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Quantum Control of Vibrational Excitations in a Heteronuclear Diatomic Molecule

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Quantum Control of Vibrational Excitations in a Heteronuclear Diatomic Molecule J. Chem. Sci., Vol. 119, No. 5, September 2007, pp. 433–440. © Indian Academy of Sciences. Quantum control of vibrational excitations in a heteronuclear diatomic molecule SITANSH SHARMA, PURSHOTAM SHARMA and HARJINDER SINGH* Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology, Hyderabad 500 032 e-mail: [email protected] MS received 28 April 2007; accepted 13 June 2007 Abstract. Optimal control theory is applied to obtain infrared laser pulses for selective vibrational exci- tation in a heteronuclear diatomic molecule. The problem of finding the optimized field is phrased as a maximization of a cost functional which depends on the laser field. A time dependent Gaussian factor is introduced in the field prior to evaluation of the cost functional for better field shape. Conjugate gradient method21,24 is used for optimization of constructed cost functional. At each instant of time, the optimal electric field is calculated and used for the subsequent quantum dynamics, within the dipole approxima- tion. The results are obtained using both Morse potential as well as potential energy obtained using ab initio calculations. Keywords. Optimal control theory; conjugate gradient method; Morse potential; ab initio calculations. 1. Introduction and phase distributions of the radiation coupled to the system. Several methods of controlling molecular Optical time-resolved techniques and frequency- dynamics, like the coherent control12–15 and optimal resolved spectroscopic techniques have helped us to control16–18 using laser fields have recently gained gain good knowledge of the electronic, vibrational popularity. While the former uses quantum mechanical and rotational dynamics in many molecules.1,2 The properties of interference of dynamical paths to growing understanding of these quantum processes regulate the production of a state, the latter provides has opened up the way to control dynamical beha- a more general prescription using calculus of varia- viour of molecular systems. With these develop- tions. ments, the long held dream of a chemist to manipulate Recently, a large number of researchers have chemical reactions, for example to generate desired focused greater attention on optimal control of quan- chemical products with high efficiency, while at the tum states, leading to an extensive theoretical and same time reducing unwanted side products, is pro- numerical work in this area. Zhu et al19 have pre- gressively becoming true. In the last two decades, sented a family of new iteration methods to solve the control of molecular dynamics via shaped laser quantum optimal control problems. They have dem- pulses has experienced considerable progress from a onstrated the efficiency of their algorithm on a typical theoretical concept3–6 to increasing number of suc- one-dimensional system consisting of the excitation cessful experiments.7–10 With the hope of quantum from the ground state to the first excited state in a computing becoming a reality in years to come; co- Morse potential of the O–H bond. Balint-Kurti et herent preparation of molecular states has gained a al20,21 have illustrated the utilization of polarization lot of importance.11 forces for optimal control of the vibrational excita- Ever since their invention, lasers have been viewed tion of the homonuclear diatomic molecule (H2), using as an ideal tool for controlling chemical processes the conjugate gradient method for optimization. because of their monochromatic nature. Growth in In this work we apply the conjugate gradient laser technology has helped us to understand that the method for the optimal control of molecular vibra- dynamics of a strongly coupled light-matter system tional excitations in the heteronuclear diatomic mole- can be influenced by alteration of temporal, spectral cule, HF. We obtain optimal fields of different pulse durations for population transfer from an initial *For correspondence vibrational state to the desired target vibrational state of the hydrogen fluoride molecule. We have treated433 434 Sitansh Sharma et al the hydrogen fluoride molecule. We have treated the T 2 Jtt[()]d, interaction of the molecule with the laser light within p =−∫ ε dipole approximation, i.e. it consists only of the in- 0 teraction of the electric field vector of the radiation with the geometry-dependent electric dipole moment and Jc the dynamical constraint, that is the Schrö- of the molecule.22,23 We have carried out calculations dinger equation is obeyed at every point of time, using Morse potential as well as a potential energy function obtained using ab initio quantum mechani- ∂Ψ()t iH|tηψi = ˆ ()〉 . cal electronic structure calculations; and compared ∂t i the results. Two pulse durations have been consid- ered: (a) 30,000 a.u. and (b) 60,000 a.u. respectively, The Grand functional to be optimized takes the fol- for the following two vibrational excitation processes, lowing shapes: where the first one is a fundamental excitation from the lowest to the next vibrational energy level and the T J[ε(t)] = |T|T|〈ψφ() ()〉−22 αε [()]d tt −ℜ 2 e second is an excitation process starting from a if 0 ∫ higher vibrational level. 0 ⎡ T ∂ ⎤ ⎢〈ψφ()|()TT〉〈 χ () t + iHttˆ ψ ()d 〉 ⎥ . (4) HF(v = 0) → HF (v = 1), if∫ f ∂t i ⎣⎢ 0 ⎦⎥ HF(v = 3) → HF (v = 4). (1) The function ψi(t) is the initial wave function propa- The conjugate gradient method for optimization gated to time T by the optimal laser field ε(t) and used in this work is described in §2. The results are φf(T) is the target vibrational state specified at the discussed in §3. Finally, we summarise the conclu- final time T. The function χf(t) can be regarded as a sions in §4. Lagrange multiplier introduced to assure satisfaction of the Schrödinger equation. The factor α0 is a posi- tive weighting parameter that specifies the weight of 2. Theoretical and computational methods the laser radiation energy to the functional. Varying the grand functional with respect to the initial wave 2.1 Theory function ψi(t), the Lagrange multiplier χf(t) and the Our aim is to design a laser pulse for population electric field ε(t) leads to the following set of equa- transfer from an initial state to a desired dynamical tions: goal at a fixed time while minimizing the pulse en- ()t ergy. The time evolution of the quantum mechanical ∂ψ i ˆ iHt,==ψψii() (0) φ i (0), (5) state ψ(t) under the influence of an external field ∂t ε(t), within the dipole approximation, is governed by the time-dependent Schrödinger wave equation ∂χ ()t iHt,TTf ==ˆ χχ() () φ (), (6) ∂t ff f ∂ ˆ itHtt= ψμεψ()=− [0 ˆ ()] () , (2) ∂t α0ε()tmt||tT|T= −ℑ ( 〈χμψψφfiif () () 〉〈 ( ) ( )), 〉 (7) ˆ ˆ where H0 and μ are the system Hamiltonian and These coupled differential equations can be solved the dipole moment operator, respectively. iteratively. In order to maximize our grand func- For this purpose we construct a cost functional, tional (4), subject to the constraints in (5) and (6) we J[ε], which depends on the optimal driving field. have used the conjugate-gradient method discussed The cost functional contains mainly three terms, below. J[ε] = Jo + Jp + Jc, (3) 2.1b Conjugate gradient method: In order to theoretically design a laser field that can selectively where, Jo contains the physical objective i.e. Jo = excite HF from its initial vibrational state to specific 2 |〈ψi (T)|ψf (T)〉| , Jp contains the penalties, i.e. vibrational excited state, calculation is carried out Quantum control of vibrational excitations in a heteronuclear diatomic molecule 435 using the conjugate gradient method21,24 by maxi- The projected search direction is Fourier trans- mizing the value of the grand functional J[ε(t)] men- formed to obtain a function of frequency, and in order tioned in (4). The gradient of J[ε(t)] including the to restrict the frequency components of the electric penalty term field within a specified range 20th-order Butter- worth band pass filter30,31 is used. The field is up- T dated using the following expression as Jtt=− [()]d2 , p ∫ ε 0 εελkkk+1()ttdtst=+ () ()(), (11) P and the dynamical constraint of Schrödinger equa- k tion with respect to the variation of ε at time t is where λ is determined by the line search and dtP () given by is a projected direction. ∂J k g kk()tstt≡=− 2()[αε [ ()] + 2.2 Ab initio molecular electronic structure method ∂ε k ()t 0 We modelled the atomic interactions using the Morse potential as well as calculated it using quan- ℑ〈mt||tT|Tχμψψφfiif() () 〉〈 ( ) ( )] 〉, (8) tum mechanical calculations (figure 1). The ab initio calculations were performed using MP2/6-31G (d, where the superscript k indicates the iteration num- p) basis set. Gaussian0332 suite of programs was ber in the optimization cycle. The factor s(t), is a used for all ab initio molecular electronic structure Gaussian envelope function is given by calculations carried out in this work. It can be seen from figure 1 that neither of the potential energy 22 st()= e−−(2)6tT/ /(T/) , (9) curves is accurate at internuclear distances much smaller than the equilibrium value. The ab initio Several groups have used s()ttT= sin(2 π / ) as enve- curve is shallower and also less harmonic. lope function. Results using such a sinusoidal factor 25 are communicated by our own group elsewhere. 3. Results The initial wave function, ψi(t), and the Lagrange multiplier, χf(t), are propagated in time using the We have chosen a heteronuclear diatomic system for second order split-operator method.26,27 The Polak– 28 studying the vibrational excitations from an initial Ribiere–Polyak search direction can be calculated using the (8) as kkkk−1 dt()ii=+ gt ()λ d () t i, where kTk k−1 ∑ g ()(tgtgtii ()− ()) i λ k = i , (10) kTk−−11 ∑ gtgt()ii () i 1 1 k k = 2, 3, … and d (ti ) = g (ti ) .
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