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Optimal control theory in adaptive simulated annealingtechnique: optimisation of laser pulse for selectivevibrational excitations and photo-dissociation of HBr+

Chandan Kumar Mondala & Bikram Natha

a Department of Chemistry, Physical Chemistry Section, Jadavpur University, Kolkata, IndiaAccepted author version posted online: 13 Feb 2013.Published online: 16 May 2013.

To cite this article: Chandan Kumar Mondal & Bikram Nath (2013) Optimal control theory in adaptive simulatedannealing technique: optimisation of laser pulse for selective vibrational excitations and photo-dissociation of HBr+,Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 111:21, 3200-3207, DOI:10.1080/00268976.2013.775515

To link to this article: http://dx.doi.org/10.1080/00268976.2013.775515

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Molecular Physics, 2013Vol. 111, No. 21, 3200–3207, http://dx.doi.org/10.1080/00268976.2013.775515

RESEARCH ARTICLE

Optimal control theory in adaptive simulated annealing technique: optimisation of laserpulse for selective vibrational excitations and photo-dissociation of HBr+

Chandan Kumar Mondal∗ and Bikram Nath

Department of Chemistry, Physical Chemistry Section, Jadavpur University, Kolkata, India

(Received 30 November 2012; final version received 2 February 2013)

An optimisation method using optimal control theory based adaptive simulated annealing technique has been explored to getoptimised laser pulses for selective vibrational excitations and photo-dissociation of the HBr+ in its ground electronic state.Potential energy curve of the system has been obtained by coupled cluster singles and doubles (CCSD) level calculation,using aug-cc-pVTZ as basis set. To have a simple pulse, a limited number of parameters is chosen as variables and thetechnique so developed works well. The study is extended with chirping field frequency to explore the effect of chirping indynamics.

Keywords: pulse shape optimisation; vibrational excitation; photo-dissociation; simulated annealing; TDFGH; frequencychirping; HBr+

1. Introduction

Design of an appropriate pulse [1–11], its parameter optimi-sation for a particular chemical process, and improvementof optimisation methods are nowadays important both fromtheoretical and experimental viewpoints. The studies giveus not only a tailor-made pulse to control a particular chem-ical process but also a deeper insight into the laser–matterinteractions.

The optimal control theory (OCT) [12,13] plays a cen-tral role in optimisation of laser pulse shape for a selectiveprocess. The theory helps us to design a laser pulse thatcan efficiently transform a system from its initial state toa desired final state. In OCT, one has to optimise a func-tion called ‘cost function’. The cost function is formulatedin such a way that in any attempt to optimise it, one notonly has to maximise the extent of achieving the target butalso minimise the integrated laser pulse intensity (a penaltyterm).

There are several techniques to optimise the cost func-tion [12,14–23] to get an optimised laser pulse for experi-mental works. Recently OCT-based genetic algorithm meth-ods [24,25] have been developed to obtain a simple shapeof optimised laser pulse that is relatively more convenientfor experiments.

In the recent past we have studied the pulse shape ef-fect on the dissociation dynamics of HBr+ molecular ion[26]. Here, our aim is to find a simple optimised laser pulsefor selective vibrational excitation and photo-dissociationof diatomic molecular ion, HBr+ . We explore the devel-opment of an optimisation method by introducing OCT

∗Corresponding author. Email: ckmondal@chemistry.jdvu.ac.in

in adaptive, step length-based simulated annealing (SA)[27,28] technique to get such simple shape of optimisedlaser pulse.

SA is a stochastic optimisation method and has someadvantages over other optimisation algorithms. Here, theoptimising function is not trapped in a local optimum andis capable of escaping from it to reach the global optimum.There is no restriction on the optimising function to bean appropriately quadratic or differentiable. The techniquehas many successful applications in several fields [29–35]of optimisation. We have applied the technique to optimiselaser pulse parameters through optimising the cost function.To get a simple and regular shape of laser pulse we consideronly a limited number of variables as parameters.

Apart from using a simple shape of laser pulse, a chirp-ing in frequency is also an easy way to control a chemicalprocess. In experiments it is even easier to use frequencychirping than a shaped laser pulse. Hence, for optimisationwe use a laser pulse having no chirping in its frequency aswell as a linear chirping on it. The chirping may be up- ordown-chirping but we test our method with both the cases.

In this paper, as a test system we have considered the di-atomic molecular ion, HBr+ in its ground electronic state.We have used Gaussian 03 package [36] for ab initio calcu-lation of the ground electronic state potential energy curveof the system. The potential was used in the energy op-erator construction. The dipole moment function was ob-tained using the same level of ab initio calculation. Thetime-dependent Schrodinger equation was solved using thefourth-order Runge–Kutta method. First, we optimise laser

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pulse for 0 → 1 vibrational excitation. Then we optimise itfor 0 → 3 and 1 → 2 vibrational excitation. Finally, we didthe same for photo-dissociation process.

The paper is organised in the following way. Section2.1 briefly outlines the system studied and related ab initiocalculations; Section 2.2 describes the OCT; Section 2.3explains the approach of SA to optimise the cost function;and Section 3 represents the results along with necessarydiscussions. A brief conclusion is given in Section 4.

2. Theory

2.1. The system

The potential energy, V (x), of HBr+ molecular ion in itsground electronic state has been calculated using coupledcluster singles and doubles (CCSD) [37] level ab initiotheory, with aug-cc-pVTZ as basis set with a UnrestrictedHartree-Fock reference state function. The variation of po-tential energies against internuclear distance (x) was fittedin an appropriate Morse function

V (x) = De

[1 − e−β(x−xe)

]2(1)

where De, β and xe are the appropriate fitting parameters.The time-independent molecular Hamiltonian for the

system is

H0 = p2x

2m+ V (x) (2)

The eigenstates and eigenvalues of H0 are obtained by usingthe Fourier grid Hamiltonian (FGH) [38,39] method and arerepresented as

H0

∣∣φ0k (x)

⟩ = ε0k

∣∣φ0k (x)

⟩, k = 1, 2, 3, . . . , N (3)

where N is an odd number of equispace grid points. Theeigenstates in the coordinate grid points in FGH methodare represented as

∣∣φ0k (x)

⟩ =N∑

p=1

∣∣xp

⟩�xω0

kp, p = 1, 2, 3, . . . , N (4)

where |xp〉 is the coordinate vector at the pth grid point, �x

is the spacing of the uniform coordinate grid and ω0kp is the

pth grid point amplitudes of kth eigenstate of H0.Now if we perturbed the system by applying external

field, ε(t), the form of Hamiltonian will be

H (x) = p2x

2m+ V (x) + μ(x)ε(t) (5)

where μ(x) is the dipole moment operator function ob-tained using similar level ab initio calculation as done in the

potential energy case. Time evolution of wavefunctionunder applied field is calculated numerically with time-dependent FGH (TDFGH) method [40,41], and is repre-sented as

|ψ(x, t)〉 =N∑

p=1

∣∣xp

⟩�xωp(t), p = 1, 2, 3, . . . , N (6)

The transition probability from initial state, φ0i , to a specific

target state, φ0f , at time t is represented as

si→f (t) = ∣∣〈ψ(x, t)| φ0f

⟩∣∣2(7)

and the dissociation probability of the system would be

Pd = 1 −nb∑

f =1

∣∣〈ψ(x, t)| φ0f

⟩∣∣2(8)

where nb is the total number of bound state of the system.We consider vibrational ground state, φ0

i=0, as the initialstate. It evolves under the applied field ε(t) and is repre-sented as ψ(x, t). To avoid non-physical reflections gen-erated due to finite grid number in numerical calculations,we have added an absorbing complex potential [42] at thestarting of the asymptotic region. The expression for theabsorbing complex potential that we have used to avoidnon-physical reflection is −iλV CAP and the total Hamilto-nian would be HCAP (x) = H (x) − iλV CAP . The form ofV CAP = (x − x ′) has been used in restricting the conditionthat it provides some non-zero values to the total Hamil-tonian after x > x ′ where x ′ is the starting point of theasymptotic region and λ is a constant.

2.2. Optimal control theory

OCT is a mathematical framework which helps us to opti-mise a process to reach to a specific target point. In pulseshaping technique the objective of the theory is to optimisea laser pulse which will transform a system from its initialstate to a specific target state efficiently. In OCT a ‘costfunction’ is defined as a measure of achievement of a targetalong with a penalty term of integrated laser intensity. Inthe case of vibrational excitation the field-dependent ‘costfunction’ J [ε(tp)] from initial state (φi) to a specific targetstate (φf ) is

J [ε(tp)] = si→f (tp) − α0

tp∫

0

[ε(t)]2dt (9)

where tp is the pulse duration. The first term si→f (tp)(Equation 7) is the transition probability from the initialstate to the specific target state at end of pulse and the

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3202 C.K. Mondal and B. Nath

second term is the penalty term. In penalty term, α0 is aconstant weighing factor which is to be chosen in such away that there is a reasonable contribution of the penaltyterm in the calculation of cost function. Clearly, to havethe maximum J [ε(tp)], we will have to have maximumtransition probability as well as minimum penalty term.

Similarly for photo-dissociation case the cost functionwould be

J [ε(tp)] = Pd − α0

tp∫

0

[ε(t)]2dt (10)

where Pd (Equation 8) is the dissociation probability.

2.3. SA approach for optimisation

The method of SA would be applied to optimise the costfunction. First, we define the cost function J for a particularset of variable parameters xi . In the random variation ofparameters, a new set of xi , denoted as x ′

i , is formed as

x ′i = xi + R�xi (11)

where R is a random number (−1 < R < 1) and �xi is thestep length.

A new cost function is defined as J ′ for the new set ofparameters. If J ′ > J , the change is accepted and if J ′ ≤ J ,the acceptance would be decided on the basis of Metropolisalgorithm,

p = e(J ′−J )/T (12)

According to this algorithm the values of ‘p’, which dependon the two cost functions and the temperature, are calculatedfirst and then compared with a random (Rp) number thatranges between 0 and 1. If p > Rp, the change would beaccepted. The temperature T should have a suitable valuethat is neither very high nor very low. In the case of very hightemperature, since most of the changes would be accepted,it is very difficult to reach a promising area. On the otherhand, if the temperature is low, most of the changes wouldbe rejected and in that case there is a high chance to gettrapped in local minima.

After some repetitions, the highest value of J and xi

are stored as J o and xoi , respectively, and after that the

temperature is reduced to T ′ as

T ′ = ηT (13)

where η is a constant term between 0 and 1. However, as thetemperature is gradually lowered, the number of acceptanceof changes would also be reduced. Hence, in the algorithmwe consider an adaptive nature of the step length �xi as itadjusts accordingly in order to maintain 50% acceptance ofchanges.

In this paper we stochastically vary laser field parame-ters using SA method to find out the maximum cost functionfor the transition and the dissociation processes. As men-tioned earlier, we consider both chirping and non-chirpingfrequency to optimise the laser pulses. For simplicity, wehave introduced a linear chirping in frequency which isdenoted as

ωk (t) = ω0 ± kct, (ωk ≥ 0) (14)

where ω0 is the frequency at the beginning of the pulseand kc is the chirp rate. The ‘ + ’ and ‘−’ sign correspondto up- and down-chirping, respectively. Therefore, the fieldamplitude would be

ε(t) = εk sin {(ω0 ± kct)t} s(t) (15)

where εk is the peak field amplitude of the laser field. Theterm s(t) is a pulse envelope function which has t1 and t2as two variable time parameters and has the form

s(t) =

⎧⎪⎪⎨⎪⎪⎩

sin2[

π2

(tt1

)]for 0 ≤ t ≤ t1

1 for t1 ≤ t ≤ t2

sin2[

π2

(tp−t

tp−t2

)]for t2 ≤ t ≤ tp

(16)

Clearly t1 and t2 determine the sharpness of the rise andfall of laser field intensity. Thus, to optimise the chirpinglaser pulse, the five variable parameters, namely, εk , ω0, kc,t1 and t2 have to be optimised.

For the non-chirping case the field amplitude varies withtime as

ε(t) = εk sin(ωkt).s(t) (17)

where ωk is the frequency of the laser field. Thus, to opti-mise the non-chirping laser pulse, εk , ωk , t1 and t2 are thefour variable parameters that need to be optimised.

3. Results and discussion

First, we report how we obtained the ground state potentialenergy curve of HBr+ that has been used in our numericalcalculation. Next, we will describe the results of vibrationaltransition for three cases, v = 0 to v = 1, v = 0 to v = 3and v = 1 to v = 2. Finally, we will discuss the result ofphoto-dissociation. All these calculations are done usingspace grid ‘x’ in the range of 1.7–10.0 a.u. (0.9–5.3 A) andthe number of grid point N = 99.

3.1. Electronic structure calculations

Figure 1 shows the potential energy curve of HBr+ inits ground electronic state, calculated by ab initio CCSDmethod, using aug-cc-pVTZ as basis set and its fitted

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Figure 1. Calculated potential energy of HBr+ in its groundelectronic state using CCSD/aug-cc-pVTZ level (in circles) andthe best fitted Morse potential (solid line).

curve to an appropriate Morse potential function, V0(x) =D0 [1 − exp {−β(x − xe)}]2. The parameters of the fit-ted Morse potential are D0 = 0.1468 a.u. (92.118 kcalmole−1), β = 0.9529 a.u. (1.801 A−1) and xe = 2.655 a.u.(1.440 A). The reduced mass of HBr+ is μ = 1827.632a.u. The fitted bond dissociation energy (D0) and the equi-librium bond length (xe) of the system agree well with thedata calculated with the help of other methods [43]. Wehave calculated the eigenstates and eigenvalues using thetime-independent FGH method which shows that there aretotal 23 bound vibrational states (nb). We have also cal-culated the variation of dipole moment of HBr+ againstinternuclear distance using the same ab initio method andfitted it to the following function:

μ(x) = μ0xe−σx4(18)

with μ0 = 0.18015 a.u. and σ = 0.00329 a.u.

3.2. v = 0 → v = 1 excitation

The v = 0 → v = 1 excitation process has been studied intwo different conditions of applied field frequency: one byintroducing a linear chirping in it using Equation (15) andother with Equation (17) where there is no such chirping.The magnitude of the peak field amplitude εk(t) was re-stricted to vary within the range of 0.0–0.03 a.u. (1.071 ×1015 W cm−2). The target time for excitation and time stepused in the calculation are 30,000 a.u. (726 fs) and 0.1 a.u.(0.0024 fs), respectively.

We obtained reasonably high values of transition proba-bilities in each case. In the case of linear up-chirping, down-chirping and non-chirping field frequencies, the transitionprobabilities in optimised pulse are obtained as 0.99098,0.99678 and 0.99814, respectively. Thus, in the case ofnon-chirping, the transition probability is found to be a lit-tle bit higher than that in any chirping case. In Figure 2

Figure 2. (a) Optimised electric field amplitude, ε(t) againsttime, (b) population of quantum states against time and (c) over-lapping of field-propagated state, ψ(tp) on the target state φ0

1 atthe end of pulse in the case of v = 0 → v = 1 excitation.

we have shown only the results obtained in optimised laserpulse, using the non-chirping field frequency. The opti-mised field amplitude against time is shown in part (a) ofFigure 2. Part (b) of the figure shows the population ofthe field-propagated initial state, v = 0, the target state,v = 1, and a higher state, v = 2, against time. From thefigure we can say that there is smooth switching of the pop-ulation from initial state v = 0 to the target state v = 1and it is completed at the end of the pulse. Part (c) of thefigure shows the overlapping of the field-propagated initialstate (v = 0) with the target state (v = 1) at the end of thepulse, which clearly reveals that there is almost completetransition from the initial state to the final target state bythe optimised laser pulse.

3.3. v = 0 → v = 3 excitation

Here, we have shown the results of the vibrational excitationfrom v = 0 to v = 3. We have explored the optimisationof laser pulse in both the chirping and non-chirping fre-quency conditions. The magnitude of εk was restricted tovary within the range of 0.0–0.06 a.u. (2.143 × 1015 Wcm−2). The target time for excitation and time step usedin the calculation are 250,000 a.u. (6047 fs) and 0.1 a.u.(0.0024 fs), respectively.

The transition probabilities in optimised laser condi-tion for the non-chirping case (0.93259) are found to be a

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Figure 3. (a) Optimised electric field amplitude ε(t) against time,(b) population of quantum states against time and (c) overlappingof field-propagated state ψ(tp) on target state φ0

3 at the end ofpulse in the case of v = 0 → v = 3 excitation.

higher than that of both up-chirping (0.12324) and down-chirping (0.75029) cases. The optimised frequency is foundas 0.011093 a.u. (2435 cm−1). In Figure 3 we have shownthe results only for the case of non-chirping frequency. Parts(a), (b) and (c) of this figure show the optimised electric fieldamplitude, populations of the different vibrational statesand the overlapping of the field-propagated initial state(v = 0) with the target state (v = 3), respectively. Here,we also observe almost complete transition from the initialstate to the target one by the laser pulse in its optimisedcondition.

3.4. v = 1 → v = 2 excitation

Here, we have shown the results of vibrational excita-tion from v = 1 to v = 2. We have explored the op-timisation of laser pulse in both the chirping and non-chirping frequency conditions. The magnitude of εk wasrestricted to vary within the range of 0.0–0.04 a.u. (1.429 ×1015 W cm−2). The target time for excitation and time stepused in the calculation are 30,000 a.u. (726 fs) and 0.1 a.u.(0.0024 fs), respectively.

The transition probabilities in optimised laser conditionfor the non-chirping case (0.99429) are again found to be alittle bit higher than that of both up-chirping (0.99254) anddown-chirping (0.98604) cases. In Figure 4 we have shown

Figure 4. (a) Optimised electric field amplitude ε(t) against time,(b) population of quantum states against time and (c) overlappingof field-propagated state ψ(tp) on target state φ0

2 at the end ofpulse in the case of v = 1 → v = 2 excitation.

the results only for the case of non-chirping frequency. Parts(a), (b) and (c) of Figure 4 show the optimised electric fieldamplitude, populations of the different vibrational statesand the overlapping of the field-propagated initial state(v = 1) with the target state (v = 2), respectively. Here,we also observe almost complete transition from the initialstate to the target one by the laser pulse in its optimisedcondition.

3.5. Photo-dissociation

For the photo-dissociation, we have considered the vibra-tional ground state, v = 0 as the initial state. After switchingon the external electric field, the population in the higherexcited state will start to increase and ultimately it goesto the dissociation continuum. The total population in thedissociation continuum measures the dissociation proba-bility (Equation 8). To incorporate the absorbing complexpotential, we have used here x ′ = 5.850 a0 (3.096 A) andλ = 0.00003 a.u. We have optimised the laser pulse for boththe chirping and non-chirping field frequency conditions toget the maximum dissociation probability. The magnitudeof εk was restricted between 0.0 and 0.15 a.u. (5.357 ×1015 W cm−2). The target time for excitation and time step

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Figure 5. (a) Dissociation probability (Pd ) against time with op-timised laser fields in the case of down-chirping (solid line), non-chirping (dashed line) and up-chirping frequency (dotted line).(b) Optimised electric field amplitude ε(t) for dissociation againsttime in the case of down-chirping frequency.

used in the calculation are 80,000 a.u. (1935 fs) and 0.1 a.u.(0.0024 fs), respectively.

In Figure 5(a) we have shown the dissociation prob-ability of HBr+ against time with optimised laser pulsefor down-chirping, non-chirping and up-chirping frequencyconditions. It was found that the dissociation probability atthe end of pulse in the case of down-chirping frequency ishigher (0.956) than that of both non-chirping (0.875) andup-chirping (0.854) cases. Figure 5(b) shows the optimisedelectric field amplitude against time for the down-chirpingcase. The higher dissociation probability in down-chirpingcase is probably for efficiently populating higher excitedvibrational states as the down-chirping frequency sweepmay be optimised to match the energy spacing betweenvibrational levels of the molecular ion.

To understand the mechanism of population transferfrom the ground vibrational state to the continuum throughthe different bound vibrational states, we have shown thepopulation of different bound vibrational states against timein Figure 6. The results of down-chirping, non-chirpingand up-chirping frequency cases are shown in parts (a), (b)and (c) of the figure. In part (a) of the figure, maximumnumbers of vibrational bound states are involved for thedissociation to occur, as energy gap in the upper boundlevel decreases and the field frequency also decreases withtime. For the up-chirping case, the least number of boundvibrational states are involved for the population transferduring time evolution. This may be due to gradual decreaseof energy gaps between bound vibrational states but the field

Figure 6. The time evolution of population in different boundvibrational states for optimal laser fields obtained using (a) down-chirping, (b) up-chirping and (c) non-chirping frequency.

frequency increases gradually which is shown in part (b) ofthe figure. Part (c) of the figure shows the time evolutionof population transfer between the vibrational states for thenon-chirping case.

To check whether pulse energy is the controlling fac-tor for the dissociation, we have calculated the pulseenergies of all three (down-chirping, non-chirping and up-chirping) cases for dissociation. The calculated pulse en-ergy values for the cases of down-chirping, non-chirpingand up-chirping frequency are found as 8.2, 9.8 and7.3 kJ, respectively. The dissociation probability valuesshow that down-chirping in frequency is more efficient fordissociation than non-chirping case even with lesser pulseenergy which signifies that here the pulse energy is not thecontrolling factor for dissociation. The probable reason forenhancing the dissociation probability may be the matchingof energy spacing in the higher vibrational levels.

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Figure 7. Comparison of transition probability for v = 0 → v =1, at different annealing steps, in the case of fixed and adaptivestep length.

3.6. Comparison of optimised frequencies inchirping and non-chirping conditions

In the case of vibrational excitation, the non-chirping opti-mised field frequencies for 0 → 1 and 1 → 2 transitions are0.011589 a.u. (2543 cm−1) and 0.011073 a.u. (2430 cm−1),respectively. The values are found to be almost equal totheir corresponding transition frequencies (0.011581 a.u.,i.e. 2542 cm−1, for 0 → 1 transition and 0.011084 a.u., i.e.2432 cm−1, for 1 → 2 transition) obtained by our theoreticalcalculation using FGH-based methodology.

For up-chirping and down-chirping conditions, theoptimised frequencies for 0 → 1 transition are foundto chirp within the range of 0.011550–0.011596 a.u.(2535–2545 cm−1) and 0.011736–0.011561 a.u. (2576–2537 cm−1), respectively, and for 1 → 2 transition, they arewithin the range of 0.011031–0.011073 (2421–2430 cm−1)and 0.011260–0.011110 (2471–2438 cm−1), respectively.Therefore, the optimised chirping frequencies in both up-and down-chirping conditions are found to vary in a nar-row range, close to the transition frequency. The closenessto the transition frequency can be justified from the view-point of resonance. To retain the closeness in the case offrequency chirping, the chirping rate is found to be verysmall.

In the case of dissociation, the optimised non-chirpingfrequency is 0.021676 (4757 cm−1) which is quite higherthan the theoretically calculated 0 → 1 transition fre-quency. In up- and down-chirping conditions, the opti-mised frequencies are found to vary within the rangeof 0.020966–0.026722 (4602–5864 cm−1) and 0.025730–0.017176 (5647–3770 cm−1), respectively. Therefore, wefind the chirping frequency is varied in a wide range with asignificant chirping rate.

3.7. Effect of adaptive step length in theoptimisation process

To show the effect of adaptive step length in the optimisa-tion processes, we have compared the optimisation resultsbetween a fixed step length and an adaptive step length. InFigure 7 we have shown the transition probabilities fromv = 0 to v = 1 for the two cases at different annealing stepsusing the non-chirping frequency. From the figure it is clearthat the convergence is relatively fast if we use the adap-tive step length during optimisation. Therefore, the use ofadaptive step length in the optimisation process improvesthe optimisation technique and it also helps to get desiredpulse shape quickly. We have obtained the same type ofresults for the other cases.

4. Conclusion

In this work we have explored the optimisation of laser pulsefor selective vibrational state (0 → 1, 0 → 3 and 1 → 2) ex-citations and photo-dissociation of HBr+ molecular ion inits ground electronic state. Potential energy of the systemhas been calculated by ab initio method at the CCSD/aug-cc-pVTZ level which is fitted well in Morse potential. OCT-based SA technique was successfully used to find an ap-propriate laser pulse for selective excitations as well as thephoto-dissociation processes. However, we have better opti-misation results in the non-chirping pulse case for selectiveexcitations while in the case of photo-dissociation, lineardown-chirping case shows better result.

The optimisation method is quite fast and may also beapplied in other vibrational excitations and with addition ofmore variable parameters. However, incorporation of addi-tional parameters will make the optimisation process slowand gives an irregular laser pulse. The use of OCT andadaptive step length in SA makes the optimisation processmore efficient. The use of OCT in the algorithm ensuresthat the laser field intensity will be optimised in such a waythat the total energy involved in the process will be mini-mum. The use of adaptive step length in the process makesus free to choose the initial values of step length. Apartfrom efficient optimisation, the method reveals the relativesignificance of variable parameters in the optimisation pro-cess from their adaptive step length. High step length of anyparameter indicates the flatness of the cost function againstthe parameter. After a few steps the adaptive step lengthof the frequency becomes the smallest one as comparedto that of the chirping rate, peak field amplitude and timeparameters. Therefore, the frequency parameter plays themost significant role in the optimisation process.

AcknowledgementsThe authors thank Dr P Chowdhuri, University of Calcuttafor necessary discussion on algorithm. The authors also thankUGC, Government of India, for financial support through CASprogramme.

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