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Pulse-shape effects and laser-induced avoided crossings in photodissociation
Volume 197, number 4,5
18 September 1992
CHEMICAL PHYSICS LETTERS
Pulse-shape effects and laser-induced avoided crossings in photodissociation Eric E. Aubanel ‘, Andrk D. Bandrauk Laboratoire de Chimie ThPorique, Fact&
‘**and Pascale Rancourt
des Sciences, Vniversite de Sherbrooke, Sherbrooke, Quebec, Canada JIK 2Rl
Received 11May 1992; in final form 22 June 1992
In a dressed state representation, laser-induced avoided crossings occur between field-molecule electronic surfaces giving rise to stable laser-induced resonances. It is shown that such states can be used to interpret short-pulse (subpicosecond and picosecond) induced molecular photodissociation. Time-dependent numerical calculations of photodissociation probabilities in Hz are used to illustrate pulse-shape effects on photodissociation. It is found that stable (long-lived) laser-induced resonances produce many kinetic energy peaks in photofragmentation, and are sensitive to pulse shapes. Short-lived resonances are insensitive to pulse details. At high intensities, nonadiabatic effects due to pulse-shape variation dominate.
induced avoided crossings between different resonant
1. Introduction
Short laser pulses now allow experimentalists to attain high peak intensities so that molecular multiphoton processes can be studied in the nonperturbative regime [ l-3 1. Thus recently, Bucksbaum and co-workers [ 41, Allendorf and Sziike [ 5 1, Dietrich and Corkum [ 6 ] have shown that with subpicosecond pulses and intensities above lOI2 W/cm’, considerable perturbation of the molecular bonds and potential surfaces of molecular ions occur under the above experimental conditions. From the theoretical side, previous nonperturbative models of field-molecule interactions have relied on time-independent (stationary) dressed state models where the photon (field) and molecular states are considered simultaneously. Such a description gives rise to crossings of field-molecule potential surfaces due to energy conservation in the photon+ molecule system. One of the useful concepts emanating from such a description is the idea of laserCorrespondence
to: A.D. Bandrauk, Laboratoire de Chimie
Theorique, Facultd des Sciences, Universite de Sherbrooke, Sherbrooke, Quebec, Canada J 1K 2R 1. ’ Members of National Center of Excellence in Molecular Dynamics. 2 1992 address: Institute for Molecular Science, Myodaiji, Okazaki, 444, Japan.
field-molecule surfaces [ 2,3,7- 141. This is illustrated in fig. 1 for Hz photodissociation. Starting from some initial vibrational level v in the ground ‘C: electronic state of that molecular ion, photodissociation to the repulsive ‘CT state can be equivalently considered as laser-induced predissociation
’
\\\ 16.6
v 6 5
-
4
T 2%
0.8
1.2 R
1.6 (A)
2.0
Fig. 1. Adiabatic (solid curves; E*(R), eq. (3) ) and diabatic (dashed curves; eq. (2) with V,z=O) surfaces for H:, 1=212.8 nm, at Iz3.2~ lOi* (a), 2.6x 10’s (b), and 5.2x lOi (c) W/ cm2. Also shown are adiabatic levels and the diabatic levels v=46.
0009-2614/92/S 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
419
Volume 197, number
4,5
CHEMICAL
PHYSICS
[ 10,l l-l 41. In the dressed representation, the fieldmolecule surface V, (‘Xz , R) +nZlw crosses the V2(‘CT , R ) + ( n - 1) fro surface as a consequence of conservation of energy. The initial bound vibrational level u interacts with continuum nuclear states through multiple absorption and emission of a photon of wavelength 2~212.8 nm (hw=47000 cm-‘) via the radiative interaction:
Zlo,(cm-‘)=1.2X10-3~(au)Z(W/cm2),
(1)
where ,u(R ) is the transition moment in atomic units, ,I$,is the constant field amplitude and OR is the electronic Rabi frequency which is a measure of the frequency of radiative transitions [ 2,3]. At high intensities, a laser-induced avoided crossing (see fig. 1) will occur transforming the original diabatic (zerofield) surfaces V,(R) and V*(R) into laser-induced adiabatic surfaces E+_(R) obtained by diagonalizing the potential matrix E(R)=
V,(R) +fm v:,(R)
E,(R)=j(V,
V,,(R) I/,(R)
>
’
+V2+kw)
+4[(V,+hW-V2)2+4V:2]“2.
(3)
The upper adiabatic surface will now support new bound states, called laser-induced resonances, since the bound and continuum adiabatic nuclear states remain coupled in the adiabatic representation through the nonadiabatic interaction ( y_ (R) 1d/ dR(y/+(R))d/dR, where v,(R) are the field-induced adiabatic electronic wavefunctions [ 1516 1. Thus adiabatic nuclear functions x+ (R) above the crossing point will correspond to laser-induced resonances which are unstable due to nonadiabatic interaction with the continuum functions x_ (R), i.e. the eigenstates of E_ (R). Such quasi-bound states, being time-independent (stationary) states, can be obtained as poles in scattering-matrix (s-matrix) coupled channel calculations [ 1l- 141. We wish to address in the present paper the time-dependent behaviour of such laserinduced resonances, in keeping with the current experiments studying these nonlinear field-induced 420
LETTERS
18 September
1992
quasibound states [ 4-61. Our results will show that the simple dressed state representation, fig. 1, is still useful in Hz photodissociation for intensities up to I= 1OL4W/cm* and picosecond ( lo-‘* s) pulses. At such high intensities, stabilization of the molecular ion in the presence of the laser field will be shown to occur due to the presence of narrow, long lived, laserinduced resonances, in accord with the semi-classical S-matrix theory of predissociation and laser-induced quasibound states [ 11,13,15 1. This stabilization will be shown to be sensitive to pulse shape effects, and will manifest itself in the anomalous kinetic energy distribution of the molecular photofragments.
2. Results and discussion The time-dependent solutions for the nuclear wavefunctions x, (R, t) and x2 (R, t) of the two diabatic potential surfaces V, (‘X:, R ) and V2(‘Xz , R) were obtained by an efftcient algorithm (second-order split operator method) for the time-dependent SchrSdinger equation [ 17 1. The radiative interactions for such calculations consisted of two types, exact: Vr:,(R, t) = jeRE,( t) cos wt ,
(4)
RWA: V,,(R, t)=~eR&,(t)
(5)
exp(iwt)/2.
The radiative matrix element ( 5 ) corresponds to the rotating wave approximation (RWA), which includes only resonant transitions. The f exp( -iwt) term occuring from the expansion of cos ot in (4) gives rise to virtual (nonresonant) corrections to RWA, eq. (5). In the dressed picture this amounts to including more photon states in fig. 1 since a classical field is a superposition of an infinite number of photon states [2,3] (see e.g. dressed state calculations at high intensities [ 9,18 ] ). We have found that up to I= lOI W/cm*, eqs. (4) and (5) give identical results for the photodissociation probabilities, thus indicating the usefulness of the two-state dressed description in fig. 1. The time-dependent calculations were thus accompanied by time-independent calculations of the S matrix for the two-channel model of fig. 1, with the diabatic dressed potential matrix defined in eq. (2). Such a calculation gives poles in the S matrix cor-
18 September 1992
CHEMICAL PHYSICS LETTERS
Volume 197, number 4,5
to RWA laser-induced resonances at energies E, - ir,, where the imaginary part r, is to be associated with the photodissociation rate via the relation r,= (21c/fi) 1TV,1*, where TV, is the RWA photodissociation amplitude from the initial vibrational state 1v) to the continuum 1c) [ 1 l- 141. The transition moment p(R) = ( 10~1erl2pa,,) = 1eR, is divergent [ 11,18 1. This creates no problem for the time-dependent calculations since at the end of the duration of the pulse, the radiation interactions (4) and ( 5 ) vanish. In the case of the stationary (time-independent) S-matrix calculations the nondiagonal radiative interaction ( 1) was truncated smoothly at around 20 A, thus giving stable resonance positions (this was checked by varying the truncation point). Finally, since the protons produced have high kinetic energy, i.e. a femtosecond time scale motion, the total dissociation probability Pd( t) was obtained by integrating I,y2(R, t) I2 beyond the right turning point of the initial vibrational level in the upper adiabatic surface E, (R). This gives also the kinetic energy distribution through the momentum (p) Fourier transform, P(E, t) = (ml p) Ixz(p, t) 12, where m is the reduced mass of the nuclear system. The first results indicating the presence of stable laser-induced resonances, i.e. adiabatic quasibound states in E, (R), are illustrated in fig. 2. In the present conditions, I = 2 12.8, so that the radiative crossing point is at 15.87 eV, which is between v= 1 and responding complex
2. Thus v= 3 which is near the crossing region, has maximum probability as a function of intensity Z, with nearly 100% photodissociation yield. v=4 has a Pd plateau between 1 and 4x lOI W/cm*, with a maximum at 2x 1013 W/cm2 and finally a plateau starting at 5 x lOI to 1OL4W/cm* with constant yield of about 70°h. P,, of u=6 rises linearly, following Fermi golden-rule behaviour: I T,, I‘a I; then follows a minimum dissociation Pd at 2.6 X 1Or3(30% yield) and a plateau converging to 50% dissociation at high intensities. All results in fig. 2 are for 100 fs pulses with 1 fs rise and fall to and from a constant field value E0 = (8%1/C) ‘I2 . The minima as we will show below from a semi-classical Smatrix analysis [ 1l131 correspond to the presence of narrow and therefore stable laser-induced resonances. Fig. 3 illustrates the effect of pulse field rise, i.e. Z&(t), on: (a) a slowly dissociating v= 4 level due to molecular stabilization at I= 2.5 x 10 I2 W/cm*; (b.i)
1 .o
1 .o
b
0.8 Pd
0.6
/!! , , , , , , 1 0.0
OE-
10
I
5E+013
lE+014 I(W/crI?)
Fig. 2. Photodissociation probability for initial levels ~~3-6, k212.8 nm, and 100 fs pulse with 1 fs rise and fall.
20
40 t
60
80
100
(fs)
Fig. 3. Photodissociation probability as a function of time, with u=4 as initial level, k212.8 nm, 100 fs pulse with 1 fs (solid curves) and 10 fs (dashed curves) rise and fall, and for 1=2.5x 10” (a), 2.0~ 10” (b.i), and 1.0X lOI (b.ii) W/cmZ.
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CHEMICAL PHYSICS LETTERS
Volume 197, number 4,5
a rapidly dissociating U= 4 level at I= 2 x 10 l3 W/ cm2; and finally (b.ii) the high intensity limit, lOI W/cm2. We note that the slower pulse rise gives usually lower dissociation yield, indicative of a more adiabatic behaviour of the photodissociation process with respect to the radiative perturbation. We emphasize that at the high intensity, I= 10 I4 W /cm2 the initial v= 4 level dissociates rapidly between 10 and 20 fs and then becomes stable, i.e. the dissociation yield becomes constant after 20 fs at this high intensity. In figs. 4-6, we show the kinetic energy distributions for the initial vibrational level ~4 at the intensity 2.5 X 1OL2 of greatest stability, Pd < 0.1; 2.0 x 1Or3:intensity of lowest stability, Pd z 1, and tinally the highest intensity 1Or4W/cm’. Results are reported for two different time duration 100 and 1200 fs pulses with the same 1 fs rise and fall. We
2 5
120
v=o1Tmr 1
I.__ P1=O
G 0 :
0 r-
80
x .c .= g D
40
2 (I
-
0
a
990
I--
TrrrrrmTr
b
a
fig. 5. Same as fig. 4, except forI=2.OX 10” W/cm2.
---l-T-pT-r
-T--T--r-rv=o 1 P1=O
t-
i.0
”
b
128
i
4.0 KE (eV)
Fig. 4. Kinetic energy distributions nm, 1=2.5 x lOI W/cm*, and (a) with 1 fs rise and fall. Also shown dissociation from diabatic levels, bility Pd. 422
4.5
---z
-i
‘LO
for initial level v=4,1= 212.8 100 fs, (b) 1200 fspulse, both are energies corresponding to and photodissociation proba-
note that for stable laser-induced resonances (1=2.5x 10” W/cm’), the shorter pulse induces nonadiabatic field transitions, i.e. the initial u=4 level dissociates via the channels u= 2-6. Thus since the dissociation rate is slow with respect to the field time variation, the initial state is redistributed over neighbouring states for short pulses (fig. 4a) as opposed to the longer pulse (fig. 4b) where only one peak corresponding to the initial state dominates. This picture changes considerably for short-lived, unstable laser-induced resonances, as occurs for the initial v=4 at 2.0~ 1013W/cm2 (fig. 5). Both shortand long-lived pulses give the same distribution, a broad peak centered around the initial level energy. Thus in this case, the dissociation is so fast that the kinetic energy distribution is insensitive to the details of the pulse. Of note is an incipient low kinetic energy distribution in both short and long time numerical experiments (fig. 5) with nodes equal to the quantum
Volume 197, number 4,s
Table 1 S-matrix poles at E,-iT,‘) and diabatic (v) -adiabatic quasi-degeneracies (k212.8 nm)
(Use)
v
& (eV)
1 (W/cm’)
&d (eV)
K (cm-‘)
hd
4 6 6
16.41 16.82 16.82
3.2x lOI 2.6 x 10” 5.2x lOI
16.35 16.89 16.80
2x1o-4 1x10-J 1x10-2
0 1 0
a) r=lifetime=5X
Fig. 6. Same as fig. 4, except for IS
I8 September 1992
CHEMICAL PHYSICS LETTERS
1.O x 1014W/cm2.
number of the initial vibrational level, u=4 in the present case! This structure becomes accentuated as one goes to higher intensity, I= 10 I4 W/cm*. Five broad peaks occur with a resonant structure at higher energy which sharpens for the longer pulse. We now turn to an interpretation of the results illustrated in figs. 2-6. Firstly, the stability (minimum Pd) of the molecule in the laser field can be correlated with the general rule derived from the semi-classical S matrix for molecular bound-continua interaction [ 1l-l 3 1: coincidences in energy, i.e. quasi-degeneracies of diabatic and adiabatic levels correspond to stable resonances characterized by the nonadiabaticity parameter, U(E) = exp (e) - 1, where t= 27c1VI2 (*/fivAF is the Landau-Zener parameter; l’,, is the radiative interaction ( 1 ), v is the nuclear velocity and M is the difference in slope of the adiabatic surfaces at the crossing point respectively. Perusal of fig. 1 shows that at I= 3.2 x 10 I* W/cm*, v= 4 is quasi-degenerate with v,~= 0, v= 6 is
10-‘2s/f
(cm-‘).
quasi-degenerate with v,,= 1 at Z=2.6x lOi W/cm2 and with vad= 0 at 5.2~ 10” W/cm*. In table 1 we show the S matrix linewidths at these quasi-degeneracies with laser-induced resonances of lifetimes much greater than the pulse lengths (see table 1) correlate with the minima in Pd observed in fig. 2. Thus for v= 4, the slow dissociation and its sensitivwavefunctions x+ (R) by the laser-induced adiabatic potential E, (R) must be operative. The adiabatic levels are above the dissociative limit (Ed= 18.07 eV) only at Z=4x 10 I4 W/cm*, but v=4 is below the adiabatic minimum energy starting at 2.6 x lOI W/cm*, near a maximum in Pd. u=6 is also below the adiabatic minimum starting at I=8 X 1Ol3 W/ cm*, and yet both levels show incomplete dissociation. Fig. 3 accentuates this fact at the high intensity, lOI4 W/cm*. The laser-induced resonance is stable ity to pulse effects (see figs. 3 and 4) is due to the presence of a stable laser-induced resonance. This state is an intermediate coupling case, being neither diabatic or adiabatic [ 1I- 13 1, since at the intensity 1=3.2X lOI* W/cm*, for vad=O, ~~0.4, whereas at I= lOI W/cm*, EX 6. Thus in the latter case, UB 1 and one is in the adiabatic limit. For the maximum dissociation (v=4), Zz2.0~ lOI W/cm*, ~~52, u > 1, one is again in the adiabatic limit. Clearly the minima in Pd are indicative of stable laser-induced resonances created by the laser-induced avoided crossing, fig. 1. As shown in figs. 4 and 5, such laserstabilized molecular states are sensitive to rapid variations in pulse shapes, resulting in nonadiabatic excitation of many vibration levels, whereas shortlived resonances are insensitive to pulse shape effects. Fig. 6 is more problematic. The incomplete dissociation at I= 10 I4 W/cm* (7OW yield) must be correlated with the adiabatic character of the laserinduced resonance (U B 1). Here the avoided cross423
Volume 197, number 45
CHEMICAL PHYSICS LETTERS
ing is at its maximum and trapping of the nuclear (no dissociation increase with time) during the laser pulse’s presence. This paradox is compounded by the broad kinetic energy structure, fig. 6. Fig. 6 suggests that the five broad peaks are to be attributed to v=O4 which have been pushed down by the radiative interaction and hence have dissociated rapidly, whereas the stable resonance in the pulse seems to correspond to trapping of the higher vibrational levels (fig. 6a shows peaks at u= 7, 11, 12, etc.). This implies considerable nonadiabatic excitation due to the pulse variation, in fact a 10 fs rise of the pulse shows much less structure than appears in fig. 6. The present numerical results require therefore a complete theory of nonadiabaticity in molecule-radiation interaction due to (i) laser pulse time variation, and (ii) laserinduced avoided crossings [ 19 1. In the present Letter we have dealt with boundcontinuum transitions, that created new laser-induced resonances. It is to be noted that in the atomatom collision problem [ 201, short pulses also induce nonadiabatic transitions as observed here.
Acknowledgement
We thank the Natural Sciences and Engineering Research Council of Canada and the National Center of Excellence in Molecular Dynamics for financing this research.
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18 September 1992
References [ 1] A.D. Bandrauk, ed., Atomic and molecular processes with short intense laser pulses, Vol. B171, NATO ASI Series (Plenum Press, New York, 1988). [ 21 M.H. Mittleman, Theory of laser-atom interactions (Plenum Press, New York, 1982) ch. 9. [ 31 A.D. Bandrauk, Molecules in Laser Fields (Marcel Dekker, New York), to be published. [4] A. Zariyev, P.H. Bucksbaum, H.G. Muller and D.W. Schumacher, Phys. Rev. A 42 ( 1990) 5500. [5] S.W. Allendorfand A. Sziike, Phys. Rev. A44 (1991) 518. [ 61 P. Dietrich and P.B. Corkum, to appear in J. Chem. Phys. [7] A.I. Voronin and A.A. Samokhin, Soviet Phys. JETP 43 (1976) 4. [S]A.M.LauandC.K.Rbodes,Phys.Rev.A 16 (1977)2392. 191 T.F. George, I.H. Zimmerman, J.-M. Yuan, J.R. Laing and P.L. De Vries, Accounts Chem. Res. 10 ( 1977) 449. [lo] J.M. Yuan andT.F. George, J. Chem. Phys. 68 (1978) 3040. [ 111 A.D. Bandrauk and M.L. Sink, Chem. Phys. Letters 57 ( 1978) 569; J. Chem. Phys. 74 ( 1981) 1110. [ 121 A.D. Bandrauk and 0. Atabek, in: Advances in chemical physics, Vol. 73, eds. J.O. Hirschfelder, R. Wyatt and R. Coalson (Wiley, New York, 1988) ch. 19. [ 121 A.D. Bandrauk and J.F. McCann, Comments At, Mol. Phys. 22 (1989) 325. [ 141 J.F. McCann and A.D. Bandrauk, Phys. Rev. A 42 ( 1990) 2806; J. Chem. Phys. 96 (1992) 903. [ 15 ] A.D. Bandrauk and M.S. Child, Mol. Phys. 19 ( 1970) 95. [ 161 H. Levebvre-Brion and R.W. Field, Perturbations in spectra ofdiatomic molecules (Academic Press, New York, 1986). [ 171 A.D. Bandrauk and H. Shen, Chem. Phys. Letters 176 (1991) 428. [ 1S] A.D. Bandrauk, E. Constant and J.M. Gauthier, J. Phys. II (Paris) I(l991) 1033. [ 191 E.E. Aubanel and A.D. Bandrauk, in preparation. [20] H.W. Lee and T.F. George, Phys. Rev. A 35 ( 1987) 4977; J. Phys. Chem. 83 (1979) 928.
18 September 1992
CHEMICAL PHYSICS LETTERS
Pulse-shape effects and laser-induced avoided crossings in photodissociation Eric E. Aubanel ‘, Andrk D. Bandrauk Laboratoire de Chimie ThPorique, Fact&
‘**and Pascale Rancourt
des Sciences, Vniversite de Sherbrooke, Sherbrooke, Quebec, Canada JIK 2Rl
Received 11May 1992; in final form 22 June 1992
In a dressed state representation, laser-induced avoided crossings occur between field-molecule electronic surfaces giving rise to stable laser-induced resonances. It is shown that such states can be used to interpret short-pulse (subpicosecond and picosecond) induced molecular photodissociation. Time-dependent numerical calculations of photodissociation probabilities in Hz are used to illustrate pulse-shape effects on photodissociation. It is found that stable (long-lived) laser-induced resonances produce many kinetic energy peaks in photofragmentation, and are sensitive to pulse shapes. Short-lived resonances are insensitive to pulse details. At high intensities, nonadiabatic effects due to pulse-shape variation dominate.
induced avoided crossings between different resonant
1. Introduction
Short laser pulses now allow experimentalists to attain high peak intensities so that molecular multiphoton processes can be studied in the nonperturbative regime [ l-3 1. Thus recently, Bucksbaum and co-workers [ 41, Allendorf and Sziike [ 5 1, Dietrich and Corkum [ 6 ] have shown that with subpicosecond pulses and intensities above lOI2 W/cm’, considerable perturbation of the molecular bonds and potential surfaces of molecular ions occur under the above experimental conditions. From the theoretical side, previous nonperturbative models of field-molecule interactions have relied on time-independent (stationary) dressed state models where the photon (field) and molecular states are considered simultaneously. Such a description gives rise to crossings of field-molecule potential surfaces due to energy conservation in the photon+ molecule system. One of the useful concepts emanating from such a description is the idea of laserCorrespondence
to: A.D. Bandrauk, Laboratoire de Chimie
Theorique, Facultd des Sciences, Universite de Sherbrooke, Sherbrooke, Quebec, Canada J 1K 2R 1. ’ Members of National Center of Excellence in Molecular Dynamics. 2 1992 address: Institute for Molecular Science, Myodaiji, Okazaki, 444, Japan.
field-molecule surfaces [ 2,3,7- 141. This is illustrated in fig. 1 for Hz photodissociation. Starting from some initial vibrational level v in the ground ‘C: electronic state of that molecular ion, photodissociation to the repulsive ‘CT state can be equivalently considered as laser-induced predissociation
’
\\\ 16.6
v 6 5
-
4
T 2%
0.8
1.2 R
1.6 (A)
2.0
Fig. 1. Adiabatic (solid curves; E*(R), eq. (3) ) and diabatic (dashed curves; eq. (2) with V,z=O) surfaces for H:, 1=212.8 nm, at Iz3.2~ lOi* (a), 2.6x 10’s (b), and 5.2x lOi (c) W/ cm2. Also shown are adiabatic levels and the diabatic levels v=46.
0009-2614/92/S 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
419
Volume 197, number
4,5
CHEMICAL
PHYSICS
[ 10,l l-l 41. In the dressed representation, the fieldmolecule surface V, (‘Xz , R) +nZlw crosses the V2(‘CT , R ) + ( n - 1) fro surface as a consequence of conservation of energy. The initial bound vibrational level u interacts with continuum nuclear states through multiple absorption and emission of a photon of wavelength 2~212.8 nm (hw=47000 cm-‘) via the radiative interaction:
Zlo,(cm-‘)=1.2X10-3~(au)Z(W/cm2),
(1)
where ,u(R ) is the transition moment in atomic units, ,I$,is the constant field amplitude and OR is the electronic Rabi frequency which is a measure of the frequency of radiative transitions [ 2,3]. At high intensities, a laser-induced avoided crossing (see fig. 1) will occur transforming the original diabatic (zerofield) surfaces V,(R) and V*(R) into laser-induced adiabatic surfaces E+_(R) obtained by diagonalizing the potential matrix E(R)=
V,(R) +fm v:,(R)
E,(R)=j(V,
V,,(R) I/,(R)
>
’
+V2+kw)
+4[(V,+hW-V2)2+4V:2]“2.
(3)
The upper adiabatic surface will now support new bound states, called laser-induced resonances, since the bound and continuum adiabatic nuclear states remain coupled in the adiabatic representation through the nonadiabatic interaction ( y_ (R) 1d/ dR(y/+(R))d/dR, where v,(R) are the field-induced adiabatic electronic wavefunctions [ 1516 1. Thus adiabatic nuclear functions x+ (R) above the crossing point will correspond to laser-induced resonances which are unstable due to nonadiabatic interaction with the continuum functions x_ (R), i.e. the eigenstates of E_ (R). Such quasi-bound states, being time-independent (stationary) states, can be obtained as poles in scattering-matrix (s-matrix) coupled channel calculations [ 1l- 141. We wish to address in the present paper the time-dependent behaviour of such laserinduced resonances, in keeping with the current experiments studying these nonlinear field-induced 420
LETTERS
18 September
1992
quasibound states [ 4-61. Our results will show that the simple dressed state representation, fig. 1, is still useful in Hz photodissociation for intensities up to I= 1OL4W/cm* and picosecond ( lo-‘* s) pulses. At such high intensities, stabilization of the molecular ion in the presence of the laser field will be shown to occur due to the presence of narrow, long lived, laserinduced resonances, in accord with the semi-classical S-matrix theory of predissociation and laser-induced quasibound states [ 11,13,15 1. This stabilization will be shown to be sensitive to pulse shape effects, and will manifest itself in the anomalous kinetic energy distribution of the molecular photofragments.
2. Results and discussion The time-dependent solutions for the nuclear wavefunctions x, (R, t) and x2 (R, t) of the two diabatic potential surfaces V, (‘X:, R ) and V2(‘Xz , R) were obtained by an efftcient algorithm (second-order split operator method) for the time-dependent SchrSdinger equation [ 17 1. The radiative interactions for such calculations consisted of two types, exact: Vr:,(R, t) = jeRE,( t) cos wt ,
(4)
RWA: V,,(R, t)=~eR&,(t)
(5)
exp(iwt)/2.
The radiative matrix element ( 5 ) corresponds to the rotating wave approximation (RWA), which includes only resonant transitions. The f exp( -iwt) term occuring from the expansion of cos ot in (4) gives rise to virtual (nonresonant) corrections to RWA, eq. (5). In the dressed picture this amounts to including more photon states in fig. 1 since a classical field is a superposition of an infinite number of photon states [2,3] (see e.g. dressed state calculations at high intensities [ 9,18 ] ). We have found that up to I= lOI W/cm*, eqs. (4) and (5) give identical results for the photodissociation probabilities, thus indicating the usefulness of the two-state dressed description in fig. 1. The time-dependent calculations were thus accompanied by time-independent calculations of the S matrix for the two-channel model of fig. 1, with the diabatic dressed potential matrix defined in eq. (2). Such a calculation gives poles in the S matrix cor-
18 September 1992
CHEMICAL PHYSICS LETTERS
Volume 197, number 4,5
to RWA laser-induced resonances at energies E, - ir,, where the imaginary part r, is to be associated with the photodissociation rate via the relation r,= (21c/fi) 1TV,1*, where TV, is the RWA photodissociation amplitude from the initial vibrational state 1v) to the continuum 1c) [ 1 l- 141. The transition moment p(R) = ( 10~1erl2pa,,) = 1eR, is divergent [ 11,18 1. This creates no problem for the time-dependent calculations since at the end of the duration of the pulse, the radiation interactions (4) and ( 5 ) vanish. In the case of the stationary (time-independent) S-matrix calculations the nondiagonal radiative interaction ( 1) was truncated smoothly at around 20 A, thus giving stable resonance positions (this was checked by varying the truncation point). Finally, since the protons produced have high kinetic energy, i.e. a femtosecond time scale motion, the total dissociation probability Pd( t) was obtained by integrating I,y2(R, t) I2 beyond the right turning point of the initial vibrational level in the upper adiabatic surface E, (R). This gives also the kinetic energy distribution through the momentum (p) Fourier transform, P(E, t) = (ml p) Ixz(p, t) 12, where m is the reduced mass of the nuclear system. The first results indicating the presence of stable laser-induced resonances, i.e. adiabatic quasibound states in E, (R), are illustrated in fig. 2. In the present conditions, I = 2 12.8, so that the radiative crossing point is at 15.87 eV, which is between v= 1 and responding complex
2. Thus v= 3 which is near the crossing region, has maximum probability as a function of intensity Z, with nearly 100% photodissociation yield. v=4 has a Pd plateau between 1 and 4x lOI W/cm*, with a maximum at 2x 1013 W/cm2 and finally a plateau starting at 5 x lOI to 1OL4W/cm* with constant yield of about 70°h. P,, of u=6 rises linearly, following Fermi golden-rule behaviour: I T,, I‘a I; then follows a minimum dissociation Pd at 2.6 X 1Or3(30% yield) and a plateau converging to 50% dissociation at high intensities. All results in fig. 2 are for 100 fs pulses with 1 fs rise and fall to and from a constant field value E0 = (8%1/C) ‘I2 . The minima as we will show below from a semi-classical Smatrix analysis [ 1l131 correspond to the presence of narrow and therefore stable laser-induced resonances. Fig. 3 illustrates the effect of pulse field rise, i.e. Z&(t), on: (a) a slowly dissociating v= 4 level due to molecular stabilization at I= 2.5 x 10 I2 W/cm*; (b.i)
1 .o
1 .o
b
0.8 Pd
0.6
/!! , , , , , , 1 0.0
OE-
10
I
5E+013
lE+014 I(W/crI?)
Fig. 2. Photodissociation probability for initial levels ~~3-6, k212.8 nm, and 100 fs pulse with 1 fs rise and fall.
20
40 t
60
80
100
(fs)
Fig. 3. Photodissociation probability as a function of time, with u=4 as initial level, k212.8 nm, 100 fs pulse with 1 fs (solid curves) and 10 fs (dashed curves) rise and fall, and for 1=2.5x 10” (a), 2.0~ 10” (b.i), and 1.0X lOI (b.ii) W/cmZ.
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a rapidly dissociating U= 4 level at I= 2 x 10 l3 W/ cm2; and finally (b.ii) the high intensity limit, lOI W/cm2. We note that the slower pulse rise gives usually lower dissociation yield, indicative of a more adiabatic behaviour of the photodissociation process with respect to the radiative perturbation. We emphasize that at the high intensity, I= 10 I4 W /cm2 the initial v= 4 level dissociates rapidly between 10 and 20 fs and then becomes stable, i.e. the dissociation yield becomes constant after 20 fs at this high intensity. In figs. 4-6, we show the kinetic energy distributions for the initial vibrational level ~4 at the intensity 2.5 X 1OL2 of greatest stability, Pd < 0.1; 2.0 x 1Or3:intensity of lowest stability, Pd z 1, and tinally the highest intensity 1Or4W/cm’. Results are reported for two different time duration 100 and 1200 fs pulses with the same 1 fs rise and fall. We
2 5
120
v=o1Tmr 1
I.__ P1=O
G 0 :
0 r-
80
x .c .= g D
40
2 (I
-
0
a
990
I--
TrrrrrmTr
b
a
fig. 5. Same as fig. 4, except forI=2.OX 10” W/cm2.
---l-T-pT-r
-T--T--r-rv=o 1 P1=O
t-
i.0
”
b
128
i
4.0 KE (eV)
Fig. 4. Kinetic energy distributions nm, 1=2.5 x lOI W/cm*, and (a) with 1 fs rise and fall. Also shown dissociation from diabatic levels, bility Pd. 422
4.5
---z
-i
‘LO
for initial level v=4,1= 212.8 100 fs, (b) 1200 fspulse, both are energies corresponding to and photodissociation proba-
note that for stable laser-induced resonances (1=2.5x 10” W/cm’), the shorter pulse induces nonadiabatic field transitions, i.e. the initial u=4 level dissociates via the channels u= 2-6. Thus since the dissociation rate is slow with respect to the field time variation, the initial state is redistributed over neighbouring states for short pulses (fig. 4a) as opposed to the longer pulse (fig. 4b) where only one peak corresponding to the initial state dominates. This picture changes considerably for short-lived, unstable laser-induced resonances, as occurs for the initial v=4 at 2.0~ 1013W/cm2 (fig. 5). Both shortand long-lived pulses give the same distribution, a broad peak centered around the initial level energy. Thus in this case, the dissociation is so fast that the kinetic energy distribution is insensitive to the details of the pulse. Of note is an incipient low kinetic energy distribution in both short and long time numerical experiments (fig. 5) with nodes equal to the quantum
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Table 1 S-matrix poles at E,-iT,‘) and diabatic (v) -adiabatic quasi-degeneracies (k212.8 nm)
(Use)
v
& (eV)
1 (W/cm’)
&d (eV)
K (cm-‘)
hd
4 6 6
16.41 16.82 16.82
3.2x lOI 2.6 x 10” 5.2x lOI
16.35 16.89 16.80
2x1o-4 1x10-J 1x10-2
0 1 0
a) r=lifetime=5X
Fig. 6. Same as fig. 4, except for IS
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CHEMICAL PHYSICS LETTERS
1.O x 1014W/cm2.
number of the initial vibrational level, u=4 in the present case! This structure becomes accentuated as one goes to higher intensity, I= 10 I4 W/cm*. Five broad peaks occur with a resonant structure at higher energy which sharpens for the longer pulse. We now turn to an interpretation of the results illustrated in figs. 2-6. Firstly, the stability (minimum Pd) of the molecule in the laser field can be correlated with the general rule derived from the semi-classical S matrix for molecular bound-continua interaction [ 1l-l 3 1: coincidences in energy, i.e. quasi-degeneracies of diabatic and adiabatic levels correspond to stable resonances characterized by the nonadiabaticity parameter, U(E) = exp (e) - 1, where t= 27c1VI2 (*/fivAF is the Landau-Zener parameter; l’,, is the radiative interaction ( 1 ), v is the nuclear velocity and M is the difference in slope of the adiabatic surfaces at the crossing point respectively. Perusal of fig. 1 shows that at I= 3.2 x 10 I* W/cm*, v= 4 is quasi-degenerate with v,~= 0, v= 6 is
10-‘2s/f
(cm-‘).
quasi-degenerate with v,,= 1 at Z=2.6x lOi W/cm2 and with vad= 0 at 5.2~ 10” W/cm*. In table 1 we show the S matrix linewidths at these quasi-degeneracies with laser-induced resonances of lifetimes much greater than the pulse lengths (see table 1) correlate with the minima in Pd observed in fig. 2. Thus for v= 4, the slow dissociation and its sensitivwavefunctions x+ (R) by the laser-induced adiabatic potential E, (R) must be operative. The adiabatic levels are above the dissociative limit (Ed= 18.07 eV) only at Z=4x 10 I4 W/cm*, but v=4 is below the adiabatic minimum energy starting at 2.6 x lOI W/cm*, near a maximum in Pd. u=6 is also below the adiabatic minimum starting at I=8 X 1Ol3 W/ cm*, and yet both levels show incomplete dissociation. Fig. 3 accentuates this fact at the high intensity, lOI4 W/cm*. The laser-induced resonance is stable ity to pulse effects (see figs. 3 and 4) is due to the presence of a stable laser-induced resonance. This state is an intermediate coupling case, being neither diabatic or adiabatic [ 1I- 13 1, since at the intensity 1=3.2X lOI* W/cm*, for vad=O, ~~0.4, whereas at I= lOI W/cm*, EX 6. Thus in the latter case, UB 1 and one is in the adiabatic limit. For the maximum dissociation (v=4), Zz2.0~ lOI W/cm*, ~~52, u > 1, one is again in the adiabatic limit. Clearly the minima in Pd are indicative of stable laser-induced resonances created by the laser-induced avoided crossing, fig. 1. As shown in figs. 4 and 5, such laserstabilized molecular states are sensitive to rapid variations in pulse shapes, resulting in nonadiabatic excitation of many vibration levels, whereas shortlived resonances are insensitive to pulse shape effects. Fig. 6 is more problematic. The incomplete dissociation at I= 10 I4 W/cm* (7OW yield) must be correlated with the adiabatic character of the laserinduced resonance (U B 1). Here the avoided cross423
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ing is at its maximum and trapping of the nuclear (no dissociation increase with time) during the laser pulse’s presence. This paradox is compounded by the broad kinetic energy structure, fig. 6. Fig. 6 suggests that the five broad peaks are to be attributed to v=O4 which have been pushed down by the radiative interaction and hence have dissociated rapidly, whereas the stable resonance in the pulse seems to correspond to trapping of the higher vibrational levels (fig. 6a shows peaks at u= 7, 11, 12, etc.). This implies considerable nonadiabatic excitation due to the pulse variation, in fact a 10 fs rise of the pulse shows much less structure than appears in fig. 6. The present numerical results require therefore a complete theory of nonadiabaticity in molecule-radiation interaction due to (i) laser pulse time variation, and (ii) laserinduced avoided crossings [ 19 1. In the present Letter we have dealt with boundcontinuum transitions, that created new laser-induced resonances. It is to be noted that in the atomatom collision problem [ 201, short pulses also induce nonadiabatic transitions as observed here.
Acknowledgement
We thank the Natural Sciences and Engineering Research Council of Canada and the National Center of Excellence in Molecular Dynamics for financing this research.
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References [ 1] A.D. Bandrauk, ed., Atomic and molecular processes with short intense laser pulses, Vol. B171, NATO ASI Series (Plenum Press, New York, 1988). [ 21 M.H. Mittleman, Theory of laser-atom interactions (Plenum Press, New York, 1982) ch. 9. [ 31 A.D. Bandrauk, Molecules in Laser Fields (Marcel Dekker, New York), to be published. [4] A. Zariyev, P.H. Bucksbaum, H.G. Muller and D.W. Schumacher, Phys. Rev. A 42 ( 1990) 5500. [5] S.W. Allendorfand A. Sziike, Phys. Rev. A44 (1991) 518. [ 61 P. Dietrich and P.B. Corkum, to appear in J. Chem. Phys. [7] A.I. Voronin and A.A. Samokhin, Soviet Phys. JETP 43 (1976) 4. [S]A.M.LauandC.K.Rbodes,Phys.Rev.A 16 (1977)2392. 191 T.F. George, I.H. Zimmerman, J.-M. Yuan, J.R. Laing and P.L. De Vries, Accounts Chem. Res. 10 ( 1977) 449. [lo] J.M. Yuan andT.F. George, J. Chem. Phys. 68 (1978) 3040. [ 111 A.D. Bandrauk and M.L. Sink, Chem. Phys. Letters 57 ( 1978) 569; J. Chem. Phys. 74 ( 1981) 1110. [ 121 A.D. Bandrauk and 0. Atabek, in: Advances in chemical physics, Vol. 73, eds. J.O. Hirschfelder, R. Wyatt and R. Coalson (Wiley, New York, 1988) ch. 19. [ 121 A.D. Bandrauk and J.F. McCann, Comments At, Mol. Phys. 22 (1989) 325. [ 141 J.F. McCann and A.D. Bandrauk, Phys. Rev. A 42 ( 1990) 2806; J. Chem. Phys. 96 (1992) 903. [ 15 ] A.D. Bandrauk and M.S. Child, Mol. Phys. 19 ( 1970) 95. [ 161 H. Levebvre-Brion and R.W. Field, Perturbations in spectra ofdiatomic molecules (Academic Press, New York, 1986). [ 171 A.D. Bandrauk and H. Shen, Chem. Phys. Letters 176 (1991) 428. [ 1S] A.D. Bandrauk, E. Constant and J.M. Gauthier, J. Phys. II (Paris) I(l991) 1033. [ 191 E.E. Aubanel and A.D. Bandrauk, in preparation. [20] H.W. Lee and T.F. George, Phys. Rev. A 35 ( 1987) 4977; J. Phys. Chem. 83 (1979) 928.