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Electronic and nuclear correlation dynamics of H2+ in an intense femtosecond laser pulse
19 June 1998
Chemical Physics Letters 289 Ž1998. 546–552
Electronic and nuclear correlation dynamics of Hq2 in an intense femtosecond laser pulse Isao Kawata, Hirohiko Kono, Yuichi Fujimura Department of Chemistry, Graduate School of Science, Tohoku UniÕersity, Sendai 980-8578, Japan Received 16 March 1998; in final form 16 April 1998
Abstract 14 Ž We investigate the wave packet dynamics of Hq Wrcm2 . femtosecond pulse by solving the 2 in a strong G 10 Ž time-dependent Schrodinger equation for a 3D model Hamiltonian molecular orientation is fixed.. As the 3D packet moves ¨ towards larger internuclear distances, the response to the laser electric field switches from the adiabatic one to the diabatic one. Electron density transfers from a well associated with a nucleus to the other well every half optical cycle, following which the interwell transition is suppressed. As a result, the electron is distributed asymmetrically. In the adiabatic region, the correlation between the electronic and nuclear motions slows down the dissociative motion and it is clearly observed in periodic interwell transitions within a half cycle. q 1998 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The development of high-power laser sources have stimulated us to study the dynamics of atoms and molecules in intense, ultrashort laser fields. For atoms interacting with intense laser fields, the electronic dynamics Žsuch as above threshold ionization. is a matter of primary concern w1x. For molecules, another kind of internal motion, namely, nuclear motion, is involved in the dynamics. Recent experiments and theories in the strong field case Ž) 10 11 Wrcm2 . have underscored the combined process of photodissociation and photoionization. Although a large number of studies have been made on the molecular dynamics in laser fields w2x, little is known about the correlation between the electronic and nuclear motions in intense laser fields. The question is what kind of role the correlation between the two
motions plays in the photodissociation and photoionization processes. In this Letter, we theoretically investigate the intense-field dynamics of the simplest one-electron w3,4x. When irradiated by an intense molecule Hq 2 laser field, this molecule either photodissociates as q Hq 2 ™ H q H , or ionizes followed by Coulomb exq q y plosion as Hq 2 ™ H q H q e . The role of each process and their interplay are discussed, with special emphasis on the electronic and nuclear dynamics in the photodissociation process. This dissociation dynamics may be revealed just by starting from the excited electronic state 1 su because in any case the dissociation proceeds via this state in Hq 2 . Thus to make the discussion as simple as possible, we deal with the dynamics starting from 1 su . We solve the time-dependent Schrodinger equa¨ Ž tion for a 3D model of Hq only the internuclear 2
0009-2614r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 6 1 - 8
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
distance is considered as a nuclear coordinate. and investigate the full dynamics in currently available ultraintense, ultrashort laser pulses. We must avoid the use of any approximations such as the Born–Oppenheimer ŽB–O. separation of electronic and nuclear degrees of freedom. To that end a wave packet method we have developed for Coulomb systems is employed w5x. The correlation between the electronic and nuclear motions is interpreted using the time-dependent surface composed of the Coulomb potentials and the dipole interaction with the intense pulse. Two phenomena caused by the intense field are examined: slow down of dissociation and ultrafast electron transfer from a well associated with a nucleus to the other well. The full dynamics in the photodissociation process is compared with that in the two-state model of 1 sg and 1 su , and is found to change from the adiabatic regime to the diabatic one as time goes.
tion between the molecule and the electric field EŽ t . of a laser pulse, V Ž R,r, z. s
In this work, we use the 3D model w3x. In the model, the following assumptions are made: the applied laser fields are linearly polarized along the z-axis; the nuclear motion is restricted to the polarization direction of the laser electric field. The electron moves in three dimensions. Due to the cylindrical symmetry of the model, the electronic degrees of freedom to be considered are the two cylindrical coordinates r and z. Here, r and z are measured with respect to the center of mass of the two nuclei. The Hamiltonian of this three-body system is Žatomic units are used.
Hˆ s y
1
E2
mp E R2
y
qV Ž R,r, z. ,
2 m p q me 4 m p me
ž
E2 Er 2
E2
1 E q
r Er
q
E z2
/
Ž 1.
where R is the internuclear distance, m e and m p are electron and proton masses and V Ž R, r , z . is the sum of the Coulomb interactions and the dipole interac-
1
1 y
R
r 2 q Ž z y Rr2 .
2 1r2
1 y
r 2 q Ž z q Rr2 .
ž
q zE Ž t . 1 q
2 1r2
me 2 m p q me
/
.
Ž 2.
The time evolution of the wave packet on 3D grids is obtained by solving the Schrodinger equation numer¨ ically without any approximation,
i
2. A 3D model and numerical method
547
Ec Ž R , r , z . Et
s Hˆc Ž R , r , z . .
Ž 3.
The method is based on the finite-difference scheme developed in Ref. w5x. Numerical difficulties concerning the singularity of Coulomb potentials are avoided by variable transformations. We would like to focus our attention on the dynamics starting from 1 su . First, the molecule is excited by a weak pump laser from the ground state Žof the vibrational quantum number Õ s 0 in 1 sg . onto 1 su . At the end of the pump process, the electronically excited component of the packet is normalized to unity. Finally, we apply an ionization pulse to the normalized packet. The ionization probability is obtained by setting absorbing boundaries for the electronic coordinates r and z w6x. The electric fields of the pump and ionization pulses are assumed to be EŽ t . s ´ Ž t .sin v t, where ´ Ž t . is the envelope function. We take the following form: ´ Ž t . s E0 sinŽ trT . for 0 F t F T and otherwise zero. For the pump pulse, v s 0.43 Ž105 nm. which corresponds to the energy gap between 1 sg and 1 su at the equilibrium internuclear distance R s 2.0 au and pulse length T is equal to 100 au Ž2.5 fs.. For the ionization pulse, v s 0.052 au Ž884 nm., E0 s 0.096 au Ž3.2 = 10 14 Wrcm2 . and T s 400 au. The ionization pulse is turned on at the end of the pump pulse.
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
548
3. Results and discussion 3.1. 3D packet dynamics In Fig. 1 we show the time-dependent probability of finding the molecule at R P Ž R ,t . s
HH< c Ž R , r , z . <
2
d r d z.
Ž 4.
The bold lines show snapshots of the nuclear motion under no ionization pulse and the thin ones show those under the ionization pulse. The time is measured from the end of the pump pulse. Due to ionization, the probability HP Ž R,t .d R under the ionization pulse is smaller than that without the pulse. It also follows from this figure that the dissociative motion under the intense field is slower than that without the field. This slowdown is due to the temporal change in V Ž R, r , z . induced by the ionization pulse. The electronic and nuclear motions in the intense field are characterized by the 1D motion along the polarization axis. The r-fixed model is known to reproduce characteristic features of the dynamics in an intense field w4x. The key to illustrate the slowdown of dissociative motion is therefore given by the time-dependent potential in two variables R and z. Contour maps for the 2D potential are shown in Fig. 2 Ž r s 1.0..
Fig. 1. Probability of finding the molecule at R in the 3D simulation. Bold lines show snapshots under no ionization pulse and the thin ones show those under the intense ionization pulse.
Fig. 2. Time-dependent Hq 2 potential as a function of z and R Ž r s1.0.: Ža. no electric field; Žb. positive electric field Ž EŽ t . s 0.09 au.. The potential energy is lower in the shaded area.
When no ionization pulse is applied ŽFig. 2a., the packet moves out toward larger internuclear distances along the two valleys given by z s "Rr2 because the electronic state is 1 su . There are no barriers along the two valleys. However, when EŽ t . ) 0, as shown in Fig. 2b, the potential surface is distorted in to having a barrier in the right valley because of the dipole interaction term in Eq. Ž2.. When EŽ t . ) 0, we call the right valley the ascending valley and the left valley the descending one wfor EŽ t . - 0, the left valley becomes the ascending onex. As will be shown in Section 3.2, the packet adiabatically follows the time-dependent electric field Žphase-adiabatic. until t , 2prv w7x. During the first half optical cycle w EŽ t . ) 0x, the greater part of density is transferred from the left well to the right one Že.g. from A to B in Fig. 2b. because at t s 0 the molecule is in the excited electronic state 1 su . The transferred component of the packet moves along the ascending valley as denoted by the arrow in Fig. 2b. As a consequence, the dissociative motion is
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
549
In Fig. 3 we show contour maps for the probability P Ž R , z ,t . s < c Ž R , r , z . < 2 d r .
H
Fig. 3. The probability obtained by integrating the 3D packet over r . Two contour maps are taken at t s87.5 Ža. and t s97.0 au Žb..
Ž 5.
These two snapshots are taken in the second half cycle. The greater part of density resides in the left Žascending. well; and this motion is in accordance with the adiabatic theorem. The population in the right well is smaller at t s 87.5 au ŽFig. 3a. than at t s 97 au ŽFig. 3b.. The distribution in the right well has a minimum ŽMin. in Fig. 3a and has a maximum ŽMax. in Fig. 3b. These two types of wavefunction alternately appear with the period of ; 24 in the second half cycle and the population in the right Žor left. well oscillates accordingly. The mechanism of the change within a half cycle is explained in Section 3.2. In the second cycle and later on, the main component remains in the left well Žalthough the pulse envelope peaks at t s 200 au.. For t G 2prv , electron does not adiabatically transfer from well to well. As a result, an asymmetric electron distribution between the two wells is formed. 3.2. Localization and delocalization due to interference between phase adiabatic states
blocked by the barrier in the ascending valley. In the second half cycle Ž EŽ t . - 0., the packet moves to the left Žascending. well: the motion is again blocked by the barrier in the left valley. As a result, during the first cycle, the dissociative motion is slowed down. Dependence of the ionization on R can be discussed by evaluating the ionization probability at a fixed internuclear distance. It has been reported that in the case of appropriately strong laser fields, ionization from 1 sg is enhanced around specific internuclear distances w8x. We have found that ionization from 1 su is also enhanced at specific internuclear distances R c . For the present ionization pulse, R c , 2 and , 7 au. The ionization probability is 0.71, but it is 0.02 if the ionization pulse is turned off at t s 2prv . Ionization occurs mainly after the second optical cycle Žalthough the packet starts from R , 2 au, the electric field then is not as strong as in the second optical cycle..
In Section 3.1 we have observed that the packet changes its form periodically within a half cycle and finally takes an asymmetric distribution between the two wells. To give clear interpretation to these phenomena, we employ a two-state model. For Hq 2, there are two close states 1 sg and 1 su , which are strongly coupled with each other by radiative interaction. The present two-state Ž1 sg and 1 su . model cannot take into account the ionization process but qualitatively explains the dynamics in the photodissociation process. By diagonalizing the electronic Hamiltonian including the radiative interaction
ž
Hˆ y y
1
E2
mp E R2
/
,
in terms of the two B–O electronic wavefunctions <1 sg : and <1 su : Žabbreviated as
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
550
where
u s 12 arctan
2²g < z
,
Ž 7.
with D Eug Ž R . s EuŽ R . y Eg Ž R .. The corresponding eigenvalues are
(
2 " D Eug Ž R . q 4 <²g < z
Ž 8. In the two-state model, the total wavefunction can be written as
/
ž
/
and PR Ž R . s cos 2 u q
ž
p
<
R <2
/ xŽ . p žu / x Ž . 1
4
< 2 R <2 4 q 2Re cos2 ux 1) Ž R . x 2 Ž R . .
q sin2
q
Ž 10b.
The third term in both equations represents the interference between <1: and <2:. The interference term becomes important as u approaches zero. Inserting Eq. Ž9. into Eq. Ž3., we obtain
E Et
x 1Ž R . s y i y
1
E2
mp E R2
1 Eu E mp E R E R
E2,1 Ž R ,t . s 12 Eg Ž R . q Eu Ž R .
ž
where the nonadiabatic coupling induced by the operator for nuclear kinetic energy, of which leading term is
q E1 Ž R ,t . x 1 Ž R .
,
is neglected. Due to its mass factor, the leading term is much smaller than EurE t. The solutions of Eq. Ž11. can be classified by using w10x
ds
2 D Eug Ž R.
²g < z
.
Ž 12 .
The adiabatic energies, E1Ž R,t . and E2 Ž R,t ., come close to each other at t s nprv Ž n s 1, 2, .... 1. The adiabatic and nonadiabatic transition probabilities at these crossing points are well described by the Landau-Zener formulae, Pad s 1 y expŽypdr4. and Pnonad s expŽypdr4., respectively. For d 4 1, Pad , 1; the phase-adiabatic picture of electronic and nuclear dynamics works well. By solving the coupled Eqs. Ž11., we obtain time-dependent populations as shown in Fig. 4. In this calculation, the internuclear distance is treated as a time-dependent parameter RŽ t .. We use the quantum-mechanical average ² R : of our 3D result as RŽ t .. The initial state at t s 0 is
Eu y
Et
x2 Ž R. ,
E Et
x2 Ž R. s y i y
1
E2
mp E R2
q E2 Ž R ,t . x 2 Ž R .
Eu y
Et
1
The factor EEuR is proportional to EŽ t .. Thus, the coupling m1p EEuR is zero at the crossing points. This is another reason for neglecting the nonadiabatic coupling induced by the operator for nuclear kinetic energy in the derivation of Eq. Ž11.. E ER
x 1Ž R . ,
Ž 11 .
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
Fig. 4. Time-dependent probabilities in the two-state model. Solid, broken and dotted lines denote the populations of
the electronic distribution is completely localized to the left at t , 84 au, it is delocalized to some extent at t , 96.0 au. From Eq. Ž11., we know that x 1 is given by the product of the phase factor expwyiH t E1Ž R,tX .d t X x and the modulus Ž x 2 is likewise given.. For any half cycle, as demonstrated by the populations of <1: and <2: in Fig. 4, the moduli are nearly constant: the interference term in Eq. Ž10. is proportional to the phase factor as cos2 ux 1) Ž R . x 2 Ž R . A cos2 u exp yi
X
X
Ht E Ž R ,t . y E Ž R ,t . 4 d t 2
1
X
.
Ž 13 . At t s 96 au, RŽ t . , 4.5 au and cos2 u , 0.25. The interference term is therefore responsible for the alternate appearances of localization and delocalization in the second half cycle. We see from Eq. Ž11. that the period of the oscillation, i.e. the difference between the points of localization at t s 84 and t s 108 au, corresponds to 2prw E2 Ž R,t . y E1Ž R,t .x Ž; 23 au at t s 96 au.. As illustrated in Fig. 3, it has been observed in the 3D model that the populations in both wells oscillate in the second half cycle. This period is nearly identical with that of the oscillation in the two-state model. The ratio in population between the two wells depends on R, as clearly seen in Fig. 3a. If x 1Ž R . s x 2 Ž R ., PLrPR is independent of R wthis is the case
551
for the parametric model where x 1Ž R . s x 2 Ž R . s d Ž R y RŽ t ..x. However, whenever x 1Ž R . / x 2 Ž R ., P LrPR depends on R. As known from Eq. Ž8., the packet <2:< x 2 : always runs in the ascending valley. The other packet <1:< x 1 : always runs in the descending valley. These two motions are nearly independent of each other as long as the adiabatic picture holds. Thus, x 1 moves faster than x 2 . The difference can be read in Fig. 3. Roughly speaking, < x 2 Ž R .< 2 takes the shape of the section cut along the line A because the dominant term in Eq. Ž10a. associated with the left well is < x 2 Ž R .< 2 Žthe interference term is small.. The dominant term in the right well is nearly expressed, in Fig. 3a, as < x 1Ž R .< 2 y 0.5 < x 1Ž R .< < x 2 Ž R .< and, in Fig. 3b, as < x 1Ž R .< 2 q 0.5 < x 1Ž R .< < x 2 Ž R .<. The interference term is relatively large where x 1Ž R . and x 2 Ž R . overlap with each other. From the locations of the points Min and Max in Fig. 3, we find that the overlap between the two functions has a maximum around R s 4.5 au. As time passes, R increases and D Eug Ž R . decreases Ži.e. d ™ 0.: the adiabatic picture breaks down and the diabatic picture takes over, i.e. < L: and < R : provide a good zeroth-order picture. In the diabatic case d < 1, u is close to pr4; consequently, the interference terms become negligible. It is known that the transition frequency between the two wells is given by D Eug J0 w2²g < z
Acknowledgements We would like to thank Prof. A. D. Bandrauk and Dr. Y. Ohtsuki for stimulating discussions. This work was supported in part by Development of High-Density Optical Pulse Generation and Advanced Material Control Techniques and also by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan Ž09894016..
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I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
References w1x C.A. Nicolaides, C.W. Clark, M.H. Nayfeh ŽEds.., Atoms in Strong Laser Fields, NATO ASI Series B, vol. 212, Plenum Press, New York, 1990; J.H. Eberly, J. Javanainen, K. Rzazewski, Phys. Rep. 204 Ž1991. 331. w2x A.D. Bandrauk ŽEd.., Molecules in Laser Fields, Marcel Dekker, New York, 1994; A. Giusti-Suzor, F.H. Mies, L.F. DiMauro, E. Charron, B. Yang, J. Phys. B: At. Mol. Opt. Phys. 28 Ž1995. 309; B. Sheehy, L.F. DiMauro, Annu. Rev. Phys. Chem. 47 Ž1996. 463. w3x S. Chelkowski, T. Zuo, O. Atabek, A.D. Bandrauk, Phys. Rev. A 52 Ž1995. 2977. w4x K.C. Kulander, F.H. Mies, K.J. Schafer, Phys. Rev. A 53 Ž1996. 2562. w5x H. Kono, A. Kita, Y. Ohtsuki, Y. Fujimura, J. Comput. Phys. 130 Ž1997. 148.
w6x R. Kosloff, D. Kosloff, J. Comput. Phys. 63 Ž1986. 363. w7x The range where the adiabatic theorem is applicable is given, e.g. in A. Messiah, Quantum Mechanics, Vol. 2, North-Holland, Amsterdam, 1970, ch. 17. w8x S. Chelkowski, T. Zuo, A.D. Bandrauk, Phys. Rev. A 46 Ž1992. R5342; T. Zuo, A.D. Bandrauk, Phys. Rev. A 52 Ž1995. R2511; M. Ivanov, T. Seideman, P. Corkum, F. Ilkov, P. Dietrich, Phys. Rev. A 54 Ž1996. 1541. w9x F.H. Mies, A. Giusti-Suzor, K.C. Kulander, K.J. Schafer in: B. Piraux, A. L’Huillier, K. Rzazewski ŽEds.., Super-Intense Laser-Atom Physics, NATO ASI Series B, vol. 316, Plenum Press, New York, 1993, p. 329. w10x Y. Kayanuma, Phys. Rev. B 47 Ž1993. 9940. w11x J.M. Gomez Llorente, J. Plata, Phys. Rev. A 45 Ž1992. R6958; M. Ivanov, P. Corkum, P. Dietrich, Laser Phys. 3 Ž1993. 375; Y. Dakhnovskii, R. Bavli, Phys. Rev. B 48 Ž1993. 11020.
Chemical Physics Letters 289 Ž1998. 546–552
Electronic and nuclear correlation dynamics of Hq2 in an intense femtosecond laser pulse Isao Kawata, Hirohiko Kono, Yuichi Fujimura Department of Chemistry, Graduate School of Science, Tohoku UniÕersity, Sendai 980-8578, Japan Received 16 March 1998; in final form 16 April 1998
Abstract 14 Ž We investigate the wave packet dynamics of Hq Wrcm2 . femtosecond pulse by solving the 2 in a strong G 10 Ž time-dependent Schrodinger equation for a 3D model Hamiltonian molecular orientation is fixed.. As the 3D packet moves ¨ towards larger internuclear distances, the response to the laser electric field switches from the adiabatic one to the diabatic one. Electron density transfers from a well associated with a nucleus to the other well every half optical cycle, following which the interwell transition is suppressed. As a result, the electron is distributed asymmetrically. In the adiabatic region, the correlation between the electronic and nuclear motions slows down the dissociative motion and it is clearly observed in periodic interwell transitions within a half cycle. q 1998 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The development of high-power laser sources have stimulated us to study the dynamics of atoms and molecules in intense, ultrashort laser fields. For atoms interacting with intense laser fields, the electronic dynamics Žsuch as above threshold ionization. is a matter of primary concern w1x. For molecules, another kind of internal motion, namely, nuclear motion, is involved in the dynamics. Recent experiments and theories in the strong field case Ž) 10 11 Wrcm2 . have underscored the combined process of photodissociation and photoionization. Although a large number of studies have been made on the molecular dynamics in laser fields w2x, little is known about the correlation between the electronic and nuclear motions in intense laser fields. The question is what kind of role the correlation between the two
motions plays in the photodissociation and photoionization processes. In this Letter, we theoretically investigate the intense-field dynamics of the simplest one-electron w3,4x. When irradiated by an intense molecule Hq 2 laser field, this molecule either photodissociates as q Hq 2 ™ H q H , or ionizes followed by Coulomb exq q y plosion as Hq 2 ™ H q H q e . The role of each process and their interplay are discussed, with special emphasis on the electronic and nuclear dynamics in the photodissociation process. This dissociation dynamics may be revealed just by starting from the excited electronic state 1 su because in any case the dissociation proceeds via this state in Hq 2 . Thus to make the discussion as simple as possible, we deal with the dynamics starting from 1 su . We solve the time-dependent Schrodinger equa¨ Ž tion for a 3D model of Hq only the internuclear 2
0009-2614r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 6 1 - 8
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
distance is considered as a nuclear coordinate. and investigate the full dynamics in currently available ultraintense, ultrashort laser pulses. We must avoid the use of any approximations such as the Born–Oppenheimer ŽB–O. separation of electronic and nuclear degrees of freedom. To that end a wave packet method we have developed for Coulomb systems is employed w5x. The correlation between the electronic and nuclear motions is interpreted using the time-dependent surface composed of the Coulomb potentials and the dipole interaction with the intense pulse. Two phenomena caused by the intense field are examined: slow down of dissociation and ultrafast electron transfer from a well associated with a nucleus to the other well. The full dynamics in the photodissociation process is compared with that in the two-state model of 1 sg and 1 su , and is found to change from the adiabatic regime to the diabatic one as time goes.
tion between the molecule and the electric field EŽ t . of a laser pulse, V Ž R,r, z. s
In this work, we use the 3D model w3x. In the model, the following assumptions are made: the applied laser fields are linearly polarized along the z-axis; the nuclear motion is restricted to the polarization direction of the laser electric field. The electron moves in three dimensions. Due to the cylindrical symmetry of the model, the electronic degrees of freedom to be considered are the two cylindrical coordinates r and z. Here, r and z are measured with respect to the center of mass of the two nuclei. The Hamiltonian of this three-body system is Žatomic units are used.
Hˆ s y
1
E2
mp E R2
y
qV Ž R,r, z. ,
2 m p q me 4 m p me
ž
E2 Er 2
E2
1 E q
r Er
q
E z2
/
Ž 1.
where R is the internuclear distance, m e and m p are electron and proton masses and V Ž R, r , z . is the sum of the Coulomb interactions and the dipole interac-
1
1 y
R
r 2 q Ž z y Rr2 .
2 1r2
1 y
r 2 q Ž z q Rr2 .
ž
q zE Ž t . 1 q
2 1r2
me 2 m p q me
/
.
Ž 2.
The time evolution of the wave packet on 3D grids is obtained by solving the Schrodinger equation numer¨ ically without any approximation,
i
2. A 3D model and numerical method
547
Ec Ž R , r , z . Et
s Hˆc Ž R , r , z . .
Ž 3.
The method is based on the finite-difference scheme developed in Ref. w5x. Numerical difficulties concerning the singularity of Coulomb potentials are avoided by variable transformations. We would like to focus our attention on the dynamics starting from 1 su . First, the molecule is excited by a weak pump laser from the ground state Žof the vibrational quantum number Õ s 0 in 1 sg . onto 1 su . At the end of the pump process, the electronically excited component of the packet is normalized to unity. Finally, we apply an ionization pulse to the normalized packet. The ionization probability is obtained by setting absorbing boundaries for the electronic coordinates r and z w6x. The electric fields of the pump and ionization pulses are assumed to be EŽ t . s ´ Ž t .sin v t, where ´ Ž t . is the envelope function. We take the following form: ´ Ž t . s E0 sinŽ trT . for 0 F t F T and otherwise zero. For the pump pulse, v s 0.43 Ž105 nm. which corresponds to the energy gap between 1 sg and 1 su at the equilibrium internuclear distance R s 2.0 au and pulse length T is equal to 100 au Ž2.5 fs.. For the ionization pulse, v s 0.052 au Ž884 nm., E0 s 0.096 au Ž3.2 = 10 14 Wrcm2 . and T s 400 au. The ionization pulse is turned on at the end of the pump pulse.
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
548
3. Results and discussion 3.1. 3D packet dynamics In Fig. 1 we show the time-dependent probability of finding the molecule at R P Ž R ,t . s
HH< c Ž R , r , z . <
2
d r d z.
Ž 4.
The bold lines show snapshots of the nuclear motion under no ionization pulse and the thin ones show those under the ionization pulse. The time is measured from the end of the pump pulse. Due to ionization, the probability HP Ž R,t .d R under the ionization pulse is smaller than that without the pulse. It also follows from this figure that the dissociative motion under the intense field is slower than that without the field. This slowdown is due to the temporal change in V Ž R, r , z . induced by the ionization pulse. The electronic and nuclear motions in the intense field are characterized by the 1D motion along the polarization axis. The r-fixed model is known to reproduce characteristic features of the dynamics in an intense field w4x. The key to illustrate the slowdown of dissociative motion is therefore given by the time-dependent potential in two variables R and z. Contour maps for the 2D potential are shown in Fig. 2 Ž r s 1.0..
Fig. 1. Probability of finding the molecule at R in the 3D simulation. Bold lines show snapshots under no ionization pulse and the thin ones show those under the intense ionization pulse.
Fig. 2. Time-dependent Hq 2 potential as a function of z and R Ž r s1.0.: Ža. no electric field; Žb. positive electric field Ž EŽ t . s 0.09 au.. The potential energy is lower in the shaded area.
When no ionization pulse is applied ŽFig. 2a., the packet moves out toward larger internuclear distances along the two valleys given by z s "Rr2 because the electronic state is 1 su . There are no barriers along the two valleys. However, when EŽ t . ) 0, as shown in Fig. 2b, the potential surface is distorted in to having a barrier in the right valley because of the dipole interaction term in Eq. Ž2.. When EŽ t . ) 0, we call the right valley the ascending valley and the left valley the descending one wfor EŽ t . - 0, the left valley becomes the ascending onex. As will be shown in Section 3.2, the packet adiabatically follows the time-dependent electric field Žphase-adiabatic. until t , 2prv w7x. During the first half optical cycle w EŽ t . ) 0x, the greater part of density is transferred from the left well to the right one Že.g. from A to B in Fig. 2b. because at t s 0 the molecule is in the excited electronic state 1 su . The transferred component of the packet moves along the ascending valley as denoted by the arrow in Fig. 2b. As a consequence, the dissociative motion is
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
549
In Fig. 3 we show contour maps for the probability P Ž R , z ,t . s < c Ž R , r , z . < 2 d r .
H
Fig. 3. The probability obtained by integrating the 3D packet over r . Two contour maps are taken at t s87.5 Ža. and t s97.0 au Žb..
Ž 5.
These two snapshots are taken in the second half cycle. The greater part of density resides in the left Žascending. well; and this motion is in accordance with the adiabatic theorem. The population in the right well is smaller at t s 87.5 au ŽFig. 3a. than at t s 97 au ŽFig. 3b.. The distribution in the right well has a minimum ŽMin. in Fig. 3a and has a maximum ŽMax. in Fig. 3b. These two types of wavefunction alternately appear with the period of ; 24 in the second half cycle and the population in the right Žor left. well oscillates accordingly. The mechanism of the change within a half cycle is explained in Section 3.2. In the second cycle and later on, the main component remains in the left well Žalthough the pulse envelope peaks at t s 200 au.. For t G 2prv , electron does not adiabatically transfer from well to well. As a result, an asymmetric electron distribution between the two wells is formed. 3.2. Localization and delocalization due to interference between phase adiabatic states
blocked by the barrier in the ascending valley. In the second half cycle Ž EŽ t . - 0., the packet moves to the left Žascending. well: the motion is again blocked by the barrier in the left valley. As a result, during the first cycle, the dissociative motion is slowed down. Dependence of the ionization on R can be discussed by evaluating the ionization probability at a fixed internuclear distance. It has been reported that in the case of appropriately strong laser fields, ionization from 1 sg is enhanced around specific internuclear distances w8x. We have found that ionization from 1 su is also enhanced at specific internuclear distances R c . For the present ionization pulse, R c , 2 and , 7 au. The ionization probability is 0.71, but it is 0.02 if the ionization pulse is turned off at t s 2prv . Ionization occurs mainly after the second optical cycle Žalthough the packet starts from R , 2 au, the electric field then is not as strong as in the second optical cycle..
In Section 3.1 we have observed that the packet changes its form periodically within a half cycle and finally takes an asymmetric distribution between the two wells. To give clear interpretation to these phenomena, we employ a two-state model. For Hq 2, there are two close states 1 sg and 1 su , which are strongly coupled with each other by radiative interaction. The present two-state Ž1 sg and 1 su . model cannot take into account the ionization process but qualitatively explains the dynamics in the photodissociation process. By diagonalizing the electronic Hamiltonian including the radiative interaction
ž
Hˆ y y
1
E2
mp E R2
/
,
in terms of the two B–O electronic wavefunctions <1 sg : and <1 su : Žabbreviated as
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
550
where
u s 12 arctan
2²g < z
,
Ž 7.
with D Eug Ž R . s EuŽ R . y Eg Ž R .. The corresponding eigenvalues are
(
2 " D Eug Ž R . q 4 <²g < z
Ž 8. In the two-state model, the total wavefunction can be written as
/
ž
/
and PR Ž R . s cos 2 u q
ž
p
<
R <2
/ xŽ . p žu / x Ž . 1
4
< 2 R <2 4 q 2Re cos2 ux 1) Ž R . x 2 Ž R . .
q sin2
q
Ž 10b.
The third term in both equations represents the interference between <1: and <2:. The interference term becomes important as u approaches zero. Inserting Eq. Ž9. into Eq. Ž3., we obtain
E Et
x 1Ž R . s y i y
1
E2
mp E R2
1 Eu E mp E R E R
E2,1 Ž R ,t . s 12 Eg Ž R . q Eu Ž R .
ž
where the nonadiabatic coupling induced by the operator for nuclear kinetic energy, of which leading term is
q E1 Ž R ,t . x 1 Ž R .
,
is neglected. Due to its mass factor, the leading term is much smaller than EurE t. The solutions of Eq. Ž11. can be classified by using w10x
ds
2 D Eug Ž R.
²g < z
.
Ž 12 .
The adiabatic energies, E1Ž R,t . and E2 Ž R,t ., come close to each other at t s nprv Ž n s 1, 2, .... 1. The adiabatic and nonadiabatic transition probabilities at these crossing points are well described by the Landau-Zener formulae, Pad s 1 y expŽypdr4. and Pnonad s expŽypdr4., respectively. For d 4 1, Pad , 1; the phase-adiabatic picture of electronic and nuclear dynamics works well. By solving the coupled Eqs. Ž11., we obtain time-dependent populations as shown in Fig. 4. In this calculation, the internuclear distance is treated as a time-dependent parameter RŽ t .. We use the quantum-mechanical average ² R : of our 3D result as RŽ t .. The initial state at t s 0 is
Eu y
Et
x2 Ž R. ,
E Et
x2 Ž R. s y i y
1
E2
mp E R2
q E2 Ž R ,t . x 2 Ž R .
Eu y
Et
1
The factor EEuR is proportional to EŽ t .. Thus, the coupling m1p EEuR is zero at the crossing points. This is another reason for neglecting the nonadiabatic coupling induced by the operator for nuclear kinetic energy in the derivation of Eq. Ž11.. E ER
x 1Ž R . ,
Ž 11 .
I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
Fig. 4. Time-dependent probabilities in the two-state model. Solid, broken and dotted lines denote the populations of
the electronic distribution is completely localized to the left at t , 84 au, it is delocalized to some extent at t , 96.0 au. From Eq. Ž11., we know that x 1 is given by the product of the phase factor expwyiH t E1Ž R,tX .d t X x and the modulus Ž x 2 is likewise given.. For any half cycle, as demonstrated by the populations of <1: and <2: in Fig. 4, the moduli are nearly constant: the interference term in Eq. Ž10. is proportional to the phase factor as cos2 ux 1) Ž R . x 2 Ž R . A cos2 u exp yi
X
X
Ht E Ž R ,t . y E Ž R ,t . 4 d t 2
1
X
.
Ž 13 . At t s 96 au, RŽ t . , 4.5 au and cos2 u , 0.25. The interference term is therefore responsible for the alternate appearances of localization and delocalization in the second half cycle. We see from Eq. Ž11. that the period of the oscillation, i.e. the difference between the points of localization at t s 84 and t s 108 au, corresponds to 2prw E2 Ž R,t . y E1Ž R,t .x Ž; 23 au at t s 96 au.. As illustrated in Fig. 3, it has been observed in the 3D model that the populations in both wells oscillate in the second half cycle. This period is nearly identical with that of the oscillation in the two-state model. The ratio in population between the two wells depends on R, as clearly seen in Fig. 3a. If x 1Ž R . s x 2 Ž R ., PLrPR is independent of R wthis is the case
551
for the parametric model where x 1Ž R . s x 2 Ž R . s d Ž R y RŽ t ..x. However, whenever x 1Ž R . / x 2 Ž R ., P LrPR depends on R. As known from Eq. Ž8., the packet <2:< x 2 : always runs in the ascending valley. The other packet <1:< x 1 : always runs in the descending valley. These two motions are nearly independent of each other as long as the adiabatic picture holds. Thus, x 1 moves faster than x 2 . The difference can be read in Fig. 3. Roughly speaking, < x 2 Ž R .< 2 takes the shape of the section cut along the line A because the dominant term in Eq. Ž10a. associated with the left well is < x 2 Ž R .< 2 Žthe interference term is small.. The dominant term in the right well is nearly expressed, in Fig. 3a, as < x 1Ž R .< 2 y 0.5 < x 1Ž R .< < x 2 Ž R .< and, in Fig. 3b, as < x 1Ž R .< 2 q 0.5 < x 1Ž R .< < x 2 Ž R .<. The interference term is relatively large where x 1Ž R . and x 2 Ž R . overlap with each other. From the locations of the points Min and Max in Fig. 3, we find that the overlap between the two functions has a maximum around R s 4.5 au. As time passes, R increases and D Eug Ž R . decreases Ži.e. d ™ 0.: the adiabatic picture breaks down and the diabatic picture takes over, i.e. < L: and < R : provide a good zeroth-order picture. In the diabatic case d < 1, u is close to pr4; consequently, the interference terms become negligible. It is known that the transition frequency between the two wells is given by D Eug J0 w2²g < z
Acknowledgements We would like to thank Prof. A. D. Bandrauk and Dr. Y. Ohtsuki for stimulating discussions. This work was supported in part by Development of High-Density Optical Pulse Generation and Advanced Material Control Techniques and also by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan Ž09894016..
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I. Kawata et al.r Chemical Physics Letters 289 (1998) 546–552
References w1x C.A. Nicolaides, C.W. Clark, M.H. Nayfeh ŽEds.., Atoms in Strong Laser Fields, NATO ASI Series B, vol. 212, Plenum Press, New York, 1990; J.H. Eberly, J. Javanainen, K. Rzazewski, Phys. Rep. 204 Ž1991. 331. w2x A.D. Bandrauk ŽEd.., Molecules in Laser Fields, Marcel Dekker, New York, 1994; A. Giusti-Suzor, F.H. Mies, L.F. DiMauro, E. Charron, B. Yang, J. Phys. B: At. Mol. Opt. Phys. 28 Ž1995. 309; B. Sheehy, L.F. DiMauro, Annu. Rev. Phys. Chem. 47 Ž1996. 463. w3x S. Chelkowski, T. Zuo, O. Atabek, A.D. Bandrauk, Phys. Rev. A 52 Ž1995. 2977. w4x K.C. Kulander, F.H. Mies, K.J. Schafer, Phys. Rev. A 53 Ž1996. 2562. w5x H. Kono, A. Kita, Y. Ohtsuki, Y. Fujimura, J. Comput. Phys. 130 Ž1997. 148.
w6x R. Kosloff, D. Kosloff, J. Comput. Phys. 63 Ž1986. 363. w7x The range where the adiabatic theorem is applicable is given, e.g. in A. Messiah, Quantum Mechanics, Vol. 2, North-Holland, Amsterdam, 1970, ch. 17. w8x S. Chelkowski, T. Zuo, A.D. Bandrauk, Phys. Rev. A 46 Ž1992. R5342; T. Zuo, A.D. Bandrauk, Phys. Rev. A 52 Ž1995. R2511; M. Ivanov, T. Seideman, P. Corkum, F. Ilkov, P. Dietrich, Phys. Rev. A 54 Ž1996. 1541. w9x F.H. Mies, A. Giusti-Suzor, K.C. Kulander, K.J. Schafer in: B. Piraux, A. L’Huillier, K. Rzazewski ŽEds.., Super-Intense Laser-Atom Physics, NATO ASI Series B, vol. 316, Plenum Press, New York, 1993, p. 329. w10x Y. Kayanuma, Phys. Rev. B 47 Ž1993. 9940. w11x J.M. Gomez Llorente, J. Plata, Phys. Rev. A 45 Ž1992. R6958; M. Ivanov, P. Corkum, P. Dietrich, Laser Phys. 3 Ž1993. 375; Y. Dakhnovskii, R. Bavli, Phys. Rev. B 48 Ž1993. 11020.