The article information
 Yuhui Pang, Binbin Wang, Yongchang Han, Shulin Conga, Yingyu Niu
 庞玉慧, 王彬彬, 韩永昌, 丛书林, 牛英煜
 Isotope E ects on TwoPhoton Population Transfer Processes of HF and DF
 HF和DF分子双光子布居转移过程中的同位素效应
 Chinese Journal of Chemical Physics, 2016, 29(3): 297302
 化学物理学报, 2016, 29(3): 297302
 http://dx.doi.org/10.1063/16740068/29/cjcp1509186

Article history
 Received on September 2, 2015
 Accepted on January 13, 2016
b. School of Science, Dalian Jiaotong University, Dalian 116028, China
Population transfer between two quantum states has been extensively studied in recent years for the important applications in molecular spectroscopy [1],chemical reaction dynamics [2],collision dynamics [3],and so on. Therefore,to obtain an efficient population transfer (close to 100%) from an initial state to a target state,various techniques are proposed,including the stimulated Raman adiabatic passage (STIRAP) [4],the temporal coherent control [5],the chirped adiabatic passage [6] etc. With the development and application of the ultraintense and ultrashort laser pulse technique,the nonlinear optical phenomena may occur during the lasermolecule interactions. The multiphoton transition process is one of the most typical nonlinear phenomena. Accordingly,the multiphoton resonance is regarded to be an alternative way to achieve efficient population transfer. Gibson showed that the adiabatic transfer of population was possible on high order multiphoton (at least 12photon) transitions [7]. Maeda et al. observed the population transfer in the Li atom between Rydberg states by high order multiphoton adiabatic rapid passage [8]. Demeter and Djotyan investigated the practical usefulness of multiphoton adiabatic transitions for the manipulation of the atomic mechanical state [9]. Recently,we have investigated the efficient population transfer process between the vibrational levels $\upsilon$ =0 and 2 of the HF system [10]. It is found that the twophoton transfer process was diabatic,in which the vibrational levels $\upsilon$ =1 and 3 may be involved. Although at some high field peak amplitudes,part of the population can reside on the intermediate state $\upsilon$ =1,a high transition probability to $\upsilon$ =2,can still be achieved via the twophoton transition process at appropriate laser parameters.
Since the isotopic substitution not only changes the mass of the system but also shifts the energy position of rovibrational levels,many researchers focus their attention on the isotope effects on various aspects of molecular dynamics in recent years. For the spectra of stimulated emission in the process of the H $_{2}$ +F $_{2}$ $\rightarrow$ HF and D $_{2}$ +F $_{2}$ $\rightarrow$ DF reactions,Basov et al. found that the emission energy in the transitions of 21 and 54 bands in the H $_{2}$ +F $_{2}$ mixture was the highest possible. However,in the D $_{2}$ +F $_{2}$ mixture,the emission energy maxima corresponded to the transitions of 32 and 87 bands [11]. For the reactive collisions of hydrogen molecular cations,Stroe and Fifirig investigated the isotope effects on the cross sections of the dissociative excitation (DE),dissociative recombination (DR),vibrational excitation and vibrational deexcitation of H $_{2}$ $^{+}$ ,D $_{2}$ $^{+}$ ,and T $_{2}$ $^{+}$ [12].They found that the DE,DR cross sections of D $_{2}$ $^{+}$ were smaller than those of H $_{2}$ $^{+}$ when the colliding election energies were below the direct DE threshold. For collision reactions in gas phase,the shape resonance phenomenon for exchange reactions of X+HOX (X=H,D) varies [13, 14] with the substitution of the H atom for D atom. For the photoassociation reactions of X+ F $\rightarrow$ XF (X=H,D),the isotope effect has been connected with the FrankCondon overlap integral [15]. However,to the best of our knowledge,the isotope effects on the population transfer process via twophoton resonance excitation have not been well studied.
In this work,to study the isotope effects on the twophoton population transfer process,we consider the DF and HF molecular systems,and set the vibrational levels $\upsilon$ =0 to 2 of the ground electronic state to be the initial and target states for each system. The transition process between these two states is of importance because it is an overtone transition under the condition of onephoton resonance [16]. We investigate the influences of the field peak amplitude and pulse duration on the twophoton population transfer processes,and present detailed comparisons between the two systems. Because the vibrational energy difference for the DF molecule is smaller than that for the HF molecule,we expect that the optimized pulse duration for the DF molecule would be longer than that for the HF molecule. We also expect that the twophoton transition process of the DF system is diabatic,similarly to the HF system [10]. Thus,the numerical calculations based on timedependent wavepacket method,which includes the influences of all vibrational levels,are performed to verify these expectations and to obtain detailed dynamics information. Moreover,in the calculations,we consider very large variation ranges for the laser intensity and the pulse duration,which might be instructive for relevant experimental research.
Ⅱ.THEORETICAL CALCULATIONIn the BornOppenheimer approximation,the timedependent Schrödinger equation (TDSE),describing the interaction of the diatomic molecule with a linearly polarized laser field,can be written as (where atomic units are used throughout unless indicated otherwise):
$ i\frac{\partial }{\partial t}\psi (t)=\hat{H}(t)\psi (t) $  (1) 
$ \hat H(t)= \hat H_{0}+\hat H_{\textrm{int}} $  (2) 
$ \begin{matrix} \hat{H}(t)={{{\hat{H}}}_{0}}+{{{\hat{H}}}_{\text{int}}}{{{\hat{H}}}_{0}}=\hat{T}+\hat{V}(r)\text{ } \\ =\frac{1}{2m}\frac{{{\partial }^{2}}}{\partial {{r}^{2}}}+\hat{V}(r) \\ \end{matrix} $  (3) 
In this study,we neglected the influence of the rotational motion of the molecule. This treatment is raised from two considerations: (ⅰ) we focus on the vibrationalstateselected population transfer process,(ⅱ) the pulse duration we considered in this work is much shorter than the rotational periods of the two molecules. For the effect of the molecular rotation on the excitation process,one can refer to Ref.[17].
The potential energy function, $\hat V(r)$ ,is given by the Morse function:
$ \hat V(r) = D_{\textrm{e}}\{ \exp[2\beta_{\textrm{e}}(rr_{\textrm{e}})]2\exp[\beta_{\textrm{e}}(rr_{\textrm{e}})] \} $  (4) 
$ \hat H_{\textrm{int}} = \mu(r)\varepsilon(t) $  (5) 
$ \mu(r) =\mu_{0}r\exp(\gamma r^{4}) $  (6) 
$ \begin{matrix} \varepsilon (t)=f(t)\cos (\omega t)\text{ } \\ ={{\varepsilon }_{0}}{{\sin }^{2}}\left( \frac{\pi t}{{{t}_{p}}} \right)\cos (\omega t) \\ \end{matrix} $  (7) 
At the initial time, $t$ =0,the diatomic molecule is assumed to be in the ground vibrational ( $\upsilon$ =0) state. The initial wave function is obtained by using the Fourier grid Hamiltonian method [19]. Starting from this initial wave function,the timedependent wave function is calculated by using the splitoperator method [20, 21, 22].
$ \begin{matrix} psi(t+\Delta t)=\exp [i\hat{H}\Delta t]\psi (t) \\ \approx \exp \left( i\hat{T}\frac{\Delta t}{2} \right)\exp [i(\hat{V}+{{{\hat{H}}}_{\text{int}}})\Delta t]\cdot \\ \exp \left( i\hat{T}\frac{\Delta t}{2} \right)\psi (t) \\ \end{matrix} $  (8) 
To perform the calculation for Eq.(8),the wave function is transformed between the momentum and coordinate grid representations to interact with the kinetic energy and potential energy operators,respectively. The fast Fourier transform (FFT) [23, 24, 25, 26] method is used to accomplish the transformation between the two representations. In the calculation,120000 uniform time grids in the region of [0,2000] fs and 256 uniform space grids in the region of [0.05,20] Bohr are used. The numerical results have been checked to be converged,by using the smaller time grid size,longer time range,the smaller space grid size,and larger space range,respectively.
The timedependent population on a specific vibrational eigenstate $v\rangle$ is given by
$ P_{v}(t) = \langle v\psi(t) \rangle^2 $  (9) 
The reduced masses of the DF and HF molecules are 3319.561 and 1744.632 a.u.,respectively. In the calculation,the vibrational levels $\upsilon$ =0 and 2 of the ground electronic state are taken to be the initial and target states,respectively,for both the DF and HF molecules. The relevant vibrational levels and transition processes are schematically illustrated in Fig. 1. The central frequency of the laser pulse used for the DF (HF) molecule is set to be 0.013062 (0.017674) a.u.,which is half of the energy difference between the vibrational levels $\upsilon$ =0 and 2.
The variation of the final population versus the field peak amplitude and pulse duration is shown in Fig. 2. The field peak amplitude varies from 0.001 a.u. to 0.06 a.u. with an interval of 0.0005 a.u. and pulse duration varies from 10 fs to 2000 fs with an interval of 10 fs. The variations of the final population on the vibrational levels $\upsilon$ =0,2,and 1 of the DF molecule are shown in Fig. 2 (a)(c),respectively,and the corresponding ones for the HF molecule are shown in Fig. 2 (d)(f),respectively. As seen in Fig. 2 (b) and (f),the final population on the target vibrational state $\upsilon$ =2 shows an oscillatory behavior dependent on the field peak amplitude and pulse duration for both the DF and HF systems. Comparing Fig. 2 (b) with (e),we found that the required pulse duration is longer than 860 fs for the DF molecule to achieve a relatively high transition probability (more than 80%),while that is only longer than 460 fs for the HF molecule. We found that there are many ``bright stripe" regions in Fig. 2(c) and Fig. 2(f). This indicates that part of the population may reside on the level $\upsilon$ =1 for the two systems during the transition process from level $\upsilon$ =0 to 2. Obviously,compared to the HF system,in the population transfer process from the initial state $\upsilon$ =0 to the target state $\upsilon$ =2,the population of the DF system may more easily reside on the intermediate state $\upsilon$ =1. We can understand this by comparing the energy differences, $E_{\upsilon=1}$  $E_{\upsilon=0}\hbar \omega$ ,for the two molecular systems. As seen in Table Ⅰ,the photon energy we used for either system is nearresonant to the transition energy between $\upsilon$ =0 and $\upsilon$ =1,and hence the population may reside on the level 1 during the twophoton absorption from $\upsilon$ =0 to $\upsilon$ =2. Moreover,the value, $E_{\upsilon=1}$  $E_{\upsilon=0}\hbar \omega$ for the DF system is quite smaller than the one for the HF system,and this is consistent with the findings that the population may more easily reside on $\upsilon$ =1 for the DF system.
In order to investigate the effects of the field peak amplitude on the final population in the twophoton population transfer for the DF and HF systems,we set the pulse duration at 1280 fs,corresponding to the vertical dashed line marked in Fig. 2. The variations of the final population on all relevant levels as a function of the field peak amplitude are shown in Fig. 3(a) for the DF molecule and Fig. 3(b) for the HF molecule,respectively. In our considered range of field peak amplitude,it shows five peaks of the final population on the target state $\upsilon$ =2 for the DF molecule in Fig. 3(a). However,it shows six peaks for the HF molecule in Fig. 3(b). This phenomenon can be explained by the twophoton Rabi oscillations [27, 28, 29, 30]. Rabi oscillation directly connects with the coupling of the relevant quantum states,and it can occur in both boundbound transitions and the continuumcontinuum interactions [28]. Many important phenomena are related with Rabi oscillation,such as the the AutlerTownes Splitting [29] and the interference structure in the photoelectron spectra [30],etc. The stronger the coupling is,the more frequent the Rabi oscillation is. We calculate the constant coefficient of the twophoton Rabi frequency for the two systems. The constant coefficient of the DF system,0.00106 a.u.,is weaker than that of the HF system,0.00126 a.u. Consequently,the oscillation period of the final population with the field peak amplitude for the DF molecule is relatively larger than the HF molecule. Thus,we can see that the coupling intensity between the $\upsilon$ =0 and $\upsilon$ =2 levels for the DF system is weaker than that for the HF system.
As mentioned above,the intermediate state $\upsilon$ =1 can be involved for both systems during the twophotonresonant transition. As seen in Fig. 3,the population residing on the level $\upsilon$ =1, $P_{1}$ ,presents several peaks at certain field peak amplitudes. For both systems,the peak values for $P_{1}$ increase with the increase of the field peak amplitude. For the DF molecule,the peak values for $P_{1}$ increase from 5.4% to 32%,while for the HF molecule,the peak values for $P_{1}$ increase from 0.9% to 13.6%. Thus,in our considered range of field peak amplitude from 0.001 a.u. to 0.06 a.u.,the probability for the population residing on the level $\upsilon$ =1 can be better suppressed for the HF molecule than for the DF molecule.
Moreover,it can be seen from Fig. 3 that the level $\upsilon$ =3 may also be involved for the DF system when the field peak amplitude is in the relatively strong range,roughly from 0.04 a.u. to 0.06 a.u. The peak probability for the population residing on level $\upsilon$ =3 is roughly 2.6% for the DF system. However,we do not find any population residing on $\upsilon$ =3 for the HF system in the same range of field amplitude.
We now investigate the variations of the final population on all relevant levels with the pulse duration for the two systems,by fixing the field peak amplitude at 0.0175 a.u.,corresponding to the horizontal dashed line in Fig. 2. As shown in Fig. 4,the population may reside on levels $\upsilon$ =1 and 3 in certain regions of pulse duration for both the two systems and the peak probability for the population residing on the level $\upsilon$ =1 decreases with the increase of the pulse duration. The peak probability for the population residing on the intermediate level $\upsilon$ =1 for the DF system decreases from 53% to 14% with the increase of the pulse duration from 400 fs to 2000 fs,while that for the HF system decreases from 32.8% to nearly 0%. The level $\upsilon$ =3 is involved for the DF molecule when the pulse duration is shorter than 600 fs,however, $\upsilon$ =3 is not involved for the HF molecule until the pulse duration is shorter than 400 fs. The maximal probability for the population residing on the level $\upsilon$ =3 of the DF molecule is roughly 28% and that of the HF molecule is roughly 4.7%. Thus,considering the population variation with the pulse duration,we can draw the same conclusion that it is more possible for the population to reside the intermediate level $\upsilon$ =1 and higher level $\upsilon$ =3 for the DF system,compared to the HF system.
By setting the field peak amplitude and pulse duration as 0.0175 a.u. and 1280 fs,respectively,we investigate the timedependent population on all relevant levels during the twophoton transfer process. With this set of parameters,the almost complete population transfer can be achieved for both systems. The population transfer processes of the DF and HF molecules are shown in Fig. 5 (a) and (b),respectively. Although nearly complete population transfer are achieved with this set of laser parameters,the population transfer processes are nonadiabatic,because the levels $\upsilon$ =1 and 3 are involved in the process for both systems. The maximal temporary probabilities for the population excited to the levels $\upsilon$ =1 and 3 for the HF molecule are roughly 30% and 15.5%,respectively,while those for the DF molecule are roughly 56% and 26%,respectively. Thus,the intermediate level $\upsilon$ =1 and higher level $\upsilon$ =3 of the DF molecule may play more important roles in the twophoton resonance process than those of the HF molecule.
The difference,between the two systems,in the twophotonresonant transition process can be understood by comparing the photon energy with the energy difference of the neighboring vibrational levels. As seen in Table Ⅰ,for the DF system,the photon energy is more nearresonant with the neighboring vibrational levels,and hence,the onephoton transitions,such as, $\upsilon$ =0 $\longleftrightarrow$ $\upsilon$ =1, $\upsilon$ =1 $\longleftrightarrow$ $\upsilon$ =2, $\upsilon$ =2 $\longleftrightarrow$ $\upsilon$ =3,may more possibly occur. This is because that compared to the case of the HF molecule,the vibrational levels in the DF molecule are displaced more evenly and closely.
Ⅳ. CONCLUSIONThe isotope effects on the twophotonresonant population transition process are investigated for the DF and HF molecular systems. The vibrational levels $\upsilon$ =0 and 2 of the ground electronic state are taken to be the initial and target states. By solving the timedependent Schrödinger equation with quantum wave packet method,we discuss the influences of the field peak amplitude and pulse duration on the twophotonresonant population transfer process in detail as well as the roles of the intermediate level $\upsilon$ =1 and the higher level $\upsilon$ =3. The population residing on these two levels may increase with the increase of the field peak amplitude,and with the decrease of the pulse duration. By comparing the two molecular system,we find that it is more possible for the population to reside on $\upsilon$ =1 and $\upsilon$ =3 for the DF molecule,because the vibrational levels in the DF molecule are displaced more evenly and closely than the corresponding ones in the HF molecule.
Ⅴ. ACKNOWLEDGMENTSThis work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education (No.20130041120053),Scientific Research Foundation for Returned Overseas Chinese Scholars,State Education Ministry,the Science and Technology Research Funds of the Department of Education of Liaoning Province (No.L2013014),Liaoning Baiqian Wan Talents Program,the National Magnetic Confinement Fusion Science Program (No.2013GB109005),the Fundamental Research Funds for the Central Universities (No.DUT15YQ105),and the National Natural Science Foundation of China (No.21473018,No.11274056,and No.11347012).
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b. 大连交通大学理学院, 大连 116028