), and rotational frequency (B e ) of all states at icMRCI + Q/ECP60MDF/32213220 levels by LEVEL program [36].
ere are few spectroscopic studies on this molecule. ese produced spectroscopic constants of present results compared with the theoretical value [37,38] that are also listed in Table 1 together. It can be seen that our results are in good agreement with the calculated results. Furthermore, there are much closer determinations by Wang and Dolg [37] than Guido and Balducci [38]. For example, the error is 3.3% for the R e of B 1 Π u state corresponding to the result. e values (2.5509 eV and 49.1833 cm −1 ) of electronic transition energy (T e ) and harmonic frequency (ω e ) of A 1 Σ u + are very close to the determinations (2.53 eV and 53 cm −1 ) by Wang, respectively. In addition, we also provided anharmonic vibrational frequency (ω e x e ) and rotational frequency (B e ) of five states first time. It indicates that using icMRCI + Q method and the larger basis set are important for studying PEC. at is to say, the PEC at icMRCI + Q/ECP60MDF/32213220 level is more accurate at present, which can be used to predict the spectral parameters of higher vibrational bands for every allowed transition system. erefore, we will apply those potentials to investigate the Franck-Condon factors (FCFs) and Einstein A coefficients in Section 3.2.
While calculating the potential energy functions, the transition dipole moment curves (TDMCs) of the allowed transitions are also investigated in theory shown in 2, which included It can be seen that the TDMCs of A 1 Σ u + ⟵X 1 Σ g + and C 1 Σ u + ⟵X 1 Σ g + firstly increase, but then C 1 Σ u + ⟵X 1 Σ g decreases until it approaches zero. On the contrary, the TDMs of B 1 Π u ⟵X 1 Σ g + and D 1 Π u ⟵X 1 Σ g + decrease with R, and then the curve of D 1 Π u ⟵X 1 Σ g + increases until it tends to zero. At the equipment internuclear distance of ground state, the values of TDMs are A 1 Σ u u., and D 1 Π u ⟵X 1 Σ g + : −3.04 a.u., respectively. At last, the molecule dissociated into two atoms with the R increase that lead the dipole moment to be a constant, which is regarded as polar compensation of each other. e reason for this situation is that the different polarity of two states (A 1 Σ u + , B 1 Π u ) at larger position tends to maximum (4.2 a.u. and 3.8 a.u., respectively). At the same time, the results of the TDMs of electronic states (C 1 Σ u + , D 1 Π u ) tend to zero.

Optical Transition Properties.
Simulation of laser conversion of photoassociation requires systematic calculations of optical transition characteristics taking into account laser absorption from ground state to vibrational level of excited state [39]. By using LEVEL [36], the FCFs and vibrational transition probabilities have been calculated for quasi target States  Figure 1: e calculated potential energy curves (PECs) at the icMRCI + Q/ECP60MDF/32213220 level.   According to Franck-Condon filtering and transitional probability, we plotted the selected vibrational levels v ′ of excited states with the population after photoassociation shown in 4, which provided a visualized description for the quantum control of photoassociation of Yb 2 . e equilibrium internuclear distance of D 1 Π u state is larger than A 1 Σ + u state that is similar to B 1 Π u and C 1 Σ u + . e ground X 1 Σ + g state is only weakly bound by van der Waals force and has larger equilibrium internuclear distance corresponding to the excited states. ere is rotational barrier of X 1 Σ + g for rotational quantum numbers (J″), already for 218, where the ground states do not support any bound levels. Above all, A 1 Σ + u and D 1 Π u are more suitable for the target states of photoassociation of Yb 2 molecule.

Dynamic Process of Selective Control for Photoassociation.
e associative process happens at short internuclear distance when a pair of colliding atoms are irradiated by a laser pulse. For selective control of photoassociation of Yb 2 , the quantitative yield and purity of target states after association are concerned. By using quantum wave packet dynamics with time-dependent potentials [34], we calculated the projections with time evolution of photoassociation. e projections of time-dependent wave functions of initial states on the target vibrational eigenstates, reflecting the photoassociative yields of Yb 2 , are investigated. In the calculations, the number of grid points is 4000, and the size of grid used was 0.02 a.u. We run the simulation with time 800000 a.u. and time step length Δt � 0.1 a.u. e projection of the wave function on vibrational eigenstates every 20000 time steps is printed. It happens that the population transfers from ground state to excited state after a pulse irradiance. In this work, the initial population is defined as 100% of ground X 1 Σ + g of Yb 2 . In 555 5, the projections of time-dependent vibrational wave function of ground state, interacting with a resonant laser pulse, on the main five vibrational eigenstates of excited states (A 1 Σ u + , B 1 Π u , C 1 Σ u + , D 1 Π u ) are plotted. at corresponds to laser pulse parameters: amplitude ε 0 � 0.169 a.u.; duration t f � 41341.37 a.u.; frequency ω match transition energy. From 5, it can be seen that, with the time evolution, the percent of yields for every state is about A 1 Σ u + : 13%, B 1 Π u : 8%, C 1 Σ u + : 9%, and D 1 Π u : 70%, respectively. Meanwhile, the corresponding yield of each state firstly increases and then decreases with time evaluation. But, when each state reaches the maximum yield, the  needed time is different, which is A 1 Σ u + (v ′ � 23): 30000 a.u., B 1 Π u (v ′ 50): 36000 a.u., C 1 Σ u + (v ′ 55): 39000 a.u., and D 1 Π u (v ′ 1): 27000 a.u., respectively. In addition, when every vibrational energy level reaches the maximum, the required time becomes longer with the level.
is indicates that the probability of optical transition of A 1 Σ u + state is greater than B 1 Σ + u and C 1 Σ u + states, which is just different from D 1 Π u . is conclusion is the same as that discussed above that the optimal target states of photoassociation are A 1 Σ u + state and D 1 Π u state. e proposed laser drive transitions to target states of photoassociative process at wavelength A 1 Σ u + : 464.7 nm, B 1 Π u : 373.1 nm, C 1 Σ u + : 361.8 nm, and D 1 Π u : 339.1 nm are obtained. For quasi target state A 1 Σ u + of photoassociation, the projections of initial wave function on the five vibrational eigenstates (v ′ 20-24) are investigated, and the results are shown in 6. At the initial time, the projections of each vibrational state are similar about 66%. By changing the laser intensity and pulse shape, we have studied the effects on yield and purity of photoassociative process. When laser intensity changed from 0.169 to 1.69 a.u. (equivalent to 1.0 × 10 14 W/cm 2 ), the yield of association was improved. For example, using Gaussian pulse made the value of v ′ 23 up to 96% at 725 fs. Furthermore, by changing the shapes of laser pulse, the developed trend of yield over time had been evaluated according to Figure 6 of ±chirp pulse. From the figure, it can be seen that, after a laser pulse, the positive chirp makes the projections of vibrational states increase, but the negative chirp makes it decrease. When the intensity of chirp pulse increases to 1.0 × 10 14 W/cm 2 , the yield of v ′ 24 is larger than other vibrational levels (v ′ 20, 21,22,23), in which the value reached maximum 99%. It is contrary to the result of Gaussian shape. Meanwhile, the needed time of the maximum yield will be shorter by using chirp pulse. Even the result by positive chirp (580 fs) is much shorter than negative chirp (653 fs). at is to say, the positive chirp pulse could effectively promote the yields of photochemical reactions, and negative chirp could inhibit them. In addition, it indicates that the purity is not very well due to the overlap of projections.
erefore, the yields of vibrational levels are relatively lower. at do not benefit our accurate quantum manipulation of photoassociation.
We came to the same conclusion with above: at initial time (t 0 fs), the selected optimal vibrational target state is v ′ 0 for target D 1 Π u of photoassociation, shown in 7. ere is a diverse oscillatory behavior observed from Figure 7(a). e reason for this behavior is that the optical target states correspond to some frequencies of vibrational wave functions. So, they lead to the diverse overlap of two wave functions of v' 0 and v' 1. e projections of five vibrational levels before 750 fs are overlap, so that we are difficult to distinguish which one is optimal vibrational target state for photoassociation. When laser intensity is 1.0 × 10 14 W/cm 2 , not only the purity but also the yield is higher up to 99% of target state v' 0 at 1570 fs in Figure 7(c). By changing the laser intensity into 1.69 a.u. of negative chirp in Figure 7(e), the excess energy was absorbed by other vibrational levels except v ′ 1, which lead to a little increase for the yields of v ′ 2, 3, and 4. In other words, the purity of target state had a sudden drop and then levels off at longer duration when increasing laser intensity. Generally, the projection of time-dependent vibrational wave function recovers the prior level after a pulse. However, the case of D 1 Π u shows a little change, which is a change between v ′ 0 and v ′ 1. e reason is that the excess energy of v' 0 after pulse was absorbed by v' 1, leading to the final population having a little increase of v' 1, 2, 3, and 4. But the finally total population keeps the same level with initial population. e calculated yield of photoassociation is further enhanced by performing optimization of the best laser pulse.
On the whole, for Yb 2 molecule, if the photoassociative process is nonadiabatic, the yields of target states are A 1 Σ u + : 0.13%, B 1 Π u : 0.08%, C 1 Σ u + : 0.09%, and D 1 Π u : 0.70%, respectively. If we use adiabatic association, the most suitable quantum manipulation of photoassociation is A 1 Σ u + and D 1 Π u states that correspond to photoassociative yields are 67% and 89% at initial time by Gaussian pulse. We also evaluated the projections interacting with positive and negative chirp laser pulse, which show that the yields of vibrational states have an increase and decrease of a little after the pulse, respectively. at is to say, different chirp pulses can be used to effectively promote or inhibit the yields of photochemical reactions. erefore, it could much more easily control the photochemical along our desired direction by changing the pulse shape. It was found that both high yield and high purity of femtosecond photoassociation of Yb 2 molecule correspond to an optimal laser pulse with intensity 1.0 × 10 14 W/cm 2 , duration 1350 fs, and frequency 29290 cm −1 by positive chirp. e maximum photoassociative yield is up to 99% of optimal target D 1 Π u state for selected level v ′ 0. ose conditions are the best situation for quantum selective control of photoassociation.

Conclusions
In this work, the potential energy curves and transition dipole moment function for excited states of Yb 2 were calculated based on CASSCF/icMRCI + Q method with ECP60MDF basis set and 32213220 active spaces. e spectroscopic constants of PECs for five states are in good agreement with the experimental determinations. Combining the PECs with TDMCs, the optical transitional characters are investigated including Franck-Condon factors and Einstein A coefficients. By Franck-Condon filtering, the selective control of laser synthesis of Yb 2 molecule is achieved by purification. e corresponding vibrational transition levels from X 1 Σ + g to excited state (A 1 Σ u + , B 1 Π u , C 1 Σ u + , D 1 Π u ) are v ′ 23, 50, 55, and 0, respectively. e laser drive transitions to target states of association at wavelengths A 1 Σ u + : 464.7 nm, B 1 Π u : 373.1 nm, C 1 Σ u + : 361.8 nm, and D 1 Π u : 339.1 nm are obtained.
Based on the transitional properties, the corresponding associative yield and purity of photoassociative process have been calculated by using quantum wave packet dynamic method. e target states of photoassociation are A 1 Σ u + state and D 1 Π u state. For A 1 Σ u + , it is indicated that using Gaussian pulse made the value of v ′ 23 up to 97% at 725 fs. After a laser pulse, the positive chirp promotes the yield of vibrational states to increase, but the negative chirp inhibits it. For D 1 Π u , when laser intensity is 1.0 × 10 14 W/cm 2 , not only the purity but also the yield is higher up to 99% for target state v' � 0 at 1350 fs. at is to say, by changing the laser parameters and pulse shapes, it could more easily control the photochemical reaction along our desired direction. ose conditions will provide an important reference and suggest a scheme for a feasible photoassociation for further experimental and theoretical researches. A feasible route to the control of photo-induced bimolecular chemical reactions will come true. Current study also may promote an important step toward the realization of highly accurate quantum manipulation and control of binary photoreactions.

Data Availability
All data that support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.