## Abstract

A three-state model for calculating the cross section of laser-induced dipole-quadrupole collisional energy transfer in Sr-Ca system is presented. The motion equations for the probability amplitudes of the three intermediate states are obtained. The expression of the cross section is derived. Various factors including field intensity, relative speed, system temperature which influence the collisional cross section are discussed to illustrate the features of the dipole-quadrupole laser-induced collisional energy transfer (LICET)process. Calculating results show that the peak of the LICET profiles moves to the red, the tuning range of the profiles obviously becomes narrower with the laser field intensity increasing and a cross section of 1.25 × 10^{−13}cm^{2} at a laser intensity of 8.29 × 10^{9}W/cm^{2} is obtained. Our results indicate that dipole-quadrupole LICET process can be the effective way to transfer energy selectively from a storage state of arbitrary parity to a target state of arbitrary parity.

©2010 Optical Society of America

## 1. Introduction

Laser-induced collisional energy transfer (LICET) is an optical phenomenon that a pair of colliding atoms transfer to the final states by the joint action of the collisional and radiative interactions. A LICET process between two atoms can be expressed by the reaction:

As plenty of the experimental studies have been made including Sr-Ca [1], Na-Ca [2], Eu-Sr [3], Li-Sr [4], Rb-K [5], Li-Sr [6], people also have proposed many theoretical models on LICET[7,8]. The two-level model proposed by Harris et al.[9] influenced most. Gallagher and Holstein presented the first detailed calculations of LICET line shapes. The molecular-based treatment demonstrated the importance of the van der Waals interaction to the low-intensity line shape [10]. Bambini and Berman demonstrated that a two-state treatment was not adequate and the intermediate state which may be populated must be treated as a real state [11]. Agresti et al. generalized two-level model into three-level perturbative theory[12]. In 1992, Sydney Geltman reexamined the behavior of cross sections in the three-state model for the two-pairs Li-Sr and Eu-Sr previously studied experimentally [13]. A. Bambini and S. Geltman gave the theory of strong-field light-induced collisional energy transfer later. They calculated the spectral profiles of LICET process induced by polarized, high-intensity laser field [14]. Based on the three-level model and considering the effects of all intermediate states, Chen et al.[15-17] developed the four-level perturbative theory between two atoms.

The previous experimental and theoretical researches on the LICET mainly concentrated on dipole-dipole process that one atom made a parity allowed transition while the other made a parity nonallowed transition. As to the research on dipole-quadrupole process that both species made parity allowed or nonallowed transitions, there were no corresponding theoretical model and just few experimental researches [18,19] were made. Actually, based on the experiment results by W. R. Green et al., we can clearly see that laser induced dipole-quadrupole collision cross section also can be large and thus permit one to consider practical collision systems having storage and target states of different parity from those that are accessible by using dipole-dipole interactions [18].

Moreover, the early calculations on the LICET process [7–9,14], based on the assumption that the relative velocity was fixed which obviously did not match the practical situation and adopted the simple set of product atomic basis states which included on the *m* = 0 substates. In this paper, the three-level dipole-quadrupole LICET model considering the relative velocity distribution function and including all the degenerate *m* states which may give more accurate details of the line shape is presented. Laser induced dipole-quadrupole collision process in Sr-Ca system is numerically calculated through immediate integration. Various factors including field intensity, relative speed, temperature which influence the collisional cross section are discussed to illustrate the features of the dipole-quadrupole LICET process. And our results indicate that dipole-quadrupole LICET process can be the effective way to transfer energy selectively from a storage state of arbitrary parity to a target state of arbitrary parity.

## 2. Theory

Figure 1 shows the LICET process between two different atoms, with those energy levels of atom A, atom B and the compound AB system that are relevant to the LICET process present and those irrelevant ignored. Taking independent atomic states of atoms A and B as basis states, these three states of the compound system can be expressed as

The wave function can be expressed by the basis states as follows:

We assume the coordinate system as Fig. 2
shown where atom B is fixed and atom A moves in a straight line with velocity *v* along the z axis. The unprimed system is fixed in space, while in the primed system, the z’ axis points the interatomic axis, and therefore rotates with respect to the unprimed system during the collision, i.e.,

*θ*=

*b*/

*R*(

*t*),and

*θ*varies form -π/2 to π/2。

The linear trajectory is assumed for *R*(*t*) = (*b*
^{2} + *v*
^{2}
*t*
^{2})^{1/2}, where *b* is an impact parameter and *v* is the relative velocity. Assuming the **E**(t) = *E*
_{0}
**a**
_{y}cos(*ωt*) is the laser field with frequency *ω*, then the interaction Hamiltonian can be written

*V*can be represented by the multipole expansion [20] where

_{AB}*l*and

*j*-1-

*l*, ${Q}_{l}^{m}\left(A\right)$ and ${Q}_{j-l-1}^{-m}\left(B\right)$ are multipole moment operators, and

*F*(

*j*,

*l*,

*m*) is given by [20,21]

*C*(

*j*-

*l*-1,

*l*,

*j*-1;

*m*, -

*m*, 0) is Clebsch-Gordan coefficient.

By making formula (3) simultaneous with the Schrödinger equation and supposing the laser field is sufficiently intense, the motion equation of formula (2) can be obtained as follows:

whereUsing a unitary transformation $a\left(t\right)=T\left(t\right)\xb7C\left(t\right)$, where

Formula (7) is a differential equation set with complex coefficients. Through solving the coupled equations as a function of impact parameter, we can obtain the probabilities of finial state. The previous calculations on the LICET process [11–13,23], adopted the simple set of product atomic basis states which included on the *m* = 0 substates. In the present work we consider three basis states, and include all possible *m* states. The transition probability is found by summing over the final level populations and averaging over the initial state populations. And then, by considering the probability density function of relative velocity between two colliding atoms in three-dimensional space under thermal equilibrium condition, the laser-induced collision cross section at given temperature can be obtained by

The probability density function F(v) can be represented by [22]

*M*is the reduced mass of the two colliding atoms,

*T*is the thermodynamic temperature and

*k*is the Boltzmann constant.

## 3. Application to Sr-Ca system

Sr-Ca system LICET process can be expressed by the following formula, and Fig. 3 illustrates those energy levels relevant to the LICET process.

The states of Sr-Ca compound system shown in Fig. 3 can be defined as:

*ω*

_{21}= −151.152 cm

^{−1}.

Assuming that the relative speed is 600m/s, the laser is at strict resonance (Δ = 0cm^{−1}) and the impact parameter *b* is equal to 0.9nm and 1.15nm respectively, through an immediate integration on formula (7), the collisional transition probability |*a*
_{3}(*t*)|^{2} versus time *t* in this Sr-Ca system is calculated as Fig. 4
shown. The results show that |*a*
_{3}(*t*)|^{2} oscillates over time, the smaller the b is, the stronger the |*a*
_{3}(*t*)|^{2} oscillates. When|*t*|≥20ps, |*a*
_{3}(*t*)|^{2} approach to a definite value, which indicate that the laser induced collision process occurs for~40ps.

In this Sr-Ca system, when the laser intensity is fixed at 0.133MW/cm^{2} and the relative speed between the two colliding atoms is 600m/s, the collisional transition probability |*a*
_{3}( + ∞)|^{2} versus the impact parameter b can be obtained through numerical integrations with different laser detuning Δ and the profiles obtained are shown in Fig. 5
. It appears that |*a*
_{3}( + ∞)|^{2} oscillates over b. To the quasimolecule system formed in the collisional process, there is an effective curve crossing at some *R _{x}*, i.e., at some

*R*the argument of the exponent in equation of motion for the probability amplitudes equals zero at

_{x}*t*such that

*R*(

*t*) =

*R*

_{x}. As the Landau-Zener theory of the curve crossing section, the transition probability is only large for b≤

*R*. At a given impact parameter

_{x}*a*

_{3}(

*t*) accumulates during each of the two times that the crossing distance is neared. And the oscillation of |

*a*

_{3}( + ∞)|

^{2}results from the adding and subtracting of the phased contributions at each of the two crossing [9]. It is also found that in the detuning case more of the contribution to the total collision cross section occurs at small impact parameter than in the strict resonance case. When laser detuning Δ>0, the maximum amplitude increases with increasing impact parameter, and when Δ<0, amplitude maximum decreased as the impact parameter increases.

A series of cross sections at various field intensities are shown in Fig. 6
and Fig. 8
. In weak field as the Fig. 6 shown, the results are characterized by a universal asymmetric line shape and the magnitude is proportion to the field intensity. The profile can be divided into two parts, the quasi-static and the anti-static wings. A qualitative understanding of this is based on the adiabatic molecular curves given in Fig. 7
. Molecular curves of state 1 and state 2 become strongly repelling van der Waals energies by the coupling between state 1 and state 2. When *ω*>*ω*
_{0} there is a such *R* where ω can equal E_{3}(*R*)-E_{2}(*R*), while in the wing of *ω*<*ω*
_{0} there is no such value of *R* which leads to the sharply reduce of the transition probability in anti-static wing. Furthermore, it is shown in Fig. 6 that as far as this Sr-Ca system is concerned, the peak of the cross section profiles is not found when the transfer laser is strictly resonant (i.e. Δ = 0cm^{−1}), but it is found when the transfer laser detunes a little from Δ = 0cm^{−1} in the quasi-static wing (in this paper, peak of the profiles is found when Δ = 1.5cm^{−1}).

As the field intensity is increased shown in Fig. 8, the resulting spectrum is shifted in frequency towards the anti-stable wing. An understanding of this is that as the laser intensity increases, the Rabi oscillation frequency in the $\text{Ca}\left(3d4{p}^{}{}^{1}\text{F}{}_{3}^{0}-4s3{d}^{}{}^{1}\text{D}{}_{2}\right)$ atomic transition increases and become dominant over the Van der Waals quasimolecule mechanism. It tends to enhance the LICET cross section as *ω* approaches to *ω*
_{31} which is <*ω*
_{32}, resulting in the bathochromic shift [13]. Figure 9
gives the shift of the peak cross section as a function of transfer laser intensity. The results show that there is a good linear relation between cross section shift and laser intensity. The magnitude of this shift to be approximately given by −5.56329 × 10^{−9}I(W/cm^{2})cm^{−1}.

Form the Fig. 8 it can be seen that with the increasing field intensity the line shape tends to become more symmetrical at its center, which is because of the cross section in the quasi-static wing lowered by the presence of a combined light and collisional shift that reduces the effective time for level crossing [23]. It is also apparent from Fig. 8 that decreasing field intensity has the effect of increasing the transition probability in the quasi-static wing. Moreover Full width at half peak of the profile becomes smaller as the field intensity increases.

The peak cross section as a function of transfer laser intensity for the Sr-Ca system is given in Fig. 10
. The collision cross section rises with laser intensity and tends to saturate. There is a maximum in our results which is interpreted as originating from the increased probability of populating the intermediate state and is not accounted for in a two-state model [13]. It is a significant departure from the two-state treatment result that the LICET peak cross section is monotone increasing with the laser intensity which does not conform to physical reality. A cross section of 1.25 × 10^{−13}cm^{2} at a laser intensity of 8.29 × 10^{9}W/cm^{2} is obtained which indicate that the laser induced dipole-quadrupole and dipole-dipole collision have comparable magnitude. So the dipole-quadrupole process also can be the effective way to transfer energy selectively from a storage state of arbitrary parity to a target state of arbitrary parity.

When the relative speed is taken as 950m/s, 1100 m/s and 1200 m/s, respectively, curves of cross section *σ* (Δ) versus laser detuning Δ through numerical integrations are shown in Fig. 11
. It can be seen that the variation in the LICET profile to laser detuning has a certain tunable range. Full width at half peak of the profile and cross section at fixed laser detuning become smaller as the relative speed increases. Moreover, the quasi-static wings approximately have the same trend at different relative speed, while the anti-static wings become steeper as the speed increases. A understanding of this is that based on the [9], relative speed is in inverse proportion to the Weisskopf radius which is in proportion to the cross section, resulting in the cross section become smaller with collisional speed reducing.

The previous theoretical works [13,15,16] on LICET cross section were based on the assumption that the relative velocity was fixed which obviously did not match the practical situation. Because all of the previous experimental investigations on LICET were made for mixed metal vapors collisional, so atoms should have many different speeds, which distributed in a certain way, around an average value that was used in the early theoretical calculation. Figure 12
shows the comparison of LICET spectrums with velocity distribution considered (solid line) and not (dashed line) at a transfer laser intensity of 1.0 × 10^{8}W/cm^{2}. It can be seen that velocity-distribution considered collision cross section larger than fixed-velocity cross section in the spectrum center and the quasi-static wing, while in anti-static wing the fixed-velocity spectrum is steeper, indicating that it is necessary to consider this distribution in the calculations.

Figure 13 shows the LICET spectrum for the Sr-Ca system at different temperatures. It is found that system temperature has no obvious effect on line shape while in quasi-static wing and especially in the spectrum center the cross section increases with the decreasing temperature. Since the average velocity decreases as the temperature increases, it leads to the Weisskopf radius rises which accounts for the cross section increasing especially in the vicinity of the spectral profiles center.

## 4. Conclusion

In this paper, by considering multipole expansion interaction Hamiltonian, a three-state model for calculating the cross section of laser-induced dipole-quadrupole collisional energy transfer in Sr-Ca system is presented. In the present work we consider three basis states, and include all possible *m* states. The transition probability is found by summing over the final level populations and averaging over the initial state populations. In view of the probability density function of relative velocity between two colliding atoms in three-dimensional space under thermal equilibrium condition, the laser-induced collision cross section at given temperature is obtained, and spectral profiles of LICET are calculated through immediate integrations. It appears that the dipole-quadrupole and dipole-dipole LICET spectrum profiles [9,13–17] have some similar features. The dipole-quadrupole Sr-Ca LICET line shape has larger tunable range than dipole-dipole Sr-Ca LICET spectrums [9].· The dipole-quadrupole profiles are strongly asymmetrical, i.e. in the antistatic wing on one side of the peak, the cross section falls rapidly; while in the quasi-static wing on the other side, the cross section is relatively large for a wide range of transfer laser detuning. Calculations show that the statistic distribution of the relative speed between two atoms has impacts on the collision cross section, indicating that it is necessary to consider this distribution in the calculations. The laser induced dipole-quadrupole collision spectrum shows the following features: (a) in the vicinity of the spectral profiles center the cross section at fixed laser detuning become smaller as the relative speed and system temperature increases; (b) the peak of the spectrum is shifted from the resonant frequency towards the antistatic region, with the shift increasing approximately linearly with laser intensity; (c) the spectral line shape becomes narrower and less asymmetric as the laser intensity increases; (d) the peak cross section tends to saturate with increasing laser intensity, and laser induced dipole-quadrupole collision cross section has a magnitude comparable to dipole-dipole collision cross section which indicates that dipole-quadrupole LICET process also can be the effective way to transfer energy selectively from a storage state of arbitrary parity to a target state of arbitrary parity.

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