分子轉動和激光脈沖對多光子激發控制的影響

馮海冉 李 鵬 鄭雨軍 王德華

(1濟寧學院物理與信息工程系,山東曲阜273155;2山東大學物理學院,濟南250100;3魯東大學物理與光電工程學院,山東煙臺264025)

1 Introduction

Molecular multiphoton processes have been researched hotspot for more than 20 years.Considerable interest has been focused on the influence of molecular-orientation on multiphoton processes.1-8Considering molecular rotational motion,the orientation of molecules relative to the polarization of a laser field is an important factor for the control of multiphoton excitations.However,most theoretical works have examined the problem by solving the time-dependent Schr?dinger equation.Few analytical studies have been conducted on the influence of molecular rotation on the control of multiphoton processes.For the past years,applications of the dynamic Lie-algebraic approach have steadily advanced.9-15Algebraic methods have been extensively used to investigate problems in nuclear physics,molecular physics,and quantum optics,etc.In the algebraic framework,the Hamiltonian system is given using a set of dynamic algebra,and the time-evolution operator of the quantum system can be directly obtained only if operators in the Hamiltonian close under communication with every element of the dynamic algebra.Computational time is saved by avoiding the solution of the time-dependent Schr?dinger equation,and analytical expression of the vibrational transition probability can be achieved using the algebraic approach.We have successfully studied the control of vibrational excitation for small molecules using the Lie-algebraic approach.16,17We have also discussed long-time average absorbed energy spectra and the average number of photons absorbed by the molecule in a normal sinusoidal laser field when rotational motions are considered.18Given that the algebraic model can be expanded to discuss the influence of molecular rotation,we can also discuss the influence of rotation on controlling infrared multiphoton excitation.Control of multiphoton vibrational excitations can enable selective vibrational transition and dissociation on demand,which helps regulate the chemical reaction and the preparation of quantum bits.

Studying the influence of molecular rotations on controlling infrared multiphoton excitation can help elucidate not only the interference effect of molecular alignment and orientation but also the impact of rotational energies,hence,the two aspects are emphases of our research and the rotational excitations are not considered here.In this study,the influence of rotation on controlling infrared multiphoton excitation in diatomic molecules is studied by the analytical algebraic approach.Transition probability with various rotational channels is analytically given using the method.To manifest the influence of rotations,we first study multiphoton resonant excitation in a normal sinusoidal laser field,considering both pure vibrational and ro-vibrational modes.We then discuss the selective multiphoton excitation of two modes in a chirped and shaped laser field.The parameters of laser pulses are also vital factors affecting quantum control.Accordingly,the influences of laser shape and initial laser phase are also studied.

2 Theoretical framework

The Hamiltonian of the system is

where Hmrepresents the Hamiltonian of free vibrational-rotational diatomic molecule and Hiis its interaction Hamiltonian with a laser field.

According to the quadratic anharmonic Lie-algebraic model,19-22

where?is Planck′s constants divided by 2π,mis the reduced mass of molecules,ris the distance between two nuclei,r0is the distance at equilibrium,lis the angular moment quantum number,ε(t)is the polarized laser field,andμis the molecular dipole moment function.

Both rotational term in the molecular Hamiltonian and the molecular dipole moment can be expanded in a series at equilibrium

where19

wherer0is the distance at equilibrium,Dis the dissociation energy,andαis the Morse parameter.

Thus the Hamiltonian operator of the system can be written as

where

whereθis the angle between the molecular orientation and the axis of the polarized field.

Given that we choose

the Hamiltonian system(Eq.(1))in the interaction picture reads as

whereχ0is the anharmonicity parameter,A0is the identity operator and I0,A-,A+have communication relations

Considering that the four operators can construct a four-dimensional Lie-algebra,the time evolution operator can be represented as23-25

The time-evolution operator UIsatisfies in the interaction picture

The set of differential equations can then be given by substituting Eq.(10)into Eq.(11)

with the initial conditions

The time-dependent population probability from the initial ro-vibrational state|vi,l＞ to the target ro-vibrational state|vf,l＞ is

where

The analytical expression of transition probability in different ro-vibrational channels is obtained,and many concrete examples can be studied using this expression.

The corresponding long-time average probability is defined as

3 Results and discussion

Here,we take OH and OD molecules as examples.All calculations are carried out using atomic units(a.u.).The parameters are taken from references,26-30namely,ω0=0.01664 a.u.,χ0=0.02323 a.u.,D=0.1614 a.u.,α=1.156 a.u.,Be=0.8598×10-5a.u.,αe=0.3253×10-5a.u.,De=0.8748×10-5a.u.for OH andω0=0.0122 a.u.,χ0=0.01645 a.u.,D=0.1636 a.u.,α=1.142 a.u.,Be=0.4556×10-5a.u.,αe=0.1321×10-5a.u.,De=0.8748×10-5a.u.for OD.ω0is the angular frequency of a anharmonic oscillator,χ0is the corresponding anharmonicity parameter,Dis the dissociation energy,αis the Morse parameter,Be,αe,andDeare the corresponding rotational constants.First,we discuss molecular multiphoton excitations and provide a concrete comparison between purely vibrational and the rovibrational cases in the first subsection.We then study the influence of rotations and laser phases in controlling infrared multiphoton excitation.The differences in the optimum laser parameters are also given in the second subsection.

3.1 Vibrational and ro-vibrational multiphoton excitations

This subsection demonstrates the influence of rotation on molecular multiphoton excitations.To study the influence of rotation,we calculate the probabilities in the purely vibrational case(molecular orientation is aligned with the field)and in the ro-vibrational case(l=1).In these cases,the laser field isε(t)=ε0sin(ωLt).The laser intensityε0is chosen as 0.0015 a.u.,and the angleθis averaged over-π/2-π/2 in the calculation.The initial state of the molecules is set at the ground state att=0.Fig.1 and Fig.2 depict the long-time average probabilities of OH and OD from the ground state to the first,second,and third excited states using Eq.(17).According to the definition of multiphoton resonance transition,31

whereωris the resonant transition frequency,ωnis the n-photon transition frequency,Ef-E0is the energy gap between the ground state and thefth excited state,these resonant transitions correspond to one,two,and three-photon transitions,respec-tively.Efis the energy eigenvalue that can be written as30,32

Fig.1 Long-time average probabilities from the ground to the first,second,and third states of OH as a function of laser frequency

Fig.2 Long-time average probabilities from the ground to the first,second,and third states of OD as a function of laser frequency

where the first two terms denote the anharmonic oscillator energy and the rigid rotor energy,and the last two terms represent vibration-rotation interaction and centrifugal distortion,respectively.Fig.1 and Fig.2 show that the average probabilities in the ro-vibrational case are much smaller than those in the purely vibrational case.We also calculate the transition probability in various rotationall-channels and find minimal changes in probability values.These findings indicate that the molecule experiences different orientations in the polarized laser field in the ro-vibrational case.Thus,the effective interaction strength in the ro-vibrational case is lower than that in the purely vibrational case.Thus,the molecular orientation in the laser field is very important to the ro-vibrational transition.This result coincides with previous ones.33,34

Few changes are observed in the values of the resonant frequencies for the two cases,but differences are still found upon more accurate calculations.The resonant frequency of the three-photon excitation in the ro-vibrational case changes from 0.014707 a.u.(in the purely vibrational case)to 0.0147069 a.u.for OH and 0.011197 a.u.(in the purely vibrational case)to 0.0111967 a.u.for OD.The shifting value of the resonant frequency is aboutωL≈0.1 cm-1,which may be due to correctional functions of the rotational energy for molecular vibrational anharmonicity.However,the rotational energy is still lower than the vibrational energy.

Fig.3 and Fig.4 show the time-dependent transition probabilities in the purely vibrational calculation(l=0)and in the ro-vibrational calculation for OH and OD molecules(l=1).The resonant probabilities are clearly found to have periodic behaviors.The resonant transition periods for the two cases are summarized in Table 1.For single-photon resonant transition,the corresponding period minimally changes in the two cases.However,the two and three-photon resonances are definitely long-time phenomena,consistent with the report of reference.35Moreover,the periods of ro-vibrational transitions become longer than those of vibrational transitions in the multiphoton resonances because the rotational energy has the corrected action on molecular anharmonicity.Non-resonant multiphoton transitions also appear earlier in the purely vibrational case than in the ro-vibrational case.Compared with OH molecule,the period of multiphoton transition for the OD molecule becomes longer in the ro-vibrational case.In other words,the resonant periods have larger changes in the OD molecule than in the OH molecule when the rotational factor is considered.The reasonis that although rotation energy has a little corrected action on molecular anharmonicity,the larger vibrational anharmonicity still cannot be overcome.Moreover,the anharmonicity of OH molecule is larger than that of OD molecule,so the influence of rotations on OD molecule is more obvious than that on OH molecule.This result is similar to a previously reported one.36

Fig.3 Average time-dependent transition probabilities for the one,two,and three-photon resonant transitions of OH as a function of time

Fig.4 Average time-dependent transition probabilities for the one,two,and three-photon resonant transitions of OD as a function of time

3.2 Control of ro-vibrational multiphoton excitations

In the previous section,we obtain the transition frequency of three-photon excitation and find the rotation energy has little corrected action on molecular anharmonicity.In order to observe correctional functions of the rotational energy and interference effect of molecular orientation on controlling multiphoton excitation,the three-photon excitations of OH and OD are taken as examples.

The laser field is

where the laser shape functions are square-sinusoidal,Gaussian,or triangular shapes,respectively.

Table 1 Resonant transition periods(unit in a.u.)of OH and OD molecules

in whichτis the laser pulse duration.Φ(t)is the phase of the laser field pulse as follows:

whereΦ0is the laser initial phase.We further calculate threephoton excitation probabilities in the purely vibrational and rovibrational cases(l=1)for comparison.When the rotation of the molecule is considered,the relationship between the molecular orientation and the polarized direction of the laser field becomes important.The maximum transition probabilities can be given as functions of the time and molecular orientation angleθ.We first calculate three-photon excitation probabilities at different laser shapes when the molecular orientation angleθis equal to zero.The laser parameters Δω1are the resonant transition frequency,and the chirped term Δω2and the laser pulse durationτcan be adjusted to obtain the optimum selective transition.

Fig.5 shows that the best selective three-photon excitation can be achieved when the laser shape is the Gaussian function.Although the three-photon excitation probability can reach ahigh value that is close to the one under the control of the square-sinusoidal and triangular laser pulse shape,oscillations of the population can be found in two cases.In addition,the three-photon excitation probability of the triangular shape case is smaller than those in the other two cases.Accordingly,we study the influence of rotations on controlling three-photon excitation under the Gaussian-shaped and chirped-laser pulse.Wecan obtain complete three-photon vibrational excitation whether in the purely vibrational or in the ro-vibrational cases.The optimum laser parameters are given in Table 2;the pulse duration isτ=18×105a.u.,and the laser intensity isImax=4.25×109W·cm-2for OH andImax=5.05×109W·cm-2for OD,respectively.Figs.6-9 show the maximum three-photon transition probabilities as functions of the time and molecular orientation angleθin both OH and OD cases.We can also see that higher transition probabilities are achieved when the rotation is considered.

Fig.5 Maximum three-photon excitation probabilities as functions of the time by the three kinds of laser shaped and chirped pulses in OH

Table 2 Optimum laser parameters(in a.u.)of OH and OD molecules

However,oscillations appear in the population which can be seen in Fig.7 and Fig.9,suggesting that rotational interference can decrease the selectivity of molecular multiphoton vibrational excitation.In addition,the highest excitation probability occurs only when the molecule is oriented along the direction of laser polarization.Fig.10 and Fig.11 show the initial laser phase dependence of the two cases,the maximum probabilities occur at different initial laser phases,which exhibit different modulation functions.In the interaction between the ultra-shortpulse and molecules,the frequency chirp is induced by the changes of laser initial phase,37-40which produces modulation actions on the maximum excitation probabilities.A comparison of Fig.10(a)and Fig.10(b)indicates that the modulation actions in the ro-vibrational case are stronger than that in the pure-vibrational case.The range in the values of the maximum probabilities is from 0.96 to 1 in the ro-vibrational case,while the maximum probabilities range from 0.975 to 0.978 in the pure-vibrational case.The same result can be seen from the Fig.11(a,b).Furthermore,oscillations are found in Fig.10(a)and Fig.11(a),which reflect the sensitivity of the initial phase modulations in the non-resonant excitation.However,oscillations become stronger when the initial laser phase is at π/2 or 3π/2,and the oscillations in OD are smaller than those in OH.This is an interesting phenomenon which is valuable to be further studied.We think that we should firstly confirm whetherthe phenomenon happens in the others or more molecules and hope that the further explanations are given in the subsequent works.

Fig.6 Maximum three-photon transition probabilities as functions of the time and molecular orientation angle θ by the Gaussian shaped and chirped pulse in OH

Fig.7 Control of three-photon vibrational transition with the molecular orientation angle using the Gaussian shaped and chirped pulse in the ro-vibrational case for OH

Fig.8 Maximum three-photon transition probabilities as functions of the time and molecular orientation angle θ by the Gaussian shaped and chirped pulse in OD

Fig.9 Control of three-photon vibrational transition with the molecular orientation angle using the Gaussian shaped and chirped pulse in the ro-vibrational case for OD

Fig.10 Initial laser phase dependence at the maximum of the selective three-photon vibrational transition probability of OH

Fig.11 Initial laser phase dependence at the maximum of the selective three-photon vibrational transition probability of OD

4 Conclusions

We analytically study the influence of rotations and laser shapes on controlling infrared multiphoton processes in diatomic molecules using the Lie-algebraic approach.Results indicate that the molecular orientation in the laser field has a greater effect on transition probability,and that the effect of rotations on infrared multiphoton excitation depends on the anharmonicity of molecules.More importantly,control of molecular alignment and orientation is necessary to obtain multiphoton selective vibrational transition because the maximum value of multiphoton transition probability occurs only when the molecule is oriented along the direction of laser polarization.Furthermore,the rotational interference may decrease the selectivity of the molecular vibrational transition.However,the correct laser shape and initial laser phase may help achieve a better multiphoton vibrational transition on demand.This approach can be extended to triatomic molecules,and bending vibration can be considered.

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