# Investigating the Selective Control of Photoassociation of Yb2

Authors: Junxia Cheng; Bo Tian; Siyu Li; Jia Wang; Shenjiang Wu
Journal: International Journal of Optics
 (2021)

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## Abstract

The 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. The corresponding vibrational transitions from X1Σ+g to the excited state (A1Σu+, B1Πu, C1Σu+, and D1Πu) are v′ = 23, 50, 55, and 0, respectively. By using quantum wave packet dynamic methods, we calculated the yields with time evaluation for the selected target states. The projections of time-dependent wave functions of initial states on the target vibrational eigenstates reflected the synthetic yields of Yb2. For target A1Σu+, 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 D1Πu state, when laser intensity is 1.0 × 1014 W/cm2, the purity and yield of target state v′ = 0 reached the maximum at 1350 fs. That is to say, changing the laser parameters and pulse shapes could control the photochemical reaction along our desired direction. These 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.


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## 1. 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 [4–6], or to explore quantum many-body systems [7, 8], even to search for topological dark matter through its impact on the fine-structure constant [9, 10]. Meanwhile, molecular clocks promise increased sensitivity to the variation of the electron-to-proton mass ratio [11–15]. 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]. The 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 1S0+3P0 and 3P0+3P0 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 1S0+3P0 and 3P0+3P0 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. This 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 Yb2 and Cd2 based on photodissociation of (171Yb)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 excited170Yb174Yb and 174Yb176Yb 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(1S0) + Li(2P1/2) 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 Yb2 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 (X1Σ+g, A1Σu+, B1Πu, C1Σu+, D1Πu) 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). Then, 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 Yb2 molecule have been calculated by changing laser parameters and pulse shapes. Finally, the conclusions are made in Section 4.


## 2. Theory and Method of Calculations

### 2.1.Ab Initio Calculation

Theab initio calculations of electronic structure for the allowed transitional electronic states (X1Σg+, A1Σu+, B1Πu, C1Σu+, D1Πu) of Yb2 have been performed with MOLPRO [28] in the D2h point group, where the symmetries of Ag, B3u, B2u, B1g, B1u, B2g, B3g, and Au representations were used. The 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. The 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 Yb2 molecule, the active space of 4 electrons in 15 orbitals (6s, 6p, 5d) generates the appropriate reference wave functions. Thus, the corresponding active spaces are (32213220).


### 2.2. Wave Packet Dynamic Method

Combining with highly correlatedab 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]. The 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]. The projections of time-dependent wave functions of initial states on the vibrational eigenstates of excited states, reflecting the photoassociation yields, were investigate. The 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 Yb2 molecule.


## 2.1.Ab Initio Calculation

Theab initio calculations of electronic structure for the allowed transitional electronic states (X1Σg+, A1Σu+, B1Πu, C1Σu+, D1Πu) of Yb2 have been performed with MOLPRO [28] in the D2h point group, where the symmetries of Ag, B3u, B2u, B1g, B1u, B2g, B3g, and Au representations were used. The 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. The 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 Yb2 molecule, the active space of 4 electrons in 15 orbitals (6s, 6p, 5d) generates the appropriate reference wave functions. Thus, the corresponding active spaces are (32213220).


## 2.2. Wave Packet Dynamic Method

Combining with highly correlatedab 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]. The 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]. The projections of time-dependent wave functions of initial states on the vibrational eigenstates of excited states, reflecting the photoassociation yields, were investigate. The 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 Yb2 molecule.


## 3. Results and Discussion

### 3.1. Potential Energies and Transition Dipole Moments of States

In1, the potential energy curves (PECs) of Yb2 molecule with a valid range 2.0 ≤ R ≤ 20 Å have been presented, which included X1Σg+ and allowed transitions from ground to excited states A1Σu+, B1Πu, C1Σu+, and D1Πu. In order to estimate the accuracy of these various PECs over the range of R, we evaluated equilibrium internuclear distance (Re), electronic transition energy (Te), dissociation energy (De), harmonic frequency (ωe), anharmonic vibrational frequency (ωexe), and rotational frequency (Be) of all states at icMRCI + Q/ECP60MDF/32213220 levels by LEVEL program [36]. There are few spectroscopic studies on this molecule. These 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 Re of B1Πu state corresponding to the result. The values (2.5509 eV and 49.1833 cm−1) of electronic transition energy (Te) and harmonic frequency (ωe) of A1Σu+ are very close to the determinations (2.53 eV and 53 cm−1) by Wang, respectively. In addition, we also provided anharmonic vibrational frequency (ωexe) and rotational frequency (Be) of five states first time. It indicates that using icMRCI + Q method and the larger basis set are important for studying PEC. That 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. Therefore, we will apply those potentials to investigate the Franck–Condon factors (FCFs) and Einstein A coefficients in Section 3.2.Figure 1
The calculated potential energy curves (PECs) at the icMRCI + Q/ECP60MDF/32213220 level.Table 1
The spectroscopic parameters of calculated and experimental values of Yb2.
StatesRe (A)Te (eV)De (eV)ωe (cm−1)ωeχe (cm−1)Be (cm−1)ReferenceX1Σg+5.112000.043215.49670.21790.0075This work5.15800.04115[a]4.1900.1733[c]A1Σu+4.21102.55090.770149.18330.01300.0105This work3.8942.530.9753[b]B1Πu4.03302.96700.533768.232610.32130.0120This work3.8733.260.6268[b]C1Σu+3.91803.04420.730163.49180.00710.01277This workD1Πu4.92403.66750.896336.41660.09850.0080This workaYixuan Wang, Michael Dolg, 1998, CCSD; bused MRCI method; cGuido M, Balducci G., 1972.While calculating the potential energy functions, the transition dipole moment curves (TDMCs) of the allowed transitions are also investigated in theory shown in2, which included A1Σu+⟵X1Σg+, B1Πu⟵X1Σg+, C1Σu+⟵X1Σg+, and D1Πu⟵X1Σg+. It can be seen that the TDMCs of A1Σu+⟵X1Σg+ and C1Σu+⟵X1Σg+ firstly increase, but then C1Σu+⟵X1Σg decreases until it approaches zero. On the contrary, the TDMs of B1Πu⟵X1Σg+ and D1Πu⟵X1Σg+ decrease with R, and then the curve of D1Πu⟵X1Σg+ increases until it tends to zero. At the equipment internuclear distance of ground state, the values of TDMs are A1Σu+⟵X1Σg+: 4.51 a.u., B1Πu⟵X1Σg+: −1.84 a.u., C1Σu+⟵X1Σg+: 1.36 a.u., and D1Πu⟵X1Σg+: −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. The reason for this situation is that the different polarity of two states (A1Σu+, B1Πu) 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 (C1Σu+, D1Πu) tend to zero.Figure 2
Transition dipole moment curves for allowed transition of Yb2 investigated in theory.


### 3.2. 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 of exited states. Firstly, we calculated the larger FCFs of every system corresponding to vibrational bands of the allowed transitions, shown in 3. It can be seen that FCFs of vibrational transitions of A1Σu+←X1Σg+ and D1Πu←X1Σg+are relatively larger than B1Πu←X1Σg+ and C1Σu+←X1Σg+. The largest values of every state are A1Σu+: 0.083, B1Πu: 0.059, C1Σu+: 0.067, and D1Πu: 0.27, which correspond to upper-level v′ = 23, 50, 55, and 0, respectively. For D1Πu←X1Σg+ system, the FCF is much larger with respect to A1Σu+←X1Σg+. On this basis, Einstein A coefficients and lifetime will be conveniently expressed by the FCF results.Figure 3
The Frank–Condon factors of transitions from the ground state to vibrational levels of excited states.The Einstein A coefficients (also called vibrational transition probabilities) of transitional systems are investigated by combining potential energy with transition dipole moment at icMRCI/ECP60MDF/32213220 level. These transition probabilitiesAv'v'', which are defined with equation Av′J′v″J″=3.136189×10−7SJ′,J″/2J′+1νv′J′v″J″3Ψν′J′rMrψv″J″2, have been calculated including main transition bands. The larger probabilities are located in the Frank-Condon region, and the largest value corresponds to a strong transition. The results of transitional systems are that A1Σu+←X1Σg+:1.07 × 107 s−1, B1Πu←X1Σg+:3.18 × 106 s−1, C1Σu+←X1Σg+:1.60 × 106 s−1, and D1Πu←X1Σg+: 4.32 × 107 s−1 correspond to v′ = 23, 50, 55, and 0 level, respectively.According to Franck–Condon filtering and transitional probability, we plotted the selected vibrational levelsv′ of excited states with the population after photoassociation shown in 4, which provided a visualized description for the quantum control of photoassociation of Yb2. The equilibrium internuclear distance of D1Πu state is larger than A1Σ+u state that is similar to B1Πu and C1Σu+. The ground X1Σ+g state is only weakly bound by van der Waals force and has larger equilibrium internuclear distance corresponding to the excited states. There is rotational barrier of X1Σ+g for rotational quantum numbers (J″), already for 218, where the ground states do not support any bound levels. Above all, A1Σ+u and D1Πu are more suitable for the target states of photoassociation of Yb2 molecule.Figure 4
The selected optimally vibrational target states of excited states of Yb2.


### 3.3. Dynamic Process of Selective Control for Photoassociation

The 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 Yb2, 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. The projections of time-dependent wave functions of initial states on the target vibrational eigenstates, reflecting the photoassociative yields of Yb2, 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. The 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 X1Σ+g of Yb2.In5555, 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 (A1Σu+, B1Πu, C1Σu+, D1Πu) are plotted. That corresponds to laser pulse parameters: amplitude ε0 = 0.169 a.u.; duration tf = 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 A1Σu+: 13%, B1Πu: 8%, C1Σu+: 9%, and D1Πu: 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 A1Σu+ (v′ = 23): 30000 a.u., B1Πu (v′ 50): 36000 a.u., C1Σu+ (v′ 55): 39000 a.u., and D1Πu (v′ 1): 27000 a.u., respectively. In addition, when every vibrational energy level reaches the maximum, the required time becomes longer with the level. This indicates that the probability of optical transition of A1Σu+ state is greater than B1Σ+u and C1Σu+ states, which is just different from D1Πu. This conclusion is the same as that discussed above that the optimal target states of photoassociation are A1Σu+ state and D1Πu state. The proposed laser drive transitions to target states of photoassociative process at wavelength A1Σu+: 464.7 nm, B1Πu: 373.1 nm, C1Σu+: 361.8 nm, and D1Πu: 339.1 nm are obtained.Figure 5
The projections of the time-dependent vibrational wave function of Yb2, interacting with a resonant laser pulse, on the main five vibrational eigenstates of excited states (A1Σu+, B1Πu, C1Σu+, and D1Πu), corresponding to laser pulse parameters: amplitude ε0 = 0.169; duration tf 41341.37; frequency ω matches transition energy (in atomic units).For quasi target stateA1Σu+ 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 × 1014 W/cm2), 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 × 1014 W/cm2, 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). That 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. Therefore, the yields of vibrational levels are relatively lower. That do not benefit our accurate quantum manipulation of photoassociation.Figure 6
The effect on the projection of theA1Σu+ state by changing pulse shapes and laser intensities: (a, b) Gaussian profile; (c, d) +chirp profile; (e, f) −chirp profile.
(a)(b)(c)(d)(e)(f)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 D1Πu of photoassociation, shown in 7. There is a diverse oscillatory behavior observed from Figure 7(a). The 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. The 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 × 1014 W/cm2, 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 D1Πu shows a little change, which is a change between v′ 0 and v′ 1. The 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. The calculated yield of photoassociation is further enhanced by performing optimization of the best laser pulse.Figure 7
The effect on the projection of theD1Πu state by changing pulse shapes and laser intensities: (a, b) Gauss pulse; (c, d) +chirp pulse; (e, f) −chirp pulse.
(a)(b)(c)(d)(e)(f)On the whole, for Yb2 molecule, if the photoassociative process is nonadiabatic, the yields of target states are A1Σu+: 0.13%, B1Πu: 0.08%, C1Σu+: 0.09%, and D1Πu: 0.70%, respectively. If we use adiabatic association, the most suitable quantum manipulation of photoassociation is A1Σu+ and D1Πu 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. That is to say, different chirp pulses can be used to effectively promote or inhibit the yields of photochemical reactions. Therefore, 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 Yb2 molecule correspond to an optimal laser pulse with intensity 1.0 × 1014 W/cm2, duration 1350 fs, and frequency 29290 cm−1 by positive chirp. The maximum photoassociative yield is up to 99% of optimal target D1Πu state for selected level v′ 0. Those conditions are the best situation for quantum selective control of photoassociation.


## 3.1. Potential Energies and Transition Dipole Moments of States

In1, the potential energy curves (PECs) of Yb2 molecule with a valid range 2.0 ≤ R ≤ 20 Å have been presented, which included X1Σg+ and allowed transitions from ground to excited states A1Σu+, B1Πu, C1Σu+, and D1Πu. In order to estimate the accuracy of these various PECs over the range of R, we evaluated equilibrium internuclear distance (Re), electronic transition energy (Te), dissociation energy (De), harmonic frequency (ωe), anharmonic vibrational frequency (ωexe), and rotational frequency (Be) of all states at icMRCI + Q/ECP60MDF/32213220 levels by LEVEL program [36]. There are few spectroscopic studies on this molecule. These 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 Re of B1Πu state corresponding to the result. The values (2.5509 eV and 49.1833 cm−1) of electronic transition energy (Te) and harmonic frequency (ωe) of A1Σu+ are very close to the determinations (2.53 eV and 53 cm−1) by Wang, respectively. In addition, we also provided anharmonic vibrational frequency (ωexe) and rotational frequency (Be) of five states first time. It indicates that using icMRCI + Q method and the larger basis set are important for studying PEC. That 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. Therefore, we will apply those potentials to investigate the Franck–Condon factors (FCFs) and Einstein A coefficients in Section 3.2.Figure 1
The calculated potential energy curves (PECs) at the icMRCI + Q/ECP60MDF/32213220 level.Table 1
The spectroscopic parameters of calculated and experimental values of Yb2.
StatesRe (A)Te (eV)De (eV)ωe (cm−1)ωeχe (cm−1)Be (cm−1)ReferenceX1Σg+5.112000.043215.49670.21790.0075This work5.15800.04115[a]4.1900.1733[c]A1Σu+4.21102.55090.770149.18330.01300.0105This work3.8942.530.9753[b]B1Πu4.03302.96700.533768.232610.32130.0120This work3.8733.260.6268[b]C1Σu+3.91803.04420.730163.49180.00710.01277This workD1Πu4.92403.66750.896336.41660.09850.0080This workaYixuan Wang, Michael Dolg, 1998, CCSD; bused MRCI method; cGuido M, Balducci G., 1972.While calculating the potential energy functions, the transition dipole moment curves (TDMCs) of the allowed transitions are also investigated in theory shown in2, which included A1Σu+⟵X1Σg+, B1Πu⟵X1Σg+, C1Σu+⟵X1Σg+, and D1Πu⟵X1Σg+. It can be seen that the TDMCs of A1Σu+⟵X1Σg+ and C1Σu+⟵X1Σg+ firstly increase, but then C1Σu+⟵X1Σg decreases until it approaches zero. On the contrary, the TDMs of B1Πu⟵X1Σg+ and D1Πu⟵X1Σg+ decrease with R, and then the curve of D1Πu⟵X1Σg+ increases until it tends to zero. At the equipment internuclear distance of ground state, the values of TDMs are A1Σu+⟵X1Σg+: 4.51 a.u., B1Πu⟵X1Σg+: −1.84 a.u., C1Σu+⟵X1Σg+: 1.36 a.u., and D1Πu⟵X1Σg+: −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. The reason for this situation is that the different polarity of two states (A1Σu+, B1Πu) 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 (C1Σu+, D1Πu) tend to zero.Figure 2
Transition dipole moment curves for allowed transition of Yb2 investigated in theory.


## 3.2. 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 of exited states. Firstly, we calculated the larger FCFs of every system corresponding to vibrational bands of the allowed transitions, shown in 3. It can be seen that FCFs of vibrational transitions of A1Σu+←X1Σg+ and D1Πu←X1Σg+are relatively larger than B1Πu←X1Σg+ and C1Σu+←X1Σg+. The largest values of every state are A1Σu+: 0.083, B1Πu: 0.059, C1Σu+: 0.067, and D1Πu: 0.27, which correspond to upper-level v′ = 23, 50, 55, and 0, respectively. For D1Πu←X1Σg+ system, the FCF is much larger with respect to A1Σu+←X1Σg+. On this basis, Einstein A coefficients and lifetime will be conveniently expressed by the FCF results.Figure 3
The Frank–Condon factors of transitions from the ground state to vibrational levels of excited states.The Einstein A coefficients (also called vibrational transition probabilities) of transitional systems are investigated by combining potential energy with transition dipole moment at icMRCI/ECP60MDF/32213220 level. These transition probabilitiesAv'v'', which are defined with equation Av′J′v″J″=3.136189×10−7SJ′,J″/2J′+1νv′J′v″J″3Ψν′J′rMrψv″J″2, have been calculated including main transition bands. The larger probabilities are located in the Frank-Condon region, and the largest value corresponds to a strong transition. The results of transitional systems are that A1Σu+←X1Σg+:1.07 × 107 s−1, B1Πu←X1Σg+:3.18 × 106 s−1, C1Σu+←X1Σg+:1.60 × 106 s−1, and D1Πu←X1Σg+: 4.32 × 107 s−1 correspond to v′ = 23, 50, 55, and 0 level, respectively.According to Franck–Condon filtering and transitional probability, we plotted the selected vibrational levelsv′ of excited states with the population after photoassociation shown in 4, which provided a visualized description for the quantum control of photoassociation of Yb2. The equilibrium internuclear distance of D1Πu state is larger than A1Σ+u state that is similar to B1Πu and C1Σu+. The ground X1Σ+g state is only weakly bound by van der Waals force and has larger equilibrium internuclear distance corresponding to the excited states. There is rotational barrier of X1Σ+g for rotational quantum numbers (J″), already for 218, where the ground states do not support any bound levels. Above all, A1Σ+u and D1Πu are more suitable for the target states of photoassociation of Yb2 molecule.Figure 4
The selected optimally vibrational target states of excited states of Yb2.


## 3.3. Dynamic Process of Selective Control for Photoassociation

The 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 Yb2, 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. The projections of time-dependent wave functions of initial states on the target vibrational eigenstates, reflecting the photoassociative yields of Yb2, 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. The 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 X1Σ+g of Yb2.In5555, 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 (A1Σu+, B1Πu, C1Σu+, D1Πu) are plotted. That corresponds to laser pulse parameters: amplitude ε0 = 0.169 a.u.; duration tf = 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 A1Σu+: 13%, B1Πu: 8%, C1Σu+: 9%, and D1Πu: 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 A1Σu+ (v′ = 23): 30000 a.u., B1Πu (v′ 50): 36000 a.u., C1Σu+ (v′ 55): 39000 a.u., and D1Πu (v′ 1): 27000 a.u., respectively. In addition, when every vibrational energy level reaches the maximum, the required time becomes longer with the level. This indicates that the probability of optical transition of A1Σu+ state is greater than B1Σ+u and C1Σu+ states, which is just different from D1Πu. This conclusion is the same as that discussed above that the optimal target states of photoassociation are A1Σu+ state and D1Πu state. The proposed laser drive transitions to target states of photoassociative process at wavelength A1Σu+: 464.7 nm, B1Πu: 373.1 nm, C1Σu+: 361.8 nm, and D1Πu: 339.1 nm are obtained.Figure 5
The projections of the time-dependent vibrational wave function of Yb2, interacting with a resonant laser pulse, on the main five vibrational eigenstates of excited states (A1Σu+, B1Πu, C1Σu+, and D1Πu), corresponding to laser pulse parameters: amplitude ε0 = 0.169; duration tf 41341.37; frequency ω matches transition energy (in atomic units).For quasi target stateA1Σu+ 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 × 1014 W/cm2), 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 × 1014 W/cm2, 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). That 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. Therefore, the yields of vibrational levels are relatively lower. That do not benefit our accurate quantum manipulation of photoassociation.Figure 6
The effect on the projection of theA1Σu+ state by changing pulse shapes and laser intensities: (a, b) Gaussian profile; (c, d) +chirp profile; (e, f) −chirp profile.
(a)(b)(c)(d)(e)(f)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 D1Πu of photoassociation, shown in 7. There is a diverse oscillatory behavior observed from Figure 7(a). The 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. The 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 × 1014 W/cm2, 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 D1Πu shows a little change, which is a change between v′ 0 and v′ 1. The 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. The calculated yield of photoassociation is further enhanced by performing optimization of the best laser pulse.Figure 7
The effect on the projection of theD1Πu state by changing pulse shapes and laser intensities: (a, b) Gauss pulse; (c, d) +chirp pulse; (e, f) −chirp pulse.
(a)(b)(c)(d)(e)(f)On the whole, for Yb2 molecule, if the photoassociative process is nonadiabatic, the yields of target states are A1Σu+: 0.13%, B1Πu: 0.08%, C1Σu+: 0.09%, and D1Πu: 0.70%, respectively. If we use adiabatic association, the most suitable quantum manipulation of photoassociation is A1Σu+ and D1Πu 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. That is to say, different chirp pulses can be used to effectively promote or inhibit the yields of photochemical reactions. Therefore, 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 Yb2 molecule correspond to an optimal laser pulse with intensity 1.0 × 1014 W/cm2, duration 1350 fs, and frequency 29290 cm−1 by positive chirp. The maximum photoassociative yield is up to 99% of optimal target D1Πu state for selected level v′ 0. Those conditions are the best situation for quantum selective control of photoassociation.


## 4. Conclusions

In this work, the potential energy curves and transition dipole moment function for excited states of Yb2 were calculated based on CASSCF/icMRCI + Q method with ECP60MDF basis set and 32213220 active spaces. The 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 Yb2 molecule is achieved by purification. The corresponding vibrational transition levels from X1Σ+g to excited state (A1Σu+, B1Πu, C1Σu+, D1Πu) are v′ 23, 50, 55, and 0, respectively. The laser drive transitions to target states of association at wavelengths A1Σu+: 464.7 nm, B1Πu: 373.1 nm, C1Σu+: 361.8 nm, and D1Πu: 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. The target states of photoassociation areA1Σu+ state and D1Πu state. For A1Σu+, 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 D1Πu, when laser intensity is 1.0 × 1014 W/cm2, not only the purity but also the yield is higher up to 99% for target state v' = 0 at 1350 fs. That is to say, by changing the laser parameters and pulse shapes, it could more easily control the photochemical reaction along our desired direction. Those 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.



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