Investigating the Selective Control of Photoassociation of Yb<sub>2</sub>

Cheng, Junxia; Tian, Bo; Li, Siyu; Wang, Jia; Wu, Shenjiang

doi:https://doi.org/10.1155/2021/6859330

International Journal of Optics

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6859330 | https://doi.org/10.1155/2021/6859330

Investigating the Selective Control of Photoassociation of Yb₂

Junxia Cheng,¹Bo Tian,²Siyu Li,¹Jia Wang,¹and Shenjiang Wu¹

Academic Editor: Sulaiman W. Harun

Received02 Jun 2021

Accepted24 Aug 2021

Published06 Sept 2021

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 (A¹Σ_u⁺, B¹Π_u, C¹Σ_u⁺, and D¹Π_u) are = 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 A¹Σ_u⁺, we used Gaussian pulse to make the yield of = 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 D¹Π_u state, when laser intensity is 1.0 × 10¹⁴ W/cm², the purity and yield of target state = 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.

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 excited ¹⁷⁰Yb¹⁷⁴Yb 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(²P_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 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, A¹Σ_u⁺, B¹Π_u, C¹Σ_u⁺, D¹Π_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). 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

The ab initio calculations of electronic structure for the allowed transitional electronic states (X¹Σ_g⁺, A¹Σ_u⁺, B¹Π_u, C¹Σ_u⁺, D¹Π_u) of Yb₂ have been performed with MOLPRO [28] in the D_2h point group, where the symmetries of , B_3u, B_2u, B_1g, B_1u, B_2g, B_3g, and A_u 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 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]. 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

In 1, the potential energy curves (PECs) of Yb₂ molecule with a valid range 2.0 ≤ R ≤ 20 Å have been presented, which included X¹Σ_g⁺ and allowed transitions from ground to excited states A¹Σ_u⁺, B¹Π_u, C¹Σ_u⁺, and D¹Π_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_ex_e), and rotational frequency (B_e) 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 R_e of B¹Π_u state corresponding to the result. The values (2.5509 eV and 49.1833 cm⁻¹) of electronic transition energy (T_e) and harmonic frequency (ω_e) of A¹Σ_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_ex_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. 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.

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 A¹Σ_u⁺⟵X¹Σ_g⁺, B¹Π_u⟵X¹Σ_g⁺, C¹Σ_u⁺⟵X¹Σ_g⁺, and D¹Π_u⟵X¹Σ_g⁺. It can be seen that the TDMCs of A¹Σ_u⁺⟵X¹Σ_g⁺ and C¹Σ_u⁺⟵X¹Σ_g⁺ firstly increase, but then C¹Σ_u⁺⟵X¹Σ_g decreases until it approaches zero. On the contrary, the TDMs of B¹Π_u⟵X¹Σ_g⁺ and D¹Π_u⟵X¹Σ_g⁺ decrease with R, and then the curve of D¹Π_u⟵X¹Σ_g⁺ increases until it tends to zero. At the equipment internuclear distance of ground state, the values of TDMs are A¹Σ_u⁺⟵X¹Σ_g⁺: 4.51 a.u., B¹Π_u⟵X¹Σ_g⁺: −1.84 a.u., C¹Σ_u⁺⟵X¹Σ_g⁺: 1.36 a.u., and D¹Π_u⟵X¹Σ_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. The reason for this situation is that the different polarity of two states (A¹Σ_u⁺, B¹Π_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 (C¹Σ_u⁺, D¹Π_u) tend to zero.

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 A¹Σ_u⁺←X¹Σ_g⁺ and D¹Π_u←X¹Σ_g⁺are relatively larger than B¹Π_u←X¹Σ_g⁺ and C¹Σ_u⁺←X¹Σ_g⁺. The largest values of every state are A¹Σ_u⁺: 0.083, B¹Π_u: 0.059, C¹Σ_u⁺: 0.067, and D¹Π_u: 0.27, which correspond to upper-level = 23, 50, 55, and 0, respectively. For D¹Π_u←X¹Σ_g⁺ system, the FCF is much larger with respect to A¹Σ_u⁺←X¹Σ_g⁺. On this basis, Einstein A coefficients and lifetime will be conveniently expressed by the FCF results.

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 probabilities A_v'v'', which are defined with equation , 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 A¹Σ_u⁺←X¹Σ_g⁺:1.07 × 10⁷ s⁻¹, B¹Π_u←X¹Σ_g⁺:3.18 × 10⁶ s⁻¹, C¹Σ_u⁺←X¹Σ_g⁺:1.60 × 10⁶ s⁻¹, and D¹Π_u←X¹Σ_g⁺: 4.32 × 10⁷ s⁻¹ correspond to = 23, 50, 55, and 0 level, respectively.

According to Franck–Condon filtering and transitional probability, we plotted the selected vibrational levels 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 D¹Π_u state is larger than A¹Σ⁺_u state that is similar to B¹Π_u and C¹Σ_u⁺. 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 D¹Π_u are more suitable for the target states of photoassociation of Yb₂ molecule.

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.

In 5 5 5 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 (A¹Σ_u⁺, B¹Π_u, C¹Σ_u⁺, D¹Π_u) are plotted. That corresponds to laser pulse parameters: amplitude ε₀ = 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 A¹Σ_u⁺: 13%, B¹Π_u: 8%, C¹Σ_u⁺: 9%, and D¹Π_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 A¹Σ_u⁺ ( = 23): 30000 a.u., B¹Π_u ( 50): 36000 a.u., C¹Σ_u⁺ ( 55): 39000 a.u., and D¹Π_u ( 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 A¹Σ_u⁺ state is greater than B¹Σ⁺_u and C¹Σ_u⁺ states, which is just different from D¹Π_u. This conclusion is the same as that discussed above that the optimal target states of photoassociation are A¹Σ_u⁺ state and D¹Π_u state. The proposed laser drive transitions to target states of photoassociative process at wavelength A¹Σ_u⁺: 464.7 nm, B¹Π_u: 373.1 nm, C¹Σ_u⁺: 361.8 nm, and D¹Π_u: 339.1 nm are obtained.

For quasi target state A¹Σ_u⁺ of photoassociation, the projections of initial wave function on the five vibrational eigenstates ( 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 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 24 is larger than other vibrational levels ( 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.

(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 0 for target D¹Π_u 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 1, which lead to a little increase for the yields of 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 D¹Π_u shows a little change, which is a change between 0 and 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.

(a)

(b)

(c)

(d)

(e)

(f)

On the whole, for Yb₂ molecule, if the photoassociative process is nonadiabatic, the yields of target states are A¹Σ_u⁺: 0.13%, B¹Π_u: 0.08%, C¹Σ_u⁺: 0.09%, and D¹Π_u: 0.70%, respectively. If we use adiabatic association, the most suitable quantum manipulation of photoassociation is A¹Σ_u⁺ and D¹Π_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. 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 D¹Π_u state for selected level 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 (A¹Σ_u⁺, B¹Π_u, C¹Σ_u⁺, D¹Π_u) are 23, 50, 55, and 0, respectively. The laser drive transitions to target states of association at wavelengths A¹Σ_u⁺: 464.7 nm, B¹Π_u: 373.1 nm, C¹Σ_u⁺: 361.8 nm, and D¹Π_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. The target states of photoassociation are A¹Σ_u⁺ state and D¹Π_u state. For A¹Σ_u⁺, it is indicated that using Gaussian pulse made the value of 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 D¹Π_u, 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.

Data Availability

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

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

The authors acknowledge the financial support from the Natural Science Foundation of China (nos. 11947126, 61701385, and 62001362), Foundation of Equipment Pre-Research Area of China (nos. 6140619010301 and 61406190501), and Shanxi Provincial Department of Education (nos. 20JS061 and 20JK0689).

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Copyright © 2021 Junxia Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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