Quantum control of photodissociation wavepackets1

Journal of Molecular Structure ŽTheochem. 461᎐462 Ž1999. 483᎐491

Quantum control of photodissociation wavepackets 夽 Kenji Mishimaa , Koichi Yamashitab,U a

b

Department of Applied Chemistry, Graduate School of Engineering, The Uni¨ ersity of Tokyo, Tokyo 113-8656, Japan Department of Chemical System Engineering, Graduate School of Engineering, The Uni¨ ersity of Tokyo, Tokyo 113-8656, Japan Received 27 June 1998; accepted 27 July 1998

Abstract Quantum wavepacket calculations have been performed to study ways of controlling the photodissociation processes of NaI and CO by using femtosecond linear-chirped laser pulses. It has been found that positively and negatively chirped pulses influence significantly the behavior of photodissociation wavepackets and photodissociation fluxes. These results arise from the ability to control wavepacket localization, the adiabatic rapid population ŽARP. transfer mechanism and an intra-pulse pump-dump process, all of which are characteristic to linear-chirped laser pulses. 䊚 1999 Elsevier Science B.V. All rights reserved. Keywords: Quantum control; Ultrashort chirped laser pulse; Wavepacket propagation; NaI photodissociation; CO photodissociation

1. Introduction The possibility of controlling chemical reactions by using properly designed laser pulses has been the subject of great interest in recent years w1,2x. Theoretically, several approaches to control reaction paths have been proposed so far. One of these is the coherent control method by Brumer and Shapiro w3x. In this method, the essence of coherent control is to utilize phase and intensity 夽 Dedicated to Professor Keiji Morokuma in celebration of his 65th birthday. U Corresponding author. E-mail: [email protected]

properties of the laser to change the character of the prepared state by using quantum interference between two pulses. Another approach is the pump-dump scheme by Tannor and Rice w4x. Their original idea was to use a pulse to create a localized wavepacket on a bound exit state. When the wavepacket moves over the excite channel of the product of interest, the wavepacket is excited by using a second pulse. Yet another approach is the optimal control theory by Kosloff et al. w5x, Rabitz et al. w6x, Wilson et al. w7x and Fujimura et al. w8x. This is a more systematic pulse optimization method. The pulse shape can be optimized under several conditions. It is usually the case, however, that the optimal control field

0166-1280r99r$ - see front matter 䊚 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 1 2 8 0 Ž 9 8 . 0 0 4 4 5 - X

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K. Mishima, K. Yamashita r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 483᎐491

has a complicated spectral and temporal structure. In the present study, we investigate an alternative way of controlling molecular photodissociation processes by using frequency swept Žchirped. ultrashort laser pulses. Kohler et al. w9x have demonstrated experimentally, as well as theoretically in both gas and condensed phases, that chirped pulses can be used to control the localization of a vibrational wavepacket on an electronically excited potential energy surface. Linearchirped laser pulses in the infrared region have been studied extensively by Bandrauk and Chelkowski w10x. They discussed the possibility of controlling unimolecular dissociations theoretically. One can expect that a pulse whose frequency is decreasing as a function of time and is resonant with transitions between the levels can effectively dissociate a molecule. Recently, they have developed a new control scheme of vibrational excitation, called Raman chirped adiabatic passage ŽRCAP. w11x, which is a complementary method to stimulated Raman scattering by adiabatic passage ŽSTIRAP. developed by Bergmann and Shore w12x. RCAP can be useful for selective excitation of highly polarizable symmetric bonds such as metal᎐metal bonds. Melinger et al. w13x have demonstrated that chirped laser pulses lead to both more efficient and selective excitation than transform-limited pulses in the adiabatic limit, that is, adiabatic rapid passage ŽARP.. Recently we have developed ARP for molecular systems, which explicitly involve molecular vibrations w14x. Cao at al. w15x have indicated that a positively chirped pulse works as a molecular ␲ pulse, which can achieve nearly a complete electronic population inversion. This paper is organized as follows. In Section 2 we summarize the methods of wavepacket calculations. In Section 3 we present our numerical results and discussion on the NaI and CO photodissociation using linear-chirped pulses. We will emphasize three characteristics of linear-chirped laser pulses, i.e. the controllability of wavepacket localization, the adiabatic rapid population ŽARP. transfer mechanism and an intra-pulse pumpdump process, which can be used to control pho-

todissociation reactions. Finally we conclude in Section 4. 2. Method of calculation The photodissociation dynamics of the NaI and CO molecules under a pulse-laser field have been studied by solving the time-dependent Schrodi¨ nger equation given by i

H1 ⭸ ␺ 1Ž t . s ⭸ t ␺2 Ž t . Vint Ž t .

ž / ž

V int Ž t . H2

␺ 1Ž t . ␺2 Ž t .

/ž /

where ␺ 1Ž t ., ␺ 2 Ž t . are the projections of the wavepacket on the ground and excited states of the system. H1 , H2 are the Hamiltonians of the ground and excited states, Hn s Tn q Vn where Tn is the kinetic energy operator and Vn is the potential energy operator. A diabatic representation of the potential energy curve is used for NaI while an adiabatic one is used for CO. We added an optical potential Vopt to V1 and V2 to suppress the unphysical reflection of wavepackets from the edge of grid w16x. Vopt s yi V0 Ž n R y n opt . r Ž n tot y n opt . if n R ) n opt Vopt s 0

otherwise

where n R is the grid number of a point on the dissociation coordinate R. The dissociation coordinate for 3.0- R - 68.5 bohr, is divided into 2048 equally-spaced grids for NaI and for 1.0- R - 6.0 bohr into 1024 grids for CO. n opt is set to 1500 for NaI and 900 for CO. The initial wavepacket, which was obtained by the relaxation method w17x with Newtonian interpolation w18x, was propagated by using the split operator method w19x. Vint Ž t . is the interaction potential, Vint Ž t . s y␮ E Ž t . q V12 Ž R .

for NaI

Vint Ž t . s y␮ E Ž t .

for CO

where ␮ is the dipole operator, EŽ t . the amplitude of the incident laser field, and V12 Ž R . the diabatic coupling potential between V1 and V2 . In this study, we treat the laser field classically

K. Mishima, K. Yamashita r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 483᎐491

485

and set EŽ t . to E Ž t . s E0 exp y␣ Ž t y t 0 . cos w ␻ Ž t .Ž t y t 0 . q ␦ x 2

␤ Ž t y t0 . q ␻ 0 2

␻Žt. s

Eo is the amplitude and ␣ determines the pulse shape in the time domain, from approximately 20 to 200 fs in this study. In the case of a linear chirp, the laser pulse is called a negatively chirped pulse for ␤ - 0, while it is called a positively chirped pulse for ␤ ) 0. The central frequency ␻ 0 was set so that it corresponds exactly to the transition energy from the ground to the excited state. For the NaI photodissociation, the flux jnŽ R,t ., which is the time-integrated norm of the wavepacket passing through a given point on Vn , was calculated by the following equation: U

jn Ž R f ,t . s

⭸␺ Ž R ,t . i ␺n Ž R ,t . n 2m ⭸R U

y␺n Ž R,t .

⭸␺n Ž R,t . ⭸R

Rf

where R f s 45.0 a.u. and the populations of the wavepackets were obtained by the following equations: Pb Ž t . s Pf Ž t . s Pi Ž t . s

Rx

HR

d R < ␺ 2 Ž R ,t . < 2 ,

i

Rf

HR

x

Rf

HR

i

d R < ␺ 2 Ž R,t . < 2 q

t

H0 j Ž R

d R < ␺ 1Ž R,t . < 2 q

2

f

,t⬘ . dt⬘,

H0 j Ž R

f

,t⬘ . dt⬘,

t

1

where R x is the crossing point between the ionic and covalent diabatic states. Pb Ž t . means a portion of the excited wavepacket, which is trapped within the quasi-bound covalent state. On the other hand, Pf Ž t . gives a portion of the excited wavepacket, which belongs to the dissociation region of the covalent state and Pi Ž t . is the sum of the populations of the wavepackets in the ionic state.

Fig. 1. Ionic and covalent potential energy curves of NaI.

3. Results and discussion 3.1. Laser control of NaI photodissociation dynamics We first apply the linear-chirped laser pulse to the control of photodissociation processes in the UV region. For alkali-metal halide molecules, the diabatic potential energy curves of the ionic ground state and the covalent dissociative excited state cross at a long nuclear distance Žsee Fig. 1.. In the NaI photodissociation reaction, Zewail and co-workers w20x have observed wavepacket oscillations trapped on the excited bound state created by a large off-diagonal coupling element between the diabatic states. Here, the NaI photodissociation is used to study the influence of femtosecond linear-chirped pulses on wavepacket dynamics. In Fig. 2, we show the time dependence of Ža. Pb Ž t ., Žb. Pf Ž t . and Žc. Pi Ž t ., respectively. In the case of a transform-limited pulse, the most prominent characteristics is the appearance of an oscillation in Pb Ž t . and Pi Ž t ., which is also observed experimentally w20x. Engle and Metiu w21x reported a detailed quantum mechanical analysis of the experiment. We understand the mechanism of this phenomenon from Fig. 3a, which shows the time-evolution of the population. When the laser is switched on, a portion of the initial wavepacket is excited to the bound excited state. The excited wavepacket moves towards dissociation along the excited potential energy curve and reaches the crossing point R x at 0.12 ps. From 0.2 to 0.4 ps, a part of the excited wavepacket transfers across the crossing point to the ionic state. The excited wavepacket, which was not transferred, continues dissociating while the transferred ionic wave-

486

K. Mishima, K. Yamashita r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 483᎐491

Fig. 2. Time-dependence of Ža. Pb Ž t ., Žb. Pf Ž t . and Žc. Pi Ž t . for transform-limited Ždotted line., positively chirped Ž ␤ s 2 = 10y5 a.u.; broken line. and negatively chirped Ž ␤ s y2 = 10y5 a.u.; solid line. pulses.

packet hits the potential wall and returns to the crossing point R x and a part of the ionic wavepacket returns to the covalent state. At 1.1 ps, the covalent wavepacket hits the potential wall and returns towards R x . These wavepacket dynamics trapped by the excited state well are the origin of the oscillation in Pb Ž t . and Pi Ž t .. We have found that positively and negatively chirped pulses significantly influence these wavepacket oscillations Žsee Fig. 2a,c.. First, we observe, in the case of a positively chirped pulse, that the period of oscillations becomes shorter,

the pulse becomes broader and the peak intensity of the oscillations decays faster than for a transform-limited pulse. Secondary, while the wavepacket oscillates and decreases regularly for a positively chirped pulse, a negatively chirped pulse exhibits a fairly irregular pattern, even producing splitting of the peaks. These phenomena originate from the energy distribution of the wavepackets generated by chirped pulses and from their incoherence. The energy distribution of a laser pulse can be obtained from the Fourier transform of the laser pulse in the time domain. By chirping a transform-limited pulse in the time domain, the energy distribution of the pulse becomes broader due to the energy-time uncertainty principle. Furthermore, the energy distribution loses its initial coherence. For example, a negatively chirp pulse generates first the highfrequency part of the wavepacket and then the low-frequency part. Because the high-energy wavepacket moves faster than the low-energy one, the generated wavepacket becomes broad and this trend is typically observed for NaI as a splitting of the peaks in oscillation Žsee Fig. 3c.. On the other hand, a positively chirped pulse generates a localized wavepacket Žsee Fig. 3b.. Kohler and Wilson have experimentally utilized the controllability of wavepacket localization in I 2 w7x and observed a splitting for NaI w22x. A strong chirp dependence in Pf Ž t ., i.e. the branching ratio of photodissociation, has also been found Žsee Fig. 2c.. Pf Ž t . for a negatively chirped pulse is approximately half that for a transform-limited or a positively chirped pulse. Pf Ž t . depends on two dynamical factors, one is the transition probability from the ionic to the covalent state at the crossing, and the other is the initial population transfer from the ionic to the covalent state. The flux changes at the crossing point in Fig. 4 indicate that the transition probability for the negatively chirped pulse is almost the same as that for the transform-limited and the positively chirped pulse, that is, chirping of pulses has no effect on the transition probability at crossing. Fig. 5 shows the time-dependence of the population in the covalent state. In the case of a negatively chirped pulse, the population

K. Mishima, K. Yamashita r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 483᎐491

487

Fig. 3. Time-evolution of the absolute value of the wavepackets for Ža. transform-limited, Žb. positively chirped Ž ␤ s 2 = 10y5 a.u.. and Žc. negatively chirped Ž ␤ s y2 = 10y5 a.u.. pulses.

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K. Mishima, K. Yamashita r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 483᎐491

Fig. 4. Time-evolution of the fluxes on the neutral potential energy curve; transform-limited Ždotted line., positively chirped Ž ␤ s 2 = 10y5 a.u.; broken line. and negatively chirped Ž ␤ s y2 = 10y5 a.u.; solid line. pulses.

The basic idea of the intra-pulse pump-dump process is illustrated as follows. In the case of a normal pump-dump process by using two lasers for a repulsive excited state the frequency of the dump laser is smaller than that of the pump laser. Therefore, a negatively chirped pulse, whose frequency components are from blue to red, can follow the wavepacket motion and thus the absorption-stimulated emission or pump-dump sequence is favored. On the other hand, a positively chirped pulse cannot enhance this process and pure absorption dominates. This intra-pulse pump-dump effect, first demonstrated by Ruhman and Kosloff w24x theoretically, has been observed experimentally quite recently. Shank and his co-workers w25᎐27x have observed a strong chirp dependence of the fluorescence yields and absorption spectra of the laser dyes CD690 in methanol and LSS750 in acetonitrile. The fluorescence yields and the absorption intensity are

Fig. 5. Time-dependence of the populations on the neutral potential energy curve; transformed-limit Ždotted line., positively chirped Ž ␤ s 2 = 10y5 a.u.; broken line. and negatively chirped Ž ␤ s y2 = 10y5 a.u.; solid line. pulses.

transfer is only half that for the transform-limited and positively chirped pulses. Therefore, the population in the covalent state, Pf Ž t ., is controlled by the population transfer from the ground ionic to the excited covalent state. We explain this characteristic population transfer by chirped pulses based on the intra-pulse pump-dump process, which is enhanced by negatively chirped pulses and prevented by positively chirped pulses. It should be noted here that Bardeen et al. w22x and Tang and Rice w23x have recently demonstrated theoretically that the shape of the pump pulse can control the branching ratio in NaI.

Fig. 6. Scheme for the quantum control of resonance states lifetimes of CO.

Fig. 7. Time-evolution of the norm in the excited state.

K. Mishima, K. Yamashita r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 483᎐491

larger for a positive chirp than for a negative chirp. 3.2. Laser control of resonance state lifetime Next we apply the linear-chirped pulse to control the lifetime of resonance states. In the excited state of CO, a well exists due to the nonadiabatic coupling between the excited B and D⬘ states, which can trap several resonance states Žsee Fig. 6.. Our purpose here is to control the lifetime of these resonance states. Fig. 7 shows the time-evolution of the excited state population of the wavepacket with respect to the linear chirp rate. In the case of a negatively chirped pulse, the populations quickly die off, while in the case of a positively chirped pulse, the population remains almost unity for a long time. The cross-correlation function, which is defined as the overlap between the time-developing wavepacket in the excited state and the eigenfunctions of the resonance states, in Fig. 8 also shows any significant correlation for a negative chirp, while for a positive chirp, there is a very strong correlation between the time-developing wavepacket and the resonance state of quantum number one. From the viewpoint of control, a negatively chirped pulse is very effective in dissociating the excited CO molecule, while a positively chirped pulse can trap the resonance states for a significantly extended time. That is, we have succeeded in controlling the wavepacket dynamics of resonance states by an appropriate selection of the parameters of the laser pulses, namely the linear chirp rate of pulses and the width of the laser pulse. The profound effect of chirped pulses on the excited state lifetime can be explained based on the adiabatic rapid passage ŽARP. population transfer mechanism w13x. We introduce here a three-state model w13x of the ground vibronic state <1: and the excited vibronic states <2: and <3:. The central frequency ␻ 0 of a chirped pulse is assumed to be ␻ 21 - ␻ 0 - ␻ 31. The dressed states < ␣ :, < ␤ : and < ␥ : originating from the interaction between the laser pulse and the three vibronic states can be obtained by diagonalizing the 3 = 3 time-dependent Hamiltonian matrix H Ž t . w13x,

0 HŽt. s

⍀ 12 Ž t . 2

⍀ 13 Ž t . 2

⌬Ž t .

0

0

⌬Ž t . q ␦

⍀ 21 Ž t . 2 ⍀ 31 Ž t . 2

⌬ Ž t . s Ž ␻ 21 y ␻ 0 . q

489

␤ t, 2

where ⍀’s are the Rabi frequencies and ␤ is the linear chirp rate. It can be shown numerically that the ground vibronic state <1: adiabatically corresponds to < ␣ : ( <2: for a positively chirped pulse and to < ␥ : ( <3: for a negatively chirped pulse. That is, chirped pulses lead to more selective excitation than unchirped pulses; a negatively chirped pulse produces higher vibrational states while a positively chirped pulse produces lower vibrational states. In the case of CO, by interacting with a dissociative potential, wavepackets generated by a negatively chirped pulse quickly dissociate, while those produced by a positively chirped pulse are trapped for a significantly longer time. Melinger et al. w13x originally demonstrated ARP on quasi-two-level atomic systems, Na vapor Ža three-level system. and on the multilevel I 2 molecule. A detailed discussion on the ARP for molecular systems, which explicitly includes molecular vibrations, is given in our recent paper w14x. 4. Conclusions In this study, we have demonstrated that three characteristics of linear-chirped laser pulses, i.e. the controllability of wavepacket localization, the adiabatic rapid population ŽARP. transfer mechanism and the intra-pulse pump-dump process, can be use to control photodissociation wavepackets. Quantum wavepacket calculations, performed for the photodissociation processes of NaI and CO by using femtosecond linear-chirped laser pulses, have revealed that positively and negatively chirped pulses influence significantly the behavior of photodissociation wavepackets and photodissociation fluxes. For the NaI photodissociation, a significant chirp dependence of the oscillation of

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K. Mishima, K. Yamashita r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 483᎐491

Fig. 8. Cross-correlation function between the time-developing wavepackets and the resonance states: Ža. negatively chirped Ž ␤ s y2 = 10y5 a.u..; and Žb. positively chirped Ž ␤ s 2 = 10y5 a.u.. pulses. The number n indicates the quantum number of resonance states. The unit of time Žhorizontal axis. is 10 2 a.u.

K. Mishima, K. Yamashita r Journal of Molecular Structure (Theochem) 461᎐462 (1999) 483᎐491

the covalent wavepacket and of the photodissociation branching ratio has been found These results arise from the controllability of wavepacket localization and the intra-pulse pump-dump process, which is enhanced by negatively chirped pulses and prevented by positively chirped pulses. In the case of the CO photodissociation, it has been predicted that a negatively chirped pulse is efficient in dissociating the resonance states that are trapped by a quasi-bound potential well in the excited state while a positively chirped pulse is effective in trapping the resonance states. This profound effect of chirped pulses on the excited state lifetime is based on the ARP transfer mechanism. Experiments to verify our theoretical predictions are anticipated. Acknowledgements Helpful discussions with Prof. Ronnie Kosloff are gratefully acknowledged. The present work is partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science, and Culture. K.Y. thanks the Sumitomo Foundation for a grant. References w1x R.Z. Zare, Science 279 Ž1998. 1875. w2x R.J. Gordon, S.A. Rice, Annu. Rev. Phys. Chem. 48 Ž1997. 601. w3x P. Brumer, M. Shapiro, Annu. Rev. Phys. Chem. 43 Ž1992. 257. w4x D.J. Tannor, S.A. Rice, J. Chem. Phys. 83 Ž1985. 5013. w5x R. Kosloff, S.A. Rice, P. Gaspard, S. Tersigni, D.J. Tannor, Chem. Phys. 139 Ž1989. 201.

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