- Email: [email protected]
Femtosecond selective control of wave packet population
Volume 17 I, number 1,2
27 July 1990
CHEMICAL PHYSICS LETTERS
Femtosecond selective control of wave packet population J.J. Gerdy ‘, M. Dantus, R.M. Bowman and A.H. Zewail ArthurAmos NoyesLaboratoryofChemical Physics‘, CaliforniaInstituteof Technology, Pasadena,CA 911IS, USA Received 13 June 1990
Femtosecond selective control of wave packet population is reported for molecular iodine. It is shown that both population and phase control of the packet motion can be observed by a 2-D pulse sequence of variable delay times and phase angles. Extension to other type of control experiments is also discussed.
Femtosecond dynamics of nuclear motion in isolated chemical reactions reveal the nature of the wave packet as the system evolves and passes through the transition states [ 1] _ An interesting question is the following: On this time scale, can one selectively control the population and phase of the packet? In this Letter, we report our observations of population control in the wave packet motion of isolated iodine molecules in the bound B state. The experiments involve the introduction of a “second dimension” to the traditional two pulse (F’TS) scheme [ 1,2 1. A three pulse sequence is used here; the first two pump pulses prepare the B state and the third pulse probes the resulting motion. It is shown that this simple sequence of pulses can build up wave packet population on the B state surface with well deJined and controllable phase difference. Delaying the second pump pulse by r, from the first introduces a phase shift or angle, 0, completely determined by ~~ and the frequencies of the wave packet wP One, therefore, has a control over the preparation process. On the other hand, controlling the probing wavelength allows us to observe the inphase and out-of-phase motion of the packet. These observations demonstrate the time scale for coherence of the motion and the selective control of population (and phase) on the femtosecond time scale. The experimental methodology is similar to other
femtosecond transition-state spectroscopy (FTS) experiments [ 11, except an additional delay line is introduced to control the time delay (q ) between the first two pulses. The sequence used is 2 ,-7,-X, ~a-& (2; ), where 1, and A’,refer to the pump pulses wavelengths and A,($ ) refers to the probing pulse wavelengths (fig. 1). Fig. 2 shows the 2-D transients obtained as a function of the delay time between A2and A, pulses and at fixed angles, 8= w,7,, where w, is a frequency that can be made to match any of the packet frequencies, oij. For 12, the oscillatory motion observed at 0=0 simply reproduces the two pulses (A,, A,) study of the motion and displays the vibration period (300 fs) observed when RL= 620 nm [ 2 1. This temporal motion in the B state of I2 (I% 0) has been fully analyzed to derive the potential using quantum [ 31 and classical [4] inversion methods. If 8 is chosen to be TCfor the frequency w1 (of the vibration period), then the phase of the first packet
’ NSF predoetoral fellow. ’ This work was supported by the NSF and the AFOSR, contribution No. 8 158.
Fig. 1. The pulsesequence used in these control experiments. The two preparation pulses are ,l, and A; and the probing pulse is n&l; ).
Elsevier Science Publishers B.V. (North-Holland)
0
Time (%)
1
Volume 171, number 1,2
CHEMICAL PHYSICS LETI-ERS
27 July 1990
side function is an approximation. The actual smooth rise comes from convoluting this signal, S, with the actual pulse shape. One notices that the expression illustrates the effect of coherence in the V, potential and the effect of the second pulse (second term when 112Cl).In the limit of q=O, one recovers the wave packet dynamics (CC, cos wtir) on V,, monitored by the standard FTS I ,A2(;lr ) experiments. In the above expression, we have averaged over the phases of the pulses, and their effect will be dealt with later [ 51. The above equation can be rewritten to highlight the effect of B on the observed transients:
Time Fig. 2. Experimental 2-D transients obtained as a function of e=o,t,, where 2x/q=300 fs, and as a function of probing time= 5, t tl. For these transients I, and A’,are 620 nm and A2is 310nm.
can be made to cancel the second packet, and the oscillatory motion is not observed, as shown in fig. 2. On the other hand, if 0 is made to be 21c,then the population is doubled and the oscillatory motion is recovered again. Fig. 2 shows the results for 8= 0, rr, 2x, 3rc, and 4x. Theoretically, the observations can be modelled by using the density matrix formalism. In the weak field, however, one can explicitly write expressions for wavefunctions at different times and derive a final result for the dependence of the laser-induced fluorescence (at 1,) on 51and r,. In this case, where we consider three states, V,, V, and V,, involved in the A,A{& experiment, we obtain [5]: S(r,, T~)=CC C${H(r) exp(io,r) 01ij +H(r-~,)q~
exp[iw,(r-r,)]}.
The sum over a! is to indicate all the states reached by A2probing for excitation from V, to the potential Vz. The sum over 4 is over states reached by A, on the potential VI from the ground state potential VW H is the Heaviside function and q is the ratio of the E field of pulse A’,to h 1.The constants {(J are given by the pulses’ shape and the transition moments involved. The discontinuous rise given by the Heavi2
where r=r1+r2, and 6 is given explicitely by 7,. In addition to the effect of the second pulse at A;, one sees that the probing of the wave packet on V, surface can be made to yield the same period oU, but with different phase shifts St, depending on the detection, a point that we will discuss below. The damping of the packet is not written explicitly, and is not relevant to this case [ 5 1. Fig, 3 shows the theoretical simulations compared with experimental observations for 8=x: and 4x. The basic features of the observations are reproduced theoretically using the scheme outlined above. Fig. 4 shows the case when the (AJ2) two-pulse sequence is used, but R, detection is changed to observe the packet at either the inner or outer turning points of the potential. As shown, the phase is 180” different, but the period is still the same, confirming the above results. Again, from the above expression if S$ = - ICand 0, then the two turning points can be probed with different phases, but the period 2rc/oV remains the same. This is entirely consistent with a quantum description [ 61 and with Letkhov’s description of the dynamics in such systems [ 71. The above observations brings to focus several important points. First, one can extend the methodology for selective manipulation of certain w;s using larger number of pulses in the sequence, as done in NMR [ 8 ] and optical analoges [ 91, Second, one may use this approach to “burn holes” at a given frequency oii and simplify the labelling of the spectra and packet dynamics. Third, the “washing out” of
CHEMICAL PHYSICSLETTERS
Volume I7 1,number 1.2
27 July 1990
Experimental
U
JWJ
6al
5wl
1zmJ
IS00
1800
2100
24ca
TimeDelay (fs) Fig. 4. Femtosecondtemporal spectra obtained for AI= 620 nm (no& pulse), The probe, A,, is a two photon at 620 nm, The inphase and out-of-phase transients were obtained by simply changing the LIF detection wavelengths to probe the different turning points, as discussed in the text.
Time (fs)
fact that in “long pulse” experiments (which can be thought of as composite of ultrashort pulses), a steady state population is formed and the actual coherence
tern are developed and become stationary. On the other hand, with ultrashort pulses, the maion is ob served and the vibrational and rotational dynamics are separable because of the different time scales. Finally, one must be cautious about differences between population and phase control in weak versus strong field limits. Mukamel and Yan [ 1l] has recently discussed differences between population buildup and changes in vibrational amplitude induced by intense pulse trains. Weiner and Nelson and co-workers [ 121 have reported on an enhancement of the amplitude of an optical phonon in a crystal using pulse train excitation and stimulated Raman techniques. Here, we use weak fields and we do not report any amplitude change of the vibration, but rather control of the wave packet timing and
of the motion (see figs. 2 and 3) is not observable. On these long time scales, the eigenstates of the sys-
population. There are many possibilities
Fig. 3. Comparison of theory and experiment. The calculations are shown for three vibration levels in the Ia B state. Time delays between the pump pulses are 150 fs (heavy) and 600 fs (light). The equation for S( r, 8) (see text) has been convoluted with the pulse profile typically 50-60 fs to obtain these calculated
transients.
coherence and buildup of population emphasizes the connection between dynamical and cw spectral approaches discussed by Heller, Kinsey and co-workers [ 101. The “washing out” of the packet oscillatory motion in the controlled 19experiments points to the
for extending to these
3
Volume 171, number 1,2
CHEMICAL PHYSICS LETTERS
experiments on isolated molecular systems the earlier phase-coherent sequence technique, developed for nanosecond [ 9 1, picosecond [ 13 1, and now femtosecond [ 141 experiments, the locking technique [ 151, and the ultrafast pulse control of evolution [ 161. The schemes of Tannor and Rice [ 171, Brumer and Shapiro [ 181, Rabitz [ 19 1, Manz and coworkers [20], and others may also be extended to this domain of experiments. In conclusion, the reported experiments demonstrate the simple concept of selective population control in wave packet molecular dynamics. The 2D feature introduced here may be extended to cover other domains of multiple pulse techniques, hopefully to a better control of reactive systems. Experiments in this direction are underway. We are grateful to Professor R.B. Bernstein many stimulating discussions.
for
References [ 1] A.H. Zewail, Science 242 (1988) 1645; A.H. Zewail and R.B. Bernstein, Chem. Eng. News 66 (1988) 24; L.R. Khundkar and A.H. Zewail, Ann. Rev. Phys. Chem. 41 (1990) 15; M. Gruebele and A.H. Zewail, Phys. Today 43 ( 1990) 24. [2] R.M. Bowman, M. Dantus and A.H. Zewail, Chem. Phys. Letters 161 (1989) 297; M. Dantus, R.M. Bowman and A.H. Zewail. Nature 343 (1990) 737. [ 31 M. Gruebele, G. Roberts, M. Dantus, R.M. Bowman and A.H. Zewail, Chem. Phys. Letters 166 (L990) 459. [4] R.B. Bernstein and A.H. Zewail, Chem. Phys. Letters 170 (1990) 321. [ 51 J. Gerdy and A.H. Zewail, to be published. [6] P.M. Fe1kerandA.H. Zewail, Phys. Rev. Letters 53 ( 1984) 501. [ 71 VS. Letokhov and V.V. Tyakht, Israel J. Chem. 30 ( 1990) 189. [ 81 R.R. Ernst, G. Bodenhausen and A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions (Oxford Univ. Press, London, 1987 ).
27 July 1990
[9] W.S. Warren and A.H. Zewail, J. Chem. Phys. 75 (1981) 5956; 78 ( 1983) 2219; 78 (1983) 2298. [ IO] D. Imre, J.L. Kinsey, A. Sinha and J. Krcnos, J. Phys. Chem. 88 (1984) 3956: E.J. Heller, Accounts Chem. Res. 14 ( 1981) 368. [ 111 S. Mukamel and Y.J. Yan, Science, submitted for publication. [ 12lA.M. Weiner, D.E. Leaird, G.P. Wiederrecht and K.A. Nelson, Science 247 ( 1990) 13 17. [ 131W.S. Warren, Science 242 (1988) 876; W.S. Warren and MS. Silver, Advan. Magn. Res. 12 ( 1988) 247; A.M. Weiner, J.P. Heritage and E.M. Kirchner, J. Opt. Sot. Am. B 5 (1988) 1563. [ 141 N.F. Scherer, A.J. Ruggiero, M. Du and G.R. Fleming, J. Chem. Phys., in press. [ 151 E.T. Slevaand A.H. &wail, Chem. Phys. Letters 110 ( 1984)
582;
E.T.Sleva, I.M. XavierJr. andA.H. Zewail, J. Opt.Am. 83 (1986) 483; E.T. Sleva, M. Glasbeek and A.H. Zewail, J. Phys. Chem. 90 (1986) 1232. [ 16] P.M. Felker and A.H. &wail, Advan. Chem. Phys. 70 (1988) 265; N. Bloembergen and A.H. Zewail, .I. Phys. Chem. 88 ( 1984) 5459; D.J. Tannor and S.A. Rice, Advan. Chem. Phys. 70 ( 1988) 441; S. Mukamel and K. Shan, J. Phys. Chem. 89 (1985) 2447. 117 D.J. Tannor and S.A. Rice, J. Chem. Phys. 83 ( 1985) 5013; D.J. Tannor, R. Kosloff and S.A. Rice, J. Chem. Phys. 85 (1986) 5805; R. Kodoff, S.A. Rice, P. Gaspard, S. Tesigni and D.J. Tannor, Chem. Phys. 139 (1989) 201. [I8 P. Brumer and M. Shapiro, Chem. Phys. Letters 126 (1986) 541; P. Brumer and M. Shapiro, Accounts Chem. Res. 22 ( 1989) 407. [ 191 H. Rabitz, in: Atomic and molecular processes with short intense laser pulses, ed. A. Bandrauk (Plenum Press, New York, 1988). [20] W. Jakubetz, J. Manz and H.-J. Schreier, Chem. Phys. Letters 165 (1990) 100; B. Hartke, J. Manz and J. Mathis, Chem. Phys. Letters 139 ( 1989) 123.
27 July 1990
CHEMICAL PHYSICS LETTERS
Femtosecond selective control of wave packet population J.J. Gerdy ‘, M. Dantus, R.M. Bowman and A.H. Zewail ArthurAmos NoyesLaboratoryofChemical Physics‘, CaliforniaInstituteof Technology, Pasadena,CA 911IS, USA Received 13 June 1990
Femtosecond selective control of wave packet population is reported for molecular iodine. It is shown that both population and phase control of the packet motion can be observed by a 2-D pulse sequence of variable delay times and phase angles. Extension to other type of control experiments is also discussed.
Femtosecond dynamics of nuclear motion in isolated chemical reactions reveal the nature of the wave packet as the system evolves and passes through the transition states [ 1] _ An interesting question is the following: On this time scale, can one selectively control the population and phase of the packet? In this Letter, we report our observations of population control in the wave packet motion of isolated iodine molecules in the bound B state. The experiments involve the introduction of a “second dimension” to the traditional two pulse (F’TS) scheme [ 1,2 1. A three pulse sequence is used here; the first two pump pulses prepare the B state and the third pulse probes the resulting motion. It is shown that this simple sequence of pulses can build up wave packet population on the B state surface with well deJined and controllable phase difference. Delaying the second pump pulse by r, from the first introduces a phase shift or angle, 0, completely determined by ~~ and the frequencies of the wave packet wP One, therefore, has a control over the preparation process. On the other hand, controlling the probing wavelength allows us to observe the inphase and out-of-phase motion of the packet. These observations demonstrate the time scale for coherence of the motion and the selective control of population (and phase) on the femtosecond time scale. The experimental methodology is similar to other
femtosecond transition-state spectroscopy (FTS) experiments [ 11, except an additional delay line is introduced to control the time delay (q ) between the first two pulses. The sequence used is 2 ,-7,-X, ~a-& (2; ), where 1, and A’,refer to the pump pulses wavelengths and A,($ ) refers to the probing pulse wavelengths (fig. 1). Fig. 2 shows the 2-D transients obtained as a function of the delay time between A2and A, pulses and at fixed angles, 8= w,7,, where w, is a frequency that can be made to match any of the packet frequencies, oij. For 12, the oscillatory motion observed at 0=0 simply reproduces the two pulses (A,, A,) study of the motion and displays the vibration period (300 fs) observed when RL= 620 nm [ 2 1. This temporal motion in the B state of I2 (I% 0) has been fully analyzed to derive the potential using quantum [ 31 and classical [4] inversion methods. If 8 is chosen to be TCfor the frequency w1 (of the vibration period), then the phase of the first packet
’ NSF predoetoral fellow. ’ This work was supported by the NSF and the AFOSR, contribution No. 8 158.
Fig. 1. The pulsesequence used in these control experiments. The two preparation pulses are ,l, and A; and the probing pulse is n&l; ).
Elsevier Science Publishers B.V. (North-Holland)
0
Time (%)
1
Volume 171, number 1,2
CHEMICAL PHYSICS LETI-ERS
27 July 1990
side function is an approximation. The actual smooth rise comes from convoluting this signal, S, with the actual pulse shape. One notices that the expression illustrates the effect of coherence in the V, potential and the effect of the second pulse (second term when 112Cl).In the limit of q=O, one recovers the wave packet dynamics (CC, cos wtir) on V,, monitored by the standard FTS I ,A2(;lr ) experiments. In the above expression, we have averaged over the phases of the pulses, and their effect will be dealt with later [ 51. The above equation can be rewritten to highlight the effect of B on the observed transients:
Time Fig. 2. Experimental 2-D transients obtained as a function of e=o,t,, where 2x/q=300 fs, and as a function of probing time= 5, t tl. For these transients I, and A’,are 620 nm and A2is 310nm.
can be made to cancel the second packet, and the oscillatory motion is not observed, as shown in fig. 2. On the other hand, if 0 is made to be 21c,then the population is doubled and the oscillatory motion is recovered again. Fig. 2 shows the results for 8= 0, rr, 2x, 3rc, and 4x. Theoretically, the observations can be modelled by using the density matrix formalism. In the weak field, however, one can explicitly write expressions for wavefunctions at different times and derive a final result for the dependence of the laser-induced fluorescence (at 1,) on 51and r,. In this case, where we consider three states, V,, V, and V,, involved in the A,A{& experiment, we obtain [5]: S(r,, T~)=CC C${H(r) exp(io,r) 01ij +H(r-~,)q~
exp[iw,(r-r,)]}.
The sum over a! is to indicate all the states reached by A2probing for excitation from V, to the potential Vz. The sum over 4 is over states reached by A, on the potential VI from the ground state potential VW H is the Heaviside function and q is the ratio of the E field of pulse A’,to h 1.The constants {(J are given by the pulses’ shape and the transition moments involved. The discontinuous rise given by the Heavi2
where r=r1+r2, and 6 is given explicitely by 7,. In addition to the effect of the second pulse at A;, one sees that the probing of the wave packet on V, surface can be made to yield the same period oU, but with different phase shifts St, depending on the detection, a point that we will discuss below. The damping of the packet is not written explicitly, and is not relevant to this case [ 5 1. Fig, 3 shows the theoretical simulations compared with experimental observations for 8=x: and 4x. The basic features of the observations are reproduced theoretically using the scheme outlined above. Fig. 4 shows the case when the (AJ2) two-pulse sequence is used, but R, detection is changed to observe the packet at either the inner or outer turning points of the potential. As shown, the phase is 180” different, but the period is still the same, confirming the above results. Again, from the above expression if S$ = - ICand 0, then the two turning points can be probed with different phases, but the period 2rc/oV remains the same. This is entirely consistent with a quantum description [ 61 and with Letkhov’s description of the dynamics in such systems [ 71. The above observations brings to focus several important points. First, one can extend the methodology for selective manipulation of certain w;s using larger number of pulses in the sequence, as done in NMR [ 8 ] and optical analoges [ 91, Second, one may use this approach to “burn holes” at a given frequency oii and simplify the labelling of the spectra and packet dynamics. Third, the “washing out” of
CHEMICAL PHYSICSLETTERS
Volume I7 1,number 1.2
27 July 1990
Experimental
U
JWJ
6al
5wl
1zmJ
IS00
1800
2100
24ca
TimeDelay (fs) Fig. 4. Femtosecondtemporal spectra obtained for AI= 620 nm (no& pulse), The probe, A,, is a two photon at 620 nm, The inphase and out-of-phase transients were obtained by simply changing the LIF detection wavelengths to probe the different turning points, as discussed in the text.
Time (fs)
fact that in “long pulse” experiments (which can be thought of as composite of ultrashort pulses), a steady state population is formed and the actual coherence
tern are developed and become stationary. On the other hand, with ultrashort pulses, the maion is ob served and the vibrational and rotational dynamics are separable because of the different time scales. Finally, one must be cautious about differences between population and phase control in weak versus strong field limits. Mukamel and Yan [ 1l] has recently discussed differences between population buildup and changes in vibrational amplitude induced by intense pulse trains. Weiner and Nelson and co-workers [ 121 have reported on an enhancement of the amplitude of an optical phonon in a crystal using pulse train excitation and stimulated Raman techniques. Here, we use weak fields and we do not report any amplitude change of the vibration, but rather control of the wave packet timing and
of the motion (see figs. 2 and 3) is not observable. On these long time scales, the eigenstates of the sys-
population. There are many possibilities
Fig. 3. Comparison of theory and experiment. The calculations are shown for three vibration levels in the Ia B state. Time delays between the pump pulses are 150 fs (heavy) and 600 fs (light). The equation for S( r, 8) (see text) has been convoluted with the pulse profile typically 50-60 fs to obtain these calculated
transients.
coherence and buildup of population emphasizes the connection between dynamical and cw spectral approaches discussed by Heller, Kinsey and co-workers [ 101. The “washing out” of the packet oscillatory motion in the controlled 19experiments points to the
for extending to these
3
Volume 171, number 1,2
CHEMICAL PHYSICS LETTERS
experiments on isolated molecular systems the earlier phase-coherent sequence technique, developed for nanosecond [ 9 1, picosecond [ 13 1, and now femtosecond [ 141 experiments, the locking technique [ 151, and the ultrafast pulse control of evolution [ 161. The schemes of Tannor and Rice [ 171, Brumer and Shapiro [ 181, Rabitz [ 19 1, Manz and coworkers [20], and others may also be extended to this domain of experiments. In conclusion, the reported experiments demonstrate the simple concept of selective population control in wave packet molecular dynamics. The 2D feature introduced here may be extended to cover other domains of multiple pulse techniques, hopefully to a better control of reactive systems. Experiments in this direction are underway. We are grateful to Professor R.B. Bernstein many stimulating discussions.
for
References [ 1] A.H. Zewail, Science 242 (1988) 1645; A.H. Zewail and R.B. Bernstein, Chem. Eng. News 66 (1988) 24; L.R. Khundkar and A.H. Zewail, Ann. Rev. Phys. Chem. 41 (1990) 15; M. Gruebele and A.H. Zewail, Phys. Today 43 ( 1990) 24. [2] R.M. Bowman, M. Dantus and A.H. Zewail, Chem. Phys. Letters 161 (1989) 297; M. Dantus, R.M. Bowman and A.H. Zewail. Nature 343 (1990) 737. [ 31 M. Gruebele, G. Roberts, M. Dantus, R.M. Bowman and A.H. Zewail, Chem. Phys. Letters 166 (L990) 459. [4] R.B. Bernstein and A.H. Zewail, Chem. Phys. Letters 170 (1990) 321. [ 51 J. Gerdy and A.H. Zewail, to be published. [6] P.M. Fe1kerandA.H. Zewail, Phys. Rev. Letters 53 ( 1984) 501. [ 71 VS. Letokhov and V.V. Tyakht, Israel J. Chem. 30 ( 1990) 189. [ 81 R.R. Ernst, G. Bodenhausen and A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions (Oxford Univ. Press, London, 1987 ).
27 July 1990
[9] W.S. Warren and A.H. Zewail, J. Chem. Phys. 75 (1981) 5956; 78 ( 1983) 2219; 78 (1983) 2298. [ IO] D. Imre, J.L. Kinsey, A. Sinha and J. Krcnos, J. Phys. Chem. 88 (1984) 3956: E.J. Heller, Accounts Chem. Res. 14 ( 1981) 368. [ 111 S. Mukamel and Y.J. Yan, Science, submitted for publication. [ 12lA.M. Weiner, D.E. Leaird, G.P. Wiederrecht and K.A. Nelson, Science 247 ( 1990) 13 17. [ 131W.S. Warren, Science 242 (1988) 876; W.S. Warren and MS. Silver, Advan. Magn. Res. 12 ( 1988) 247; A.M. Weiner, J.P. Heritage and E.M. Kirchner, J. Opt. Sot. Am. B 5 (1988) 1563. [ 141 N.F. Scherer, A.J. Ruggiero, M. Du and G.R. Fleming, J. Chem. Phys., in press. [ 151 E.T. Slevaand A.H. &wail, Chem. Phys. Letters 110 ( 1984)
582;
E.T.Sleva, I.M. XavierJr. andA.H. Zewail, J. Opt.Am. 83 (1986) 483; E.T. Sleva, M. Glasbeek and A.H. Zewail, J. Phys. Chem. 90 (1986) 1232. [ 16] P.M. Felker and A.H. &wail, Advan. Chem. Phys. 70 (1988) 265; N. Bloembergen and A.H. Zewail, .I. Phys. Chem. 88 ( 1984) 5459; D.J. Tannor and S.A. Rice, Advan. Chem. Phys. 70 ( 1988) 441; S. Mukamel and K. Shan, J. Phys. Chem. 89 (1985) 2447. 117 D.J. Tannor and S.A. Rice, J. Chem. Phys. 83 ( 1985) 5013; D.J. Tannor, R. Kosloff and S.A. Rice, J. Chem. Phys. 85 (1986) 5805; R. Kodoff, S.A. Rice, P. Gaspard, S. Tesigni and D.J. Tannor, Chem. Phys. 139 (1989) 201. [I8 P. Brumer and M. Shapiro, Chem. Phys. Letters 126 (1986) 541; P. Brumer and M. Shapiro, Accounts Chem. Res. 22 ( 1989) 407. [ 191 H. Rabitz, in: Atomic and molecular processes with short intense laser pulses, ed. A. Bandrauk (Plenum Press, New York, 1988). [20] W. Jakubetz, J. Manz and H.-J. Schreier, Chem. Phys. Letters 165 (1990) 100; B. Hartke, J. Manz and J. Mathis, Chem. Phys. Letters 139 ( 1989) 123.