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Chapter 15

Laser-Phase Control of Dissociative Ionization of Molecules: Exact Non-BornOppenheimer Simulations for H +

2

1

A. D. Bandrauk , J. Levesque, and S. Chelkowski

1

Laboratoire de Chimie Théorique, Facultédes Sciences, Université de Sherbrooke, Québec J1K 2R1, Canada Visiting professor: Department of Physics, University of California, Santa Barbara, C A 93106

Exact non-Born-Oppenheimer numerical solutions of the time-dependent Schroedinger equation for a1DH molecule in an intense two-color (ω + ω) laser field have been obtained in order to clarify and identify the impor -tant mechanisms for pulse control of electron-nuclear dy -namics. This benchmark simulation allows for a study of unexpected asymmetries in electron ionization and pro -ton dissociation. A quantum regime is identified where electron quasistatic tunnelling is shown to be the domi -nant contribution to asymmetric and non-classical ioniza -tion in competition with proton dissociation. It is fur -ther shown that the quasistatic model of electron tunneling ionization and above barrier proton dissociation is a very useful concept in understanding intense field dissociative ionization and the concomitant counterintuitive (i.e. non -classical) behavior of electrons and nuclei in intense short laser pulses. +

2

Laser control of nuclear motion in molecules is a growing area of re search due to the ever improving laser technology making available laser pulses with variable and controlable amplitude, intensity and phase (jf). Thus various laser coherent control schemes have been proposed to control

© 2002 American Chemical Society

221


222 photochemical products in chemical reactions by coherent superpositions of electronic or (and) nuclear states (2). We have focused on using the simple superposition of a field of frequency UJ and its second harmonic 2a; (i.e. UJ + 2uj) coherent superposition of laser fields. Such a superposition creates a periodic but nonsymmetric electromagnetic field (Fig. 1(a)). Pre vious control scenarios, such as UJ + 3UJ (2) or 2UJ -h 4a; (3) always create symmetric fields which allow for laser control of molecular processes by interference of symmetry conserving quantum pathways. The UJ + 2UJ su perposition on the other hand leads to interference between nonsymmetry conserving quantum pathways. This leads to control of angular distribu tions of atomic ionization processes (4-6), of molecular photodissociation (5), the directional control of photocurrents in quantum wells (?) and in semiconductors (8). In the molecular case, we have shown previously that by using UJ -f 3a; or UJ 4- 2a; coherent superposition one can control ionization (9) and also enhance high order harmonic generation (10) in the simplest one-electron molecules H j , Hg" and recently the two-electron molecule H2 (11). These results were obtained from exact 3-D or 1-D numerical solutions of the time-dependent Schroedinger equation, T D S E for static nuclei (i.e. in the Born-Oppenheimer approximation). Recently we have performed the first non-Born-Oppenheimer simulation of dissociative ionization for a 1-D model of (13-14) and have applied this numerical methodology for the T D S E to propose a new method of imaging nuclear wave functions (13-14) and to study laser phase directional control of dissociative ionization of (15-16). Previous experimental attempts to control molecular dissociative ion ization using a UJ + 2a; coherent superposition scheme were reported for H2 and H D molecules (17-18). Strong asymmetries or anisotropics of positively charged nuclear fragments and ionized electrons were found. Unexpectedly, both positively charged nuclear fragments and negatively charged electrons were found to be preferentially emitted in the same direction (i.e. in a "counterintuitive", non-classical direction). Thus for the case of the most asymmetric combined electric field (see inset in F i g . 1(a), phase 0 = 0 or 7r), one would expect the positively charged nuclei to be preferentially ejected in the direction of maximum positive electric field, following pre cepts of classical mechanics, whereas negatively charged electrons would be accelerated in the opposite direction (i.e. downfield), contrary to what is observed experimentally. (16-18). In the experimental interpretation of this unusual asymmetry, it was assumed that electrons behave " normally" and the anomalous angular distribution was attributed to complex nuclear effects (i.e. to "abnormal" protons). We have succeeded in explaining these unusual experimental results in two ways (15-16). First by solving the complete dynamic, non-Born-Oppenheimer T D S E for a 1-D H j , we +

223

0.020

< Laser Control and Manipulation of Molecules Downloaded from pubs.acs.org by UNIV OF CALIFORNIA SAN DIEGO on 03/19/17. For personal use only.

h-

O UU

0.015

a. CO

>CD cc

0.010 4

LU LU O H LLi

O

a: h-

o LU _J LU Q LU N

O

2

4 E

6 e|

/E

8 p h o t o n

10 12 14 16 18 20 22 24 E

p h o

,on

=

1

- 65eV 1

Figure 1: Electron kinetic energy, ATI, spectrum for i / J in a laser field with frequencies u> (X = 1064nm) and 2u (X = 532nm), relative amplitude f=0.5 and two different phases: a)(f> = 0 , b)(j) = f in equation (1). Energies are in photon numbers.


224 can obtaim the exact kinetic energy spectra of the electrons, called A T I (Above Threshold Ionization (19)) spectra and proton dissociation spectra, called A T D (Above Treshold Dissociation (20)) spectra. We have thus ob tained the same kind of electron-proton correlated asymmetries as seen in experiment (i.e. we obtain from the numerical T D S E solutions of the ex act electron-proton time-dependent wave-functions a preferential emission of electrons and protons in the same direction). Thus for a coherent laser field superposition described by E(t) = Eo(t) • (cos(wt) + /cos(2u;* + {z,R,t)

(5)

for simultaneous electron^) and proton (R) motion. The numerical solu tion of the 1-D T D S E , equation (5), gives the exact non-Born-Oppenheimer electron-proton wavefunction ip(z,R,t). Projection onto exact solutions of a free electron in a laser field, called Volkov states (12), allows us to com pute the complete electron A T I spectra (see Fig. 1). The asymmetric proton A T D spectra are calculated by integrating out the electron coordinate z af ter projecting on the atomic Is orbitals localized at z = thus giving two R-dependent functions +OO

o il> (z±-)fl>(z,R,t)dz u

/

.

(6)

The Fourier transform of the latter gives the total proton kinetic energy spectra, which at low energies (Fig. 2) corresponds to the A T D spectra

227 whereas at higher energies one obtains C E (Coulomb Explosion) spectra due to C R E I (Charge Resonance Enhanced Ionization) (11,26).

a) ATI spectra As discussed in the introduction, perusal of the A T I spectra during the dissociative ionizaton of in a two-color (1064nm 4- 532nm) laser pulse Laser Control and Manipulation of Molecules Downloaded from pubs.acs.org by UNIV OF CALIFORNIA SAN DIEGO on 03/19/17. For personal use only.

1 3

2

with IQ = = 4.4 x 1 0 W / c m and relative amplitude / = 0.5 (equation 3) shows surprising forward at (j> = 0 and backward at (f> = f anisotropics or asymmetries (Fig. 1). The intensity I = 4.4 x 1 0 W / c m was chosen so that the peak maximum intensity of the two overlapping color fields at phase (p = 0 becomes I = I ( l + f) = 1 0 W / c m . We provide next a simple explanation of these anomalies using the quasistatic atomic tunnelling model illustrated in F i g . 3(a). A t first the electron ionizes with a tunneling rate r ( s ) at time to that depends on the instantaneous field E(t) (5,21). + 1 3

2

0

2

1 4

2

0

_ 1

2 T { t o )

"Wo)

=a

GXP

a§ '

3E(t )\ 0

I

H

where the J's in (7) are ionization potentials. In the second step the electron moves in the laser field (without the Coulomb attraction V(z)) as a clas sical particle starting at rest at the tunneling time £0 and position z = ZQ (Fig. 3(a)) which is determined by the initial condition V(z to) — —I , in the 1-D potential 1

V(z,t) = -

1 2

(z +

. +zE(t)

p

.

1)2

(8)

We next solve the Newton equations of motion in the two-color laser field only, E q . (1), with initial conditions appropriate for tunneling v(t ) = 0 o

,

z = z(t ) 0

= z

0

.

(9)

We thus get the following solutions for the electron position z(t) z(t) = z + z 0

osc

+ v (t - t ) a

0

,

(10)

E f f z sc = ~ 4 [ cos(ut) + 7 cos(2otf + (/))- cos(ut ) - - cos(2atfo + )] to 4 4 0

0

. (11)

The instantaneous electron velocity v(t) is readily found to be (by integrat ing once the equations of motion), v(t) = v (t ) - — [ sin(u;£) 4- { sin(2ut + )] UJ 2 d

Q

,

(12)

228

0.0 l

m

a

x

=1x10

1

4

Wcm"

j

_

/

2

l

m

a

x

=1x10

1

5

Wcm-

/

2

1 j

1

en

Energy (u,


d

1

-1.0

X

t

* °-

\ / v

/ - 8 - 6 - 4

-2

0

2

4

-

6

-4

/

-2

0

2

4

z | (u.a.) e

Figure 3(a): Quasistatic potential (la.u.=27.2eV) for a ID H atom at intensity I — 1 0 W e m where underbarrier ionization occurs at to by tunneling and at intensity I = 10 Wcm~ where overbarrier ionization occurs at to. 1 4

- 2

15

b)

0

2

9

0.3

1 0

• 2

4

. 6

8

10

. 12

14

» 16

18

20

Figure 3(b): Forward (P ) and backward (P-) ionization probabilities and relative probability (P /(P+ + P-)) called asymmetry ratio, for the hydrogen atom in the same field as in Fig. 1(a). Counterintuitive corresponds to underbarrier tunneling whereas intuitive is the overbarrier classical regime. I x corresponds to the maximum field intensity reached by the two-color field (for 0 = 0 and f = 0.5, I x = 2.25 Jo +

+

}

ma

ma

229 v (to) = — [sin(wio) + J sin(2urt + )] 0


d

•

(13)

^d(^o) is called the "drift velocity" and is the result of applying the initial condition, equation (9), after the tunneling time to, (5,21). After the turnoff of the pulse, the oscillatory terms in equation (12) average out to zero so that the final electron velocity, as measured by experiment, coincides with the drift velocity Vd(to). Similarly, neglecting oscillatory terms in equation (10) for long times, one obtains the measured electron position z(t) = v (to)(t-t ) d

+z

0

0

.

(14)

Thus the two results, equations (13) and (14) determine in this simple oneelectron model the measurable forward/backward electron asymmetry. In particular, equation (13) predicts a clear asymmetry for (j) — | , since in this case the last term in (13) becomes an even cosine function, so that v (to) 7^ ^d(-^o)- The symmetry about the field maximum (minimum) (Fig. 1(b)) for cj) = - is thus broken. This can be attributed to the fact that the net field, equation (1), is not symmetric about each field maxima (minima) as seen in the inset of Fig. 1(b). As an example, around each field maxima, there is a sharp rise of the field to the left, and a slow descent into a kink. Since ionization is expected to occur mainly at the field maxima (minima) according to the tunneling expression (7), the asymmetry in the ionization can thus be attributed to the local asymmetry of the electric field around its maxima (minima), without assistance from the permanent Coulomb fields of the nuclei (protons). The case of the phase 4 x 1 0 W / c m , the overbarrier ionization regime. In the first, weak field case, the tunneling electron has zero velocity as it leaves the atom, v (to) = 0 so that it will be under maximum influence of the atom Coulomb field and the electron field E which will decrease with d

1 4

1 4

d

2

2


230 time and eventually bring the electron back after half a cycle (Fig. 3(a)). In the strong field overbarrier limit, the electron has sufficient energy to completely overcome the concerted refocusing effect of the Coulomb and electric fields, equation (8), F i g . 3(a). The standard tunneling model in fact predicts a decrease of phase sensitivity and response of tunneling ion ization with increasing laser intensity (17). This is corroborated by the calculated A T I asymmetry for the H-atom illustrated in F i g . 3(b). Thus for I < 4 x 1 0 W / c m , the asymmetry as measured by the rates of the integrated forward, P+, and backward P- probabilities, is large (i.e. coun terintuitive) around I ~ 1 0 W / c m , and becomes intuitive (classical) for I > 4 x 1 0 W / c m . The separation between the counterintuitive (under barrier) tunneling regime and the intuitive (overbarrier) classical regime occurs at the critical intensity Jo where the ionization process occurs when the initial atomic Is state becomes degenerate with the maximum of the (i ) total potential, equation(8). This gives the result that J = and since the ionization potential I = 0.67a.u. for a 1-D H atom, one obtains readily I = 0.0125a.u. = 4 x 1 0 W / c m , in agreement with Fig. 3(b). 1 4

2

1 4

1 4

2

2

4

c

p

1 4

2

c

b)ATD spectra Finally, we comment on the proton kinetic energy, or A T D spectra illustrated in Fig. 2. The asymmetry in the 0 = 0 case is as expected mainly in the forward direction, showing that protons behave classically contrary to previous interpretations (17-18). The intuitive asymmetry is nevertheless less for the protons (Fig. 4(a)) than the counterintuitive electronic result, Fig. 1(a). This is evidently related to the greater quantum nature of the electron. However for the case 0 = f , F i g . 4(b), one finds an unexpected asymmetry for the proton dissociation. Since from the field structure (inset in Fig. 1(b)) one has equal maxima-minima field strenghts, one would have expected a symmetric proton dissociation. It is to be observed that in both 0 = 0 and f cases, the electron and proton asymmetries are in the same direction, with the electron asymmetry about twice larger than the proton asymmetry. Two factors must be operative: a)Coulomb interaction between electrons and protons, or (and) b) the local asymmetry of the net field amplitude around each maximum-minimum of these fields as discussed above for electrons. In order to understand more the asymmetry in the calculated A T D spectra in presence of ionization, we illustrate in F i g . 4, the calculated A T D spectra for pure A T D , i.e. a two surface simulation, with the E coupled radiatively to the E state, without ionization. Comparing fig ures 4 and 2, one observes that the asymmetries calculated with ionization (Fig. 2) correspond better with the v = 5 results in the two surface cal culations (Fig. 4). This last figure shows that strong asymmetries occur 2

P

2

U

231

/forward

v


7 6 5

4 3 2 1

0

8

7 6^ 5

Y

J

v=5 (H)

(a)

, b a c k w a r d

P =0.72 P=0.26 +

/ b a c k w a r d

iw\

v

v=5

d>=7t/2

w

3 2 1 -\ 0 y forward 0.28 b a c k w a r d 0.24 / P . v=1 =0.17$=Q P=0.77(c) 0.20 forward 0.16 0.12 P =0.0038 P =0.0101 -r 0.08 0.04 0.00 0.28 forward V=1 cf)=7T/2 (d) 0.24 0.20 -I P + =0.014 P =0.0099 0.16 0.12 b a c k w a r d 0.08 0.04 0.00 i 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Proton energy in H+p channel (eV) :

Figure 4 Proton kinetic energy (ATD) spectra in the dissociative H+p channel calculated using the two-surface model, the same laser parameters as in Figs. 2,3 (X = 1064 + 532nm ) were used but with different wave-packet initializations: (a),(b) initialization from v=5; (c),(d) initialization from v=l. >

232 for the higher vibrational levels. This can be readily understood from the two-surface dressed staste representation of the u + 2u photodissociation processes illustrated in F i g . 5. Thus the v = 1 state can dissociate to the S potential after 3 photon absorption to the \u, —3,0) repulsive state and interfere with the E potential via the corresponding \g, - 1 , - 1 ) dressed state (we use the notation |t/>,ni,n ) for the electronic tj) state with n\ photons uj\ (1064nm) and n photons U2 (532nm)). The v = 5 state disso ciation interferes via two isoenergy pathways with final states \g, —2,0) and |u,0, —1) corresponding to two 1064nm photons (rt\ — 2,n2 = 0) and one 532nm photon (m = 0 , n = 1) Furthermore since the latter one photon (532nm) dissociation of the v = 5 state is about 70 times larger than its two photon (1064nm) dissociation, the preponderance of the higher v levels to A T D is thus responsible for the enhanced backward-forward asymmetry at 0 = 0 (Fig. 2(a)). The same reasoning applies at 0 = 7r/2 since in Fig. 2(a), P / P _ ~ 0.2 for v = 5 and 1.4 for v = 1, F i g . 4, without ionization. Summarizing the comparison of Figs. 2 and 4 at 0 = 0 we conclude that the v — 5 A T D with its dominant forward asymmetry follows the classical intuitive model, whereas the v = 1 level A T D is counterintuitive, with the backward asymmetry dominant. Both v = 5 and v = 1 A T D asymmetries for H f are reduced by a factor of two in the presence of ionization which was simulated assuming a Franck-Condon distribution from an H2, v = 0 initial molecule. Thus the dissociative ionization result, F i g . 2, represents the average over vibrational levels of excited from H2. This interpretation over an average vibrational level distribution of HJ also explains the results for the 0 = 7r/2 case. Thus in the dissociative-ionization case, F i g . 2(b), P + / P _ ~ 0.6 whereas it is 0.2 for v = 5 and 1 for v = 1 in the purely dissociative (no ionization) two-state calculation, F i g . 4. For the 0 = 0 case, Posthumus et al. (27), have proposed previously that at the high intensities used here, the electron is always localized at the down-field nucleus, i.e. in the opposite direction of the field. For the 0 = 0 case, since the negative field is on for a longer time than the large positive field (see F i g 1(a)) the electron will be therefore localized in the opposite direction of the negative field most of the time thus forcing the proton to be ejected in the negative field direction, i.e. in the counterintuitive direction, in agreement with the purely dissociative v = 1 level (Fig. 4(c)), but not in agreement with the v = 5 level (Fig. 4(a)) nor the exact dissociative ionization result, Fig. 2(a). One can surmise that the v = 1 case is closer to a quasistatic picture since a larger number of photons are involved than the v = 5 level dissociation, so that the v = 1 behaviour follows the Posthumus quasistatic model whereas the v = 5 level falls more within the multiphoton regime (3). The 0 = - case is more problematic since the quasistatic model predicts 2

U

2

5

2


2

2

+


233

g,o,o> 0)

E 3 C c o o

u,-3,0>

sz

a.

o Q. LU

lg,-l,-l>

-6 6

8

10

R (a.u.) Figure 5: Dressed states for a;(1064nm) -f 2u;(532nm) dissociation of v = 1 and v = 5 vibrational levels, v = 1 dissociates to two isoenergy states: \g, — 1, —1) and |u, —3,0) whereas v = 5 dissociates to the two isoenergy states |t£,0, — 1) and |g, — 2,0).


234 a complete symmetric dissociation. The v = 1 level results (Fig.4(d)) agree with the quasistatic model but not the v — 5 case, (Fig. 4(b)), which shows a very large backward (5:1) asymmetry. The dissociative-ionization comparative result. F i g . 2(b), is a backward/forward asymmetry of 1.6, reflecting the average over low and high vibrational levels. One tempting explanation is the local asymmetry of the net field, illustrated in the inset of Fig. 1(b). The sharp rise from negative to positive amplitude of the electric field to the left of each maximum will inhibit an adiabatic following of the proton motion from the positive to negative amplitude to the right of field maxima. Thus protons will remain always longer under the influence of the negative field amplitude, thus being ejected predominantly backwards in this symmetric field, as obtained in the simulation, F i g . 2(b) and 4(b). In conclusion, we have shown that at high intensities (J > 1 x 1 0 W / c m ) where radiative processes are no longer perturbative the quasistatic model offers a simple explanation for the calculated asymetries in both electron and proton emission in a; + 2a; fields. 1 4

2

Aknowledgments We thank N S E R C (Natural Sciences and Engineering Research of Canada) and C I P I (Canadian Institute for Photonic Innovations) for financial sup port of this research on laser control and manipulation of molecules.

References

1. T . Brabec,F. Krausz. Rev. Mod. Phys., 72, 545, 2000. 2. M. Shapiro, P. Brumer. Acc. Chem. Res., 22, 407, 1989. 3. E . Aubanel, A . D . Bandrauk. Chem. Phys. , 194, 159, 1995.

Chem.

Phys. Lett, 229, 169, 1994;

4. N . B . Baranova, B . Y . Zeldovitch. J. Opt. Soc. Am. B, 8, 27, 1991. 5. D.W.Schumacher, P . H . Bucksbaum. Phys. Rev. A, 54, 4271, 1996. 6. K.J. Schafer, K.C. Kulander. Phys. Rev. A, 45, 8026, 1992. 7. E . Dupont, P . B . Corkum, H . C . Liu, M. Buchanan, Z . R . Wasilewski. Phys. Rev. Lett, 74, 3596, 1995. 8. R . Atanasov, J . E . Sipe, H . Van Driel Phys. Rev. 1996.

Lett, 76, 1703,

235 9. T . Zuo, A . D . Bandrauk. Phys. Rev. A, 54, 3254, 1996. 10. A . D . Bandrauk, S. Chelkowski, H . Y u , E. Constant. Phys. Rev. A, 56, R2357, 1997.


11. A . D . Bandrauk, H . Y u . Int. J. Mass Spectrom., 192, 379, 1999. 12. S. Chelkowski, C . Foisy, A . D . Bandrauk. 1998.

Phys. Rev. A, 57, 1176,

13. S. Chelkowski, P . B . Corkum, A . D . Bandrauk Phys. Rev. Lett., 82, 3416, 1999. 14. A . D . Bandrauk, S. Chelkowski. Chem. Phys. Lett., 2001. 15. A . D . Bandrauk, S. Chelkowski. Phys. Rev. Lett., 84, 3562, 2000. 16. S. Chelkowski, M. Zamojski, A . D . Bandrauk. 023409, 2001. 17. B.Sheehy, B . Walker, L.F. Di Mauro. 1995.

Phys.

Rev. A, 63,

Phys. Rev. Lett., 74, 4799,

18. M . R . Thompson, J. Posthumus, L.J. Frasinski J. Phys. B, 30, 5755, 1997. 19. Atoms in Intense Laser Fields. New-York, 1992. 20. Molecules in Laser Fields. New-York, 1994.

M . Gavrila, ed., Academic Press,

A . D . Bandrauk, ed., M. Dekker Publ.,

21. P . B . Corkum. Phys. Rev. Lett., 71, 1994, 1993. 22. M. Ivanov, P . B . Corkum, T.Zuo, A . D . Bandrauk. Phys. Rev. Lett., 74, 2933, 1995. 23. See contribution of G . Gerber et al to this volume. 24. See contribution of R . Lewis et al to this volume. 25. A . D . Bandrauk in. The Physics of Electronic and Atomic Collisions. Y. Itikawa et al, ed., AIP Conf. Proc., vol. 500, A I P , New-York, 2000, p.1020. 26. P . B . Corkum, P. Dietrich. Comments At. Mol. Phys., 28, 357, 1993. 27. J . H . Posthumus, M . R . Thompson, A.J. Giles, K. Codling. A54, 955, 1996.

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