J. Phys. Chem. 1993,97, 12625-12633
12625
Laser Control Scheme for State-Selective Ultrafast Vibrational Excitation of the HOD Molecule J. Manz and G. K. Paramonov'vt Institut jlir Physikalische und Theoretische Chemie, Freie Universitiit Berlin, Takustrasse 3, 141 95 Berlin. Germany Received: May 21, 1993'
An efficient approach to controlling the population dynamics with ultrashort infrared laser pulses is developed and demonstrated for a two-dimensional model of the HOD stretching vibrations in the electronic ground state. The results of computer sipulations show that, in principle, any specified vibrational target level may be populated with probability close to 100%.
1. Introduction Efficient and activecontrolof molecular dynamics with properly chosen optimallaser fields is an important aim of laser chemistry1-' as well as a subject of much current interest. The purposes of laser control range from steering a molecule to a specified target state to manipulating the outcome of chemical reactions. Among the various schemes of laser control which have been proposed up to now, there are the following: a pump-and-dump scheme*,9via an excited electronic state, an active pumppump control scheme,lo and a coherent scheme for controlling unimolecular reactions with weak continuous-wave (CW)fields.Il-13 Strong field control schemes are based on optimization of a single driving laser pulseIkl9 or pulse sequences,20v21 or they employ the general theory of optimal contr0122-~~or use chirped laser pulses.*6~2~Theoretical foundations are also laid to take into account nonideality of a real experiment: a theory of optimal control of uncertain quantum systems insensitive to errors in the molecular Hamiltonianor to errors in the initial state of a quantum system has been developed by Rabitz and co-workers.28 A molecular control theory based on the Liouville space density matrix formalism has been advanced recently by Wilson and co-workers29~30which allows one to deal with mixed states, imperfect laser fields, and environmental or solvent effects. Some of the theoretical schemes, e.g., the coherent control scheme of Brumer and Shapiro,l1-I3have already been demonstrated experimentally.3'-34 Other schemes, e.g., the strong field control schemes, call for more demanding laser pulse amplitudes and durations, or they require laser fields with very complex structure. Available experimental techniques3s37 may prove to be useful in constructinglaser fields with the required s t r u c t ~ r e s . ~ ~ In many approaches for manipulating the yields of chemical reacti0ns,3~-4'the first and very important step of optimal laser control is selective excitation of molecular vibrations of the reactants (see also refs 48-50). Of special interest for controlling both unimolecular and bimolecular reactions is the possibility of populating any prescribed vibrational target level on a picosecond (ps)/femtosecond (fs) time scale with probability close to 100%. For example, this proved to be a very important intermediate step in picosecond laser-controlled isomerization.20 Ultrafast preparation of excited molecules with selectively populated vibrational levels may also be useful for molecular spectroscopy, investigationof intramolecular vibrational energy redistribution (IVR),and relaxation processes. In this paper, we develop a laser control scheme for stateselective ultrafast vibrational excitation of the HOD molecule in the electronic ground state. Infrared (IR) laser pulses of 1-ps duration are used in computer simulations for controlling the population dynamics in a two-dimensional model of the HOD t Permanent address: Institute of Physics, Belarus Academy of Sciences, Scarina aye. 70, 220602 Minsk, Republic of Belarus. Abstract published in Aduunce ACS Absrr~crs,November 1, 1993.
0022-3654/93/209112625%04.00/0
stretching vibrations. A very similar control scheme may be realized also by using 500-fs IR-laser pulses. The present approach is the extension of our strong field control schemeslc21 developed previously for state-selective excitation and lasercontrolled isomerization of one-dimensional quantum systems. The paper is organized as follows. In section 2,we describe a two-dimensional model of the HOD stretching vibrations, emphasizing the respective energy level structure and their labeling. The interaction of the HOD molecule with the laser field is described in section 3. The laser control scheme for stateselective vibrational excitation is presented in section 4,while the results are summarized in section 5 .
2. Model and Vibrational-Level Structure We consider a two-dimensional model for stretching vibrations in a nonrotating and nonbending HOD molecule in the electronic ground state, includingkineticcoupling of the OH and OD bonds. The model is adapted from ref 49;here we summarize rather briefly some properties and nomenclature which are important for the subsequent applications. The coordinates describing the OH and OD bonds are denoted rH and rD, respectively, with the associated conjugate momenta being PH and PD. The atomic masses are M H = 1.008amu, MO = 16.00 amu, and M D = 2.014 amu. Atomicunitsare used unless the other is indicatedexplicitly. The coupled vibrational stretches of the HOD molecule are represented by the molecular Hamiltonian
= 7 f +~ 7 f + ~ 7fH.D + Do
(1) where 7 f and ~ 7 f describe ~ individual bonds OH and OD, respectively; HH,Drepresents the coupling; and Do = 0.1994 hartree is the well depth. Explicitly = pH2/2mH + vH(rH) (2) with reducedmass mH = M H M o / ( M H+ Mo)and Morsepotential 7fH
vH(rH) DO{~XP[-B(~H -roll - 11' - DO (3) with equilibrium distance ro = 1.821~0 and Morse parameter /3 = 1.189~0-I.The eigenfunctions IkH) of the individual OH stretching vibrations, satisfying the Schriidinger equation
(4)
for kH = 0, 1, ,..,K H ,are the well-known Morse oscillator wave functions. The maximum quantum number KH is given by the integer closest from below to
& G / h @ - 0.5
(5) and the total number of bound states of the individual OH bond equals K H + 1. Equivalent expressions hold for the individual OD bond, with eigenfunctions and maximum quantum number denoted by IfD) and LD, respectively.
0 1993 American Chemical Society
Manz and Paramonov
12626 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993
%,,I
8
The kinetic coupling operator in the molecular Hamiltonian (1) is %,,D = (cos e/MO)PHPD
(KH
e..,
+ 1)(LD + 1)
0
(6)
with the fixed bending angle 0 = 104.5'. Within the finite expansion approach49 used below, the molecular Hamiltonian Hml (1) supports the vibrational eigenstates I+y) which may be ordered accordingto increasingenergies E, and labeled by the energetic-level quantum number u = 1, 2,
a c,
(7)
-2-
-
VibrationaleigenstatesI+,) are approximatedby finite expansions
I+,)
=
ZD
G~,J~H)Iu
(8)
The energiesE"and coefficients CknrWare obtained, as usual, by inserting expansion (8) into the SchriMinger equation
7fmolllcI,) = EUW")
(9)
operating with (l'~l(PHI, and solving the eigenvalue problem. The eigenstates are classified49as bound states: E, C E,, resonances: E,
> EdiB
(10) (1 1)
where Ediu is the lowest threshold energy for dissociation. In the present paper, a scheme for ultrafast control of the population dynamics is developed only for bound states of the HOD molecule, see Figure 1. It is instructive to label the bound eigenstates by the ordered pairs (m,n) of quantum numbers m and n, indicating approximately m vibrational quanta in the OH bond (the first quantum number) and n vibrational quanta in the OD bond (the second quantum number). This labeling is determined by the dominant expansion coefficient CkHIDPin (8): if ICkH.m,Dln,J2 dominates, the assignment is as follows u
-
-
E,
-3-
-
-4
6
0
1 2
-
Figure 1. Energy-level structure and labeling for thestretching vibrations of the two-dimensional model of the HOD molecule in the electronic ground state. The target states to be populated selectively are indicated by asterisks.
(m,n)
(12)
Imn)
(13)
E(lu,")
(14)
The u (m,n) assignment is presented in Figure 1 for the 50 lowest vibrational levels of the HOD molecule, together with their energies. The ( m > 0, n = 0) states will be referred to as local OH states, the ( m = 0, n > 0 ) states, as local OD states, and the (m,m)states, having equal numbers of vibrational quanta in each bond, as "symmetric" states. Sometimes, vibrationalstates will also be labeled u(m,n) by using both energetic u and local mode (m,n)quantum numbers. The manifolds of local OH and OD states will sometimesalso be referred to as local OH and OD modes, respectively. 3. Interaction with the Laser Field
The excitation of vibrational stretches in the HOD molecule by a laser field is described semiclassically within the electric dipole approximation. The molecule interacting with the laser field is represented by the Hamiltonian
7f = a m , + 7fint(t) The interaction term Hint(t)has the form
(15)
for the HOD molecule in the electronic ground state by a bond dipole model5'
= rHP0 ew(-rH/r*) + '& exp(-rD/r*) (l 7, with ~ro= 1.85 D / A and r* = 0.6 A. p(rH,rD)
The HOD molecule is supposed to be placed in the xz plane, with the I axis being the bisector (see Figure 1). With the x-polarized electric field of the laser, the interaction operator Hi,,,(t) takes the form
where C#I
7fint(t) = 4 8 ) = ( m - 0)/2, and
COS 4
(18)
c((rH,rD) = Po[rH exp(-rHlr+) - rD exp(-rD/r*)l (19) The laser fields to be used below for controlling state-selective excitation of the HOD molecule are composed, in general, of several ultrashort subpulses. The correspondingglobal electric field strength is
The subpulw i starts at t = tor,its duration is tpr,and its current time interval is
+ t,
(21) ct is the electric field amplitude, WI is the laser carrier frequency, and is the phase. The specific, sin2-type,shape of the driving laser pulses1c16J8 used in this study is not of primary to, It Itp,
7fint(t) = -~(rH~D)*eo4t) (16) where eo is the polarization unit vector of the electric field r ( t ) and the electric dipole moment function fi(rH,rD) is represented
P(~H,~D)
Laser Control of HOD Excitation
The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12627
importancesimilar results can be obtained with other shapes (Gaussian, for example; see also refs 19, 52, and 53). Initially, at t = 0, the HOD molecule is supposed to be in the vibrational ground state:
= = lm=O,n=O) (22) The solution I*(t) ) of the time-dependent SchrMinger equation I*(t=O))
d i h d * ( t ) ) = [%mol + %int(t)I*(t))
(23)
starting from the vibrational ground state and comprising only bound states (10) can be expanded in terms of molecular eigenstates J+d) with eigenenergies Ed
energy,49 and they may serve to store energy in mode-selective dissociation.55 A. Local OD and OH States. A yasonable approach to stateselective vibrational excitation of the HOD molecule is multiphoton resonance pumping, proved successfulin one-dimensional models (see refs 4, 15, 17, 18, 25, and 52). Accordingly, the target state 24(0,7), for example, may be excited from the initial one 1 (0,O)with the seven-photon resonance transition
-
(0,O) (7 photons) pumped by a single laser pulse
-
(0,7)
(30)
e ( t ) = t l sin’(rt/t,,) cos(w,t)
(31)
- -- -
via the supposingly dominant excitation pathway with time-dependent probability amplitudes &(t) and corresponding initial conditions A,(t=O) =, ,a (25) Insertion of the expansion (24) into the Schriidinger equation (23) and operating, as usual, with (+J, yield the differential equation for the probability amplitudes
(0,O) (0,l) ... (0,6) (0,7) (32) To yield the complete localization of population at the target level (0,7), the driving laser pulse (3 1 ) should be optimized with respect to the field amplitude €1 and the carrier frequency wl,the phase being of no importance when only one state (0,O)is initially populated. This is achieved by using the optimizationprocedure’* which consists of two main steps: amplitude optimization and frequency optimization. 1 . Amplitude Optimization. In the first step, the laser carrier frequency w1 is fixed to the exact seven-photon resonance, w1 = w;= =
The dipole moment matrix elements (+ u ; l ~ l $ d )are evaluated from (8) and (19) as in ref 54. The solution A,(t) of the time-dependentalgebraic Schriidinger equation (26) with initial conditions (25) yields the populations of the states u as well as the populations of the corresponding (m,n) states
-
pu(t) = I A U ( ~ ) I ’ p(m,n)(r) (27) with the u -., (m,n) correspondence specified in Figure 1. We shall also use the total population of all local OH states,
as well as the total population of all local OD states,
which are instructive for describing the population dynamics of the HOD molecule when developing the optimal laser control scheme.
4. Laser Control of the State-Selective Vibrational Excitation The general purpose of optimal state-selective ultrafast laser control of molecular excitation may be formulated as follows: Starting from a certain initial state, any prescribed target state should be excited selectively, with probability close to loo%, by laser pulses which are as short as possible. At the same time, the structure of the pulses should be as simple as possible in order to allow experimental realization. In the present study, the HOD molecule is supposed to be initially in the vibrational ground state 1 (0,O). As target states, we shall consider, exemplarily, the local OD state 24(0,7), the localOHstate23(5,0) and thesymmetricstate26(3,3). Selective excitation of local states is an important intermediate step to selective control of the dynamics of unimolecularand bimolecular reactions. Low-lying symmetricstates (being not very interesting in themselves) are important for developing a general strategy for selective excitation of high-lying states or resonances (1 1 ) with m = n; these states may have lifetimes which are at least 3 orders of magnitude longer than local states of about the same
- E(,,,)]/7
(33) Then, numerical solution of the Schriidinger equation (26) at increasing values of the laser pulse amplitude €1 gives the final population of the target level (0,7) by the end of the pulse,
= p(0,7)(r=tp1) (34) versus the pulse amplitude €1. The value of the amplitude corresponding to the first local maximum of 7$: is considered a first approximation to the optimal laser pulse amplitude $07)
q t .
2. Frequency Optimization. In the second step, the laser pulse amplitude el is fixed to the optimal value obtained in the first step, while the laser carrier frequency w1 is varied in a certain domain so as to obtain the maximal final population f$’7,of the target level. The correspondingvalue of the laser carrier krequency is considered a first approximation to the optimal frequency Then, both steps are repeated several times until1 convergence of the total optimization procedure. The optimal values of the laser pulse amplitude and the carrier frequency, yielding complete population of the target level (0,7), are: cyPt = 0.10244 hartree/eao, which corresponds to 526.78 MV/cm or to the pulse peak intensity ZI= 368.3 TW/cm*, and woPt = 0.99978 which corresponds to 0.01141 hartree/h or to 2504.21 cm-I. The final population of the target level is = 0.9926. The population dynamics for the optimal sevenphoton pumping of the (0,O) (0,7) transition is demonstrated in Figure 2A for the initial state (0,O)and for the target state (0,7), for the total population of all local OD states (29) which is represented by curve “0-D” and for the total population of all the other states (curve “OTHERS”). The correspondingoptimal laser field (3 1) yielding complete population of the target level (0,7) by the end of the pulse is shown in Figure 2B. It is seen from Figure 2 that the excitation of local OD states during the pulse dominates the excitation of all the other states. This indicates the dominant role of the excitation pathway (32) and very weak competition of any “side transitions”,
,
el7,
-
-
-
-
(OJ) (1,1), (092) (192) (0,3) (193), (35) As a consequence, the population dynamics of the present twodimensional model system is similar to the one-dimensionalone. On the other hand, states ( m > 0, n > 0) are also populated ‘a.
Manz and Paramonov
12628 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 A 1
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Figure 2. Parts A and B show selective population of the local OD state 24(0,7) of the HOD molecule by a single optimal laser pulse with constant carrier frequency. (A) Population dynamics: (O,O), initial state; (0.7),target state; 0-D,total population of all local OD states; curve OTHERS, total population of all the other states, except those indicated explicitly. (B) Optimal laser pulse yielding complete localization of population. The pulse parameters are ?,I = 1 ps, eFt= 526.78 MV/cm, u F = 2504.21 cm-l. Parts c and d show optimal five-photon excitation of the local OH state 23(5,0) by a single laser pulse with constant carrier frequency. (C) Population dynamics: (O,O), initial state; (5,0), target state; 0-H, total population of all local OH states; curve OTHERS, total population of all the other states, except those indicated explicitly. (D) Optimal laser pulse yielding maximal final population at the target level. The pulse parameters are tpl = 1 ps, eFt = 392.8 MV/cm, coypt = 3426.93 cm-l.
during the pulse. These states do not belong to the local OD mode. Therefore, the total excitation process is not completely mode-selective. It is seen from Figure 2A that, only for t > 0.8tpl,almost all population is localized in the local OD mode, and by the end of the pulse only the target state (0,7) is populated. The final distribution of population is almost completely both mode-selective and state-selective. The situation changes when a single laser pulse (31) is used for selective excitation of the local OH state 23(5,0), which has slightly less energy than the previous target state 24(0,7). By analogy,the excitation scheme may be the five-photon transition
-
-
( 5 photons) ($0) (36) which proved to be successful in a one-dimensional model of the local OH bond.21 The supposingly dominant excitationpathway is
(0,O)
- -- -
(0,O) (1,O) with resonant frequency
...
wits = [E(,,,)-
(4,O)
(5,O)
(37)
(38) The optimizationprocedure described above gives the following optimal parameters of the driving laser pulse: optimal amplitude p P t = 0.076 387 hartree/eao, which corresponds to 392.8 MV/ cm or to the pulse peak intensity 11= 204.78 TW/cm2, and optimal frequency oypt= 0.99976 wf", which corresponds to 0.015 61 hartree/h or to 3426.93 cm-l. The corresponding population dynamics is illustrated in Figure 2C for the initial state (O,O), for the target state (5,0), for the total population of all local OH states (28) represented by curve 0-H, and for the total population of all the other states (curve OTHERS). The driving laser pulse yielding maximum final population at the target level (5,O) is presented in Figure 2D. It is seen from Figure 2C that complete /5
localization of population at the target level (5,O) cannot be achieved by using a single laser pulse for pumping the five-photon resonance. Energy levels (m > 0, n > 0) which do not belong to the local OH mode (represented by curve OTHERS) also acquire substantial population both during the excitationprocess and by the end of the pulse. The final population of the target level is only 0.7789, while the rest of the population is localized mainly at the levels 9(2,1) and 15(3,1): = 0.1533 and = 0.0563. This indicates that the "side transitions"
e;,)
e$1,
-
-
(290) (291) and (390) (391) (39) compete against the excitation pathway (37), and the population dynamics of the two-dimensional system is no longer similar to the one-dimensional one. The final distribution of population is neither mode-selective nor state-selective. State-selective excitation of the target level (5,O) can be achieved by using the sequential multiphoton pumping proposed previously2'for state-selective excitationof high-lying vibrational levelsin a one-dimensionalmodel of the isolated OH bond. Within the sequential approach,21the overall five-photon transition (36) is divided into two sequential steps
-
- -
-
(0,O) (3 photons) (3,O) (2 photons) (5,O) (40) The sequential process (40) can be controlled by superposition of two subpulses:
-
The first subpulseshould be optimized for the individual transition
-
(0,O)
(3 photons)
(3,O)
(43)
The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12629
Laser Control of HOD Excitation while the second subpulse should be optimized separately for the individual transition
-
-
(2 photons) (5,O) (44) starting from the level (3,O)as initial state. The optimization procedure gives the following optimal parameters of the subpulses. The first subpulse has the optimal amplitude efPt = 0.034 063 hartree/eao, which correspondsto 175.16 MV/cm or to the pulse peak intensity ZI= 40.719 TW/cmZ, and the optimal frequency wyt = 1.0003 UT,which corresponds to 0.016 43 hartree/h or to 3606.016 cm-1. The first subpulse has duration ?,I = 1 ps and = 0.9999. yields the final population The second subpulse has the same duration tp2 = 1 ps, the optimal amplitude = 0.009 405 4 hartree/eao which corresponds to 48.364 MV/cm or to the pulse peak intensity 1 2 = 3.1045 TW/cm2, and the optimal frequency = 1.0003 w y , which corresponds to 0.0144 hartree/h or to 3160.53 cm-l. The second subpulse yields the final population p’l” = 0.9853, starting from the initial state (3,O). The rest of th5fpopulation is localized predominantly at the level 16(4,0): the final population = 0.0142. Apparently, the selectivity of the second step is somewhat less than the selectivity of the first one. Finally, the superpositionof the two individual subpulses (4 1) and (42) gives the global laser field
(3,O)
e;o, e(?)
= el sin2(?rt/t,,) cos(o,t) e2
+
sin2[?r(t- to2)/tp2]cos(w2t
+ cpz)
(45)
which yields almost complete localization of population at the target level (5,O) starting from the initial level (0,O). In the general case, the time delay to2 and the phase (p2 of the second subpulse should also be taken into account. The corresponding population dynamics are presented in Figure 3, parts A, C, E, and G, for four cases with equal phases (p2 = 0 but different time delays to2 of the second subpulse of the global laser field (45). The respective global fields (45) are shown in the bottoms (Figure 3B,D,F,H) together with the sin2 envelopes of subpulses (dashed curves). The main conclusions to be drawn from Figure 3 are the following. First, the final population of the target level (5,O) depends very weakly on the time delay to2 of the second subpulse, when it is varied in a rather wide domain, from t,l to 0.3tP,,without any additional optimization of subpulses of the global field (45). On the other hand, delays shorter than 0.3rPl cause substantial decrease of the final population of the target level. Second, at short delay times of the second subphase, to2 < tP1/2,the global field (45) has a very complicated structure and may even seem to be inaccessible to experimental realization. But this apparent complexity is really not a problem, since all global fields presented in Figure 3 are nothing but superpositions of two (sin2)subpulses. Increasinglystructured global fields yield morecomplicated population dynamics but almost the same final population of the target level (within the above-specified domain of r02). Third, the process of sequential state-selective multiphoton excitation (40) is also almost completely mode-selective: only local OH states (curve 0-H) acquire substantialpopulation during the excitation process at the expense of decreasing population of the initial state (0,O). Indeed, the population dynamics for the (0,O)state and for the local OH mode, including only local OH states (28), is very similar to a two-level behavior in all four cases of Figure 3: at t 0.5 ps the population P(o,o)= POH,and at t > 0.75 ps almost all population is localized in the local OH mode (POH 1).
Fourth, comparison of the unsuccessful attempt to populate level 23(5,0) with probability close to 100% by a single laser pulse with the successful approach via two sequential laser pulses yields (at least) two very important advantages of the sequential scheme: it improves the selectivity from = 0.7789 to 0.9853, and it reduces the maximal intensity from -205 to -40 TW/cm2. Similar reduction of the required intensity may also be achieved by application the sequential scheme to other transitions or by using more subpulses for the same overall transition. The sequential multiphoton excitation (40) also depends very weakly on the phase (p2 of the second subpulse. An additional analysis showed, for example, that at a fixed delay time to2 = 0.5 ps the final population of the target level (5,O) changes from 0.9880 to 0.9876 if (p2 changes within the interval --17 < < u. An evident extension of the techniqueof sequentialmultiphoton excitation can be used for the state-selective excitation of highlying local OH and OD states. The respective global laser field (20) may consist of several subpulses, each subpulse being optimized for the individual sequential transition. The phases pl of sequential subpulses have minor influence on the final population of the high-lying target level; Le., they may be arbitrary. B. SymmetricStates. The target symmetric state 26(3,3) has three vibrational quanta in each bond. On first glance, one may assume that it can be excited by pumping the three-photon transition resonance
qsl0,
-
- - -
(0,O) (3 photons) via the symmetric excitation pathway
(3,3)
(46)
(0,O) ( 1 9 1 ) (2,2) (3,3) (47) However, the respective dipole moment matrix elements ( n , mllrlm+l, n+l ) (48) are usually 2 orders of magnitude smaller than matrix elements
(n,m)lrlm+l, n) or (n,mllrlm, n + l )
(49) Therefore, the excitation pathway (47) requires very strong laser fields. But these exceedingly strong fields also stimulate many nonresonant transitions involving larger dipole moment matrix elements (49) which destroy the selectivity of the process. This situation is illustrated in Figure 4A, where time-dependent populationsare shown for the case when transition (46) is pumped in the exact three-photon resonance with a very strong single laser pulse presented in Figure 4B. Although the field strength is more than 10 times greater than in the case of Figure 2A,B, the population of the target resonant level (3,3) is much less than the total population of all nonresonant levels (curve OTHERS), in contrast to the results presented in Figure 2A. That means, in fact, that the excitation pathway (47) is not dominant. This dilemma cannot be solved by using the sequentialapproach within the symmetric excitation pathway (47), for example, by dividing the overall transition (46) into three sequential one-photon transitions over the symmetric states, because even the lowest symmetric state 5(1,1) cannot be excited’ selectively from the vibrational ground state 1(0,O)by pumping the one-photon resonance. Another feasible approach for selective excitation of the target symmetric state (3,3) is the six-photon overall transition
-
-
(0,O) (6 photons) (3,3) (50) comprising intermediate transitions with large dipole moment matrix elements (49). The six-photon overall transition (50) pumped by a single laser pulse turns out to be much more efficient than the three-photon transition (46). The time-dependent populations for the case when transition (50) is pumped in the exact six-photon resonance are presented in Figure 4C, and the respective single laser pulse is shown in Figure 4D. Comparison
12630 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993
Manz and Paramonov c
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Figure 3. Selective population of the local OH state 23(5,0) of the HOD molecule by sequences of two optimal laser subpulses with different time delays. (A, C,E, and G) Population dynamics: (O,O), initial state; (3,0), intermediate target state; (5,0), target state; 0-H,total population of all local OH states. (B, D, F, and H)The respective global laser fields and the sin2 envelopes of subpulses. The parameters of the optimal subpulses are = 3160.53 cm-l. The time delays are to2 = 1 ps (A, = 175.16 MV/cm, w F = 3606.016 cm-1, = 48.364 MV/cm, and t,l = tp2 = 1 ps, B), to2 = 0.75 ps (C, D), to2 = 0.5 ps (E, F), and to2 = 0.3 ps (G, H).
with the three-photon resonance pumping of Figure4A,B indicates that the six-photon transition (50) requires a 5 time smaller field strength, yet it results in higher population of the target symmetric state (3,3). On the other hand, there are 10 possible excitation pathways from the initial state (0,O)to the final state (3,3) for thesix-photonoverall transition (50). These excitation pathways
compete against each other, and if no care is taken of choosing thedominant excitation pathway, the stateselectivityof the process is rather low. This is clearly seen from Figures 4C, where the total population of all nonresonant states (curve OTHERS) is much greater than the population of the target resonant state (3,3). Other possible reasons for the rather low state selectivity
Laser Control of HOD Excitation
The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12631 E
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Figure 4. Parts A and B show resonant three-photon excitation of the target symmetric state 26(3,3) by a single laser pulse with a constant carrier frequency w p = [E(3,3)- E(o,o)]/~.(A) Population dynamics: (O,O), initial state; (3,3), target state; OTHERS, total population of all the other states except the initial and target states. (B) The driving laser pulse: t,l = 1 ps, €1 = 6.535 GV/cm, WI = 6298.46 cm-*. Parts C and D show resonant six-photon excitation ofthe target symmetric state 26(3,3) by a single laser pulse with constant carrier frequency w:" = [E(3,3)-E(o,o)]/6. (C) Population dynamics: the level labeling is as in Figure 4A. (D) The driving laser pulse: t,l = 1 ps, €1 = 1.307 GV/cm, w1 = 3149.37 cm-'. Parts E, F,G, and H show selective population of the symmetric target state 26(3,3) of the HOD molecule by sequences of two optimal laser subpulses with different time delays. (E, G) Population dynamics: (O,O), initial state; (3,0), intermediate target state; (3,3), target state; OTHERS, total population of all the other states except those indicated explicitly. (F, H) The respective global laser fields and the sin2envelopes of subpulses. The parameters of the optimal subpulses are tpl = t,2 = 1 ps, cypt = 175.16 MV/cm, = 3606.016 cm-1, e;pt = 164.7 MV/cm, and wqpt = 2695.07 cm-I. The time delays are to2 = 1 ps (E, F) and 202 = 0.6 ps (G, H).
of the process at hand are destructiveinterferenceof the excitation pathways and large frequencydetuningsof sequentialtransitions
Im,n)
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Im+l, n) Im+l, n + l ) (51) in OH and OD branches, respectively. This is confirmed by the
fact that even the lowest symmetric state (1,l) cannot be excited selectively from the ground state (0,O) by pumping the two-photon resonance. State-selective excitation of the target symmetric state (3,3) is, therefore, a rather difficult task. Fortunately, the goal can
12632 The Journal of Physical Chemistry. Vol. 97, No. 48, 1993
be achieved easily with the sequential scheme, which also selects automatically the dominant excitation pathway. The overall sixphoton transition (50) is divided into two sequential steps
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(0,O) (3 photons) (3,O) (3 photons) (3,3) (52) The sequential process (52) can be controlled by superposition of two subpulses c(t)
= c, sin2(nt/tp,) cos(w,t)
+
c2 sin2[n(t - tO2)/tp2]cos(w2t) (53)
which are optimized separately for the individual 3-photon transitions of the sequence (52). The phases of subpulses have minor influence on the final population of the target level (3,3) and are not taken into account in the global field (53). The supposingly dominant excitation pathway is
- - - - - -
(3,3) (54) The duration of 1 ps is taken for both subpulses in global field (53), so the optimal parameters of the first subpulse are the same as in the previous case of the two-step transition (40), while the second subpulse has optimal amplitude = 0.032 029 hartree/ eao, which corresponds to 164.7 MV/cm or to the pulse peak intensity 12 = 36.002 TW/cmZ, and optimal frequency wqpt = 1.00009 OF,which corresponds to 0.012 28 hartree/h or to 2695.07 cm-1. The second subpulse yields the final population = 0.9868, starting from the initial state (3,O). The rest of the population is localized predominantly a t the level 20(3,2) with the final population = 0.0123. The selectivity of the second step is somewhat less than the selectivity of the first step, which gives = 0.9999 starting from the vibrational ground state. The population dynamics for the two-step sequential transition (52) is presented in Figure 4E,G for two different time delays of the second subpulse to2 = tpl and to2 = 0.6tpl,respectively. The respective global fields (53) are shown in the bottoms (Figure 4F,H) together with the sinZ envelopes of subpulses (dashed curves). The final population of the target level (3,3) weakly depends on the time delay 202 of the second subpulse within the domain from fpl to 0.6tpl, which is 2 times shorter than the respective domain for the two-stepsequential transition (40). Delays shorter than 0.5tPl result in substantial decrease of the final population ofthe target level (3,3). In theoptimalsituation, modeselectivity of the overall transition (52) takes place only in the first step, and the final distribution of population is almost completely state selective.
(0,O) (LO)
(2,O)
(3,O)
W )
e;3)
5. Conclusion In the present model study, laser control of state-selective vibrational excitation of the HOD molecule is simulated by means of simple (sin2) laser pulses with constant carrier frequency wi and the pulse duration tPi = 1 ps (the full width at half maximum (FWHM) is 500 fs). The specific, sin2-type, shape of subpulses constituting the global laser field (20) is not of primary importance-a very similar control scheme also can be developed by using other shapes of subpulses, for example Gaussian. There is also no critical dependence of the developed control schemeon theduration of subpulses. For example, 2 times shorter laser pulses, FWHM = 250 fs, can be used in the developed scheme, but they are more demanding to the field strength. The present control scheme allows substantial reduction of the required maximum field strength by using more sequential laser subpulses for the same overall transition. Another advantage of the proposed control scheme is very weak dependence of the final population of the specified target level on the time delays of sequential subpulses as well as on their phases. Both these features are favorable for experimental
Manz and Paramonov realization of our control scheme. In particular, the phase insensitivity indicates that subpulses of the global field may even be generated by different lasers. Experimentally, it would not be necessary to achieve a control over the relative phases of the ultrashort subpulses which easily may suffer phase chirp due to some phase-modulation processes. An evident extension of the sequential control scheme should make it possible to populate selectively high-lying vibrational levels of the HOD molecule as well as similar dihydride model systems. Wearecurrently in the process of developingour scheme for the state-selective population of the vibrational levels with energetic number u = 50. In principle, any prescribed target level can be populated with probability close to 100% by using the properly optimized global laser field in analogous model simulations. Taking into account that the individual subpulses in principle may be generated with the available experimental infrared te~hnology,~~3* it may thus be also possible to superimpose these subpulses to form the global field (20) and to approach the experimental realization of the proposed control scheme. At the same time, we realize that an actual experiment producing the ultrashort and very strong laser pulses required for the ultrafast control scheme nowadays is still a rather difficult problem. We also realize that the realisticexperimental situation, imperfection of laser fields, for example, may require some modifications of the proposed ultrafast control scheme to be considered in close connection with the relevant experiment. Acknowledgment. We thank Prof. W. Jakubetz and Mr. E. Kades for their kind emergency service when we lost our dipole matrix elements. Generous financial support by the Deutsche Forschungsgemeinschaft (project Ma 5 15/9), by Freie Universitgt Berlin, and by the Fonds der Chemischen Industrie is also gratefully acknowledged. The computations were carried out on our HP 750 work stations. References and Notes (1) Ben-Shaul, A,; Haas, Y.; Kompa, K. L.; Levine, R. D. Lnsers and Chemical Change, Springer: Berlin, 1981. (2) Crim, F. F. Annu. Rev. Phys. Chem. 1984,35, 647. (3) Lupo, D. W.; Quack, M. Chem. Rev. 1987,87, 181. (4) Brumer, P.; Shapiro, M. Acc. Chem. Res. 1989, 22, 407. (5) Crim, F. F. Science 1990, 249, 1387. (6) Levine, R. D.; Zewail, A. H.; El-Sayed, M. A. J. Phys. Chem. 1991,
95, 7961. (7) Brumer, P.; Shapiro, M. Annu. Rev. Phys. Chem. 1992, 43, 257. (8) Tannor, D. J.; Rice, S.A. J. Chem. Phys. 1985,83, 5013. (9) Tannor, D. J.; Kosloff, R.;Rice, S . A. J. Chem. Phys. 1986,85,5805. (10) Potter,E.D.;Herek,J.L.;Petersen,S.;Liu,Q.;Zewail,A.H.Nature 1992, 355, 66. (1 1) Brumer,P.; Shapiro, M. Faraday Discuss. Chem. Soc. 1986,82,177. (12) Shapiro, M.; Brumer, P. J. Chem. Phys. 1986,84, 4103. (13) Shapiro, M.; Brumer, P. J . Chem. Phys. 1991, 95, 8658. (14) Paramonov, G. K.; Sawa, V. A. Phys. Lett. A 1983, 97, 340. (1 5) Paramonov, G. K.;Sawa, V.A.; Samson, A. M. Infrared Phys. 1985, 25, 201. (16) Paramonov, G. K. Chem. Phys. Lett. 1990, 169, 573. (17) Jakubetz, W.; Just, B.; Manz, J.; Schreier, H.-J. J. Phys. Chem. 1990, 94, 2294. (18) Paramonov, G. K. Phys. Lett. A 1991,152, 191. (19) Joseph, T.; Manz, J. Molecular Phys. 1986, 58, 1149. (20) Combariza, J. E.; Just, B.; Manz, J.; Paramonov, G. K. J. Phys. Chem. 1991,95,10351. Combarizi, J.E.;Manz, J.;Paramonov,G.K.Faruday Discuss. Chem. SOC.1991, 91, 358. (21) Just, B.;Manz, J.; Paramonov, G. K. Chem. Phys. Lett. 1992,193, 429. (22) Shi, S.; Woody, A,; Rabitz, H. J. Chem. Phys. 1988, 88, 6870. (23) Shi, S.;Rabitz. H. Chem. Phys. 1989, 139, 185. (24) Shi, S.; Rabitz, H. J . Chem. Phys. 1990, 92, 2927. (25) Jakubetz, W.;Manz, J.;Schreier,H.-J. Chem. Phys.Lert. 1990,165. 100. (26) Chelkowski, S.;Bandrauk, A. D.; Corkum, P. B. Phys. Rev. Lett. 1990,65, 2355. Chelkowski, S.;Bandrauk. A. D. Chem. Phys. Lett. 1991, 186, 264. (27) Just, B.; Manz, J.; Trisca, I. Chem. Phys. Lett. 1992, 193, 423. (28) Dahleh, M.; Peirce, A. P.;Rabitz, H. Phys. Rev. A 1990, 42, 1065. (29) Yan, Y. J.;Gillian, R. E.; Whitnell, R.M.; Wilson, K. R.; Mukamel, S.J . Phys. Chem. 1993, 97, 2320.
Laser Control of HOD Excitation (30) Kohler, B.; Krause, J. L.; Raksi, F.; Rose-Petruck, C.; Whitnell, R. M.; Wilson, K. R.; Yakovlev, V. V.; Yan, Y. J. J . Phys. Chem., this issue. (31) Chen. C.; Yin, Y. Y.; Elliott, D. S . Phys. Rev. Letr. 1990, 64, 507. (32) Chen, C.; Elliott, D. S.Phys. Reo. Lett. 1990, 65, 1737. (33) Park, S.M.; Lu,S.P.;Gordon, R. J. J. Chem. Phys. 1991,94,8622. (34) Lu,S.P.; Park, S.M.; Xie, Y.; Gordon, R. J. J. Chem. Phys. 1992, 96, 6613. (35) Corkum, P. B. Opr. Lerr. 1983, 8, 514. (36) Corkum. P. B.;Ho, P. P.; Alfano, R. R.; Manassah, J. T. Opt. Lett. 1985, 10, 625. (37) Rolland, C.; Corkum. P. B. J. Opt. Soc. Am. E 1988, 5, 641. (38) Several useful experimental techniques, including laser pulse shaping technology, aredescribed by Wilsonandco-workers in ref 29. Seealso: Kohler, B.; Krause, J. L.; Raksi, F.; Rose-Petruck, C.; Whitnell, R. M.; Wilson, K. R.; Yakovlev, V. V.; Yan. Y. J. In Reaction Dynamics in Clusters and Condensed Phases; 26th Jerusalem Symposium; Jortner, J., Levine, R. D., Pullmann, B., Eds.; Kluwer: Dordrecht, 1993, in press. (39) Engel, V.; Schinke, R. J . Chem. Phys. 1988.88.6831. (40) Imre, D. G.;Zhang, J. Chem. Phys. 1989,139, 89. (41) Vander Wal, R. L.; Scott, J. L.; Crim, F. F. J. Chem. Phys. 1990, 92. _.803. .~ (42) Bar, 1.; Cohen, Y .; David, D.; Rosenwaks,S.; Valentini, J. J. J . Chem. Phys. 1990, 93, 2146. (43) Vander Wal, R. L.; Scott, J. L.; Crim, F. F.; Weide, K.; Schinke, R. J . Chem. Phys. 1991, 94, 3548.
The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12633 (44) Sinha, A.; Hsiao, M. C.; Crim, F. F. J. Chem. Phys. 1990,92,6333; 1991. 94, 4928. (45) Crim, F. F.; Hsiao, M. C.; Scott, J. L.; Sinha, A.; Vander Wal, R. L. Phil. Trans. R . Soc. London A 1990,332,259, (46) Bronikowski, M. J.;Simpson, W. R.;Girard, 8.;a r e , R. N. J . Chem. Phys. 1991,95, 8647. (47) Amstrup, B.; Henrihen, N. E. J . Chem. Phys. 1992, 97, 8285. (48) Manz, J. In Molecules in Physics, Chemistry and Eiology; Maruani, J., Ed.; Kluwer: Dordrecht, 1989; Vol. 3, p 365. (49) Hartke, B.; Manz, J.; Mathis, J. Chem. Phys. 1989, 139, 123. (50) Combariza, J. E.; Daniel, C.; Just, B.; Kades, E.; Kolba. E.; Manz, J.; Malisch, W.; Paramonov, G.K.;Warmth, B. In Isotope E’ecrs in GasPhase Chemistry; Kaye, J. A., Ed.; ACS Symposium Series; American Chemical Society: Washington, 1992; Vol. 502, p 310. (51) Lawton, R. T.; Child, M. S.Mol. Phys. 1980, 40, 773. (52) Dolya, Z. E.; Nazarova, N. B.; Paramonov, G. K.; Sawa, V. A. Chem. Phys. Lett. 1988, 145, 499. (53) Holthaus, M.; Just, B. Phys. Reu.A, in press. Breuer, H.-P.;Holthaus, M. J. Phys. Chem.. this issue. (54) Jakubetz, W.; Manz, J.; Mohan, V. J. Chem. Phys. 1989,90,3686. ( 5 5 ) Bisseling, R. H.; Kosloff, R.; Manz, J.; Mrugata, F.; Remelt, J.; Weichselbaumer, G.J. Chem. Phys. 1987, 86, 2626.
