6636
J . Phys. Chem. 1988, 92, 6636-6647
Time Evolution of Single- and Two-Photon Processes for a Pulse-Mode Laser Stewart 0. Williams and Dan G. Imre* Department of Chemistry, University of Washington, Seattle, Washington 981 95 (Received: May 5, 1988)
The advent of femtosecond lasers has provided us an opportunity to better characterize nuclear motion during fast chemical processes, since, unlike conventional spectroscopics which probe final-state distributions, we are now able to study reaction dynamics (in real time) while the reaction is taking place, as demonstrated in recent experiments on ICN and NaI by Zewail et al. In an effort to provide insight into events in the laboratory, we present a study of a two-state system, the ground and excited states, coupled by a pulsed laser system. A system of coupled equations is obtained and then used to investigate the time evolution of single- and two-photon processes on both repulsive and bound excited states. A consistent set of results are presented for short and long pulses in weak field, as well as the strong-field limit.
resolution. We will discuss a scheme by which such high-resolution spectra can be obtained while at the same time preserving temporal resolution on the order of the shortest laser pulse. Section I1 describes the theoretical treatment of two adiabatic surfaces coupled by a laser pulse. In section IIa we present an overview of the interpretation of the results derived from the time dependent formalism in the CW limit. Section IIb addresses the theoretical framework for the pulsed laser study. In section 111 we present results for the case where the laser is turned on very rapidly and remains on. Section IV contains results for a short pulse (in time). In both sections I11 and IV, we examine the time evolution of the wave function and population on the excited state, and the time evolution of the Raman spectrum for both on- and off-resonance excitation. In each case, results for a repulsive and bound (Morse and harmonic) excited state are described. In section V we investigate the consequences introduced by using a high-powered laser excitation source; that is, we investigate the effect of a strong laser field on single- and two-photon processes.
I. Introduction In a recent publication',2 we examined the two-photon processes for continuous wave (CW) excitation. We employed a time-dependent f o r m a l i ~ m ~to- ~examine properties of Raman wave functions which, as we have shown, is what is created in the lab under such excitations. Thus, we have used a time-dependent approach to arrive at a time-independent quantity, the Raman wave function. The applications of time-dependent formalisms to CW excitation situations has produced many misconceptions.6 In many respects these are well-founded. On one hand, the time-dependent equations appear to imply that the experiment prepares at t = 0 a localized wavepacket. On the other hand, these equation^^,^ are valid only in the limit of very long pulse or C W excitation. In this limit, 1 = 0 cannot even be defined. Despite this apparent paradox, the time-dependent formulation is exact, to first-order perturbation theory for absorption and second order for Raman. The contradiction does not stem from any new approximations or inexactness of the equations, but rather its origin is in the interpretation we impose on the equation. The advantage that this formalism offers is in its intuitive nature. It makes a simple connection between a spectrum obtained with a C W light source, in the frequency domain, and the dynamics which would have occurred were the molecule excited with a very short pulse laser. The localized wavepacket picture is exactly what happens in the lab when the exciting light source is a 6 pulse in time. The time-dependent formulation turns our attention from energy level diagrams to dynamics, yet it gives a somewhat confusing account as to what really happens in the lab. In an effort to clear up this fundamental question, we present here a study of what happens in the lab from the C W limit to the short-pulse limit. We present a study of one- and two-photon spectroscopy for pulsed excitation. We address the same question we raised for C W excitation in ref 1, namely, what is prepared on the excited-state surface and how does it evolve in time? If we could both timeand frequency-resolve the Raman spectrum, how would it evolve in time? The study involves a fully quantum mechanical (QM) treatment for diatomic molecules, of two potential energy curves coupled by a classically treated laser field. Our aim is to enhance our intuitive approach to experimental situations by presenting a number of studies involving both bound and repulsive excited states. We believe that a starting point for the interpretation of short-pulse laser excitation should be. knowing what the laser prepares. We will examine throughout the paper the wave function on the excited state and its time evolution. The Raman spectrum will be used as a probe of the time evolution of the excited-state wave function. The Raman/emission spectra in the following sections are calculated assuming high-frequency
where I$(O)) = h121xi)is the initial wavepacket, wl2 is the transition dipole moment, and ]xi) is the ground-state wave function and 1$(t)) = dH+#(0)) is the evolving wavepacket on the excited state, propagated by the excited state Hamiltonian, HeX(h=1) The interpretation generally associated with eq 1 is that at time t = 0, the laser transfers the wave function, I$(O)), to the excited-state surface. Since I$(O)) is not an eigenstate of the excited-state Hamiltonian, Hex,it becomes an evolving wavepacket, I $ ( t ) ) , on the excited-state surface. The absorption spectrum is then given by the Fourier transform of the autocorrelation function ($(O)I$(t)). This interpretation gives the wrong impression-that the CW laser prepares a moving wavepacket. But it is obviously not the case. This is a very useful description of the equation since it makes the task of interpreting the spectrum quite easy. A spectral line can immediately be correlated with the motion of the wavepacket along the coordinate associated with that spectral line. When we examine the equations for two-photon transitions, e.g., Raman/emission processes, it becomes clear that what the C W laser prepares is not a moving wavepacket. The Raman amplitude into final vibrational level aY is given by'-5
( 1 ) Williams, S. 0.;Imre, D. G. J . Phys. Chem. 1988, 92, 3363. (2) Williams, S. 0.;Imre, D. G. J . Phys. Chem. 1988, 92, 3374. (3) Tannor, D. J.; Heller, E. J. J . Chem. Phys. 1982, 77, 202. (4) Heller, E. J.; Sundberg, R. L.; Tannor, D. J. Chem. Phys. Lett. 1982, 93, 586. ( 5 ) Lee, S. Y.; Heller, E. J. J . Chem. Phys. 1979, 71. (6) Rama Krishna, M . V . ; Coalson, R. Chem. P h y s . 1988, 120, 327.
where wL and oiare the laser frequency and initial ground-state energy, respectively. I@r) = p211xr), where Ixr) is the final vibrational eigenstate of the ground state. This equation can be written in the form'
0022-3654/88/2092-6636$01.50/0
11. Theory and Numerical Techniques
a. The CW Limit. In this section, we present a brief summary of the time-dependent formalism for single- and two-photon spectroscopy for the case of C W excitation. Here the absorption spectrum is given
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6637
Time Evolution of Single- and Two-Photon Processes
pE(O)l$,,(O)) is suddenly promoted to the excited state where it is propagated by the excited-state Hamiltonian. Another interesting limit to consider is a pulse which turns on a t t = 0 and remains constant for all times. In that case, eq 6 is given by
NOWwe define the Raman wave function,
t )) = C L -P'e-"..t
which is a simple Franck-Condon overlap Then air a ( between two vibrational wave functions. Thus the interpretation of this equation suggests that the first photon prepares the state which is quite different from a moving wavepacket. In fact, IqR)is time independent.' Equation 3 shows that the way this state is created is by the laser "transforming" the moving wavepacket at the frequency w (w = wL *),thus killing all frequency components of 1c$(t))and leaving only one-the one which is on-resonance with the laser, creating a time-independent wave function. The relation between this wave function and the moving wavepacket is clear. I+R) is nothing more than one of many frequency components encompassed by I $ ( t ) ) . Thus, I$R) cannot have any amplitude in regions which I $ ( t ) ) does not visit. As we have shown in ref 1, despite the fact that the Raman wave function is time independent, it is not an eigenfunction of the excited-state Hamiltonian. When we discuss pulsed excitation, we will present the generation of the Raman wave function by the laser from a slightly different point of view but with similar end results. b. Pulsed Laser. We now turn to excitation with a pulsed laser. We consider a model diatomic-like system with two one-dimensional potentials representing the ground state, HBr,and the excited state, Hex. The ground state is taken to be harmonic, while the excited state is either a Morse or harmonic potential for the bound-to-bound transition or an exponential for the bound-free transition. The two surfaces are coupled by a transition dipole operator &t), where I.L is the transition moment and E ( t ) is the electric field. In this case, E ( t ) is a pulse of light with central frequency w and pulse shape A ( t ) , so E ( t ) = eiurA(t).The radiation field is treated classically. In our studies, the system starts in the ground electronic state in u" = 0 ($gr) and there is zero amplitude in the excited state ($ex = 0). The field couples the two surfaces and transfers amplitude from the ground to the excited state and also from the excited state back to the ground state. We ensure that at all times the probability of being on the excited state ( $exl$ex)5 0.01 to avoid saturation effects. The Hamiltonian for this system is then a 2 X 2 matrix of operators.
+
(4)
The time-dependent Schradinger equation is then ( h = 1)
At t = 0, is an eigenstate of Hgrand $, = 0. Since our main interest is in observing the excited-state wave function in time, we can transform eq 5 into two integral equations Provided we are in the weak-field limit, which for +ex and is equivalent to first-order perturbation6,' theory, then
and I$gr(t)
= e'"'Wgr(0))
To interpret this equation, it is beneficial to examine two cases. For a &function pulse in time, eq 6 reduces to I+cx(t)
= (-i/h)e-'~e"PE(o)llCg~(o))
(7)
The physical picture that eq 7 presents is exactly analogous to the interpretation we impose on eq 1, the C W limit absorption spectrum equation, The modified wavepacket Id(0)) = ~~
(7) Tannor, D. J.; Kosloff, R.; Rice, S.A. J. Chem. Phys. 1986,85, 5805.
Pl$gr(t))
dt
(8)
We can choose the ground-state energy, wo = 0. Then (9)
which is equivalent to eq 3 for the Raman wave function in the C W limit. Note the difference between eq 7 and 9; in the former, the ground-state wave function is promoted once, whereas in the latter a small amplitude of the ground-state wave function is continuously arriving on the excited state as long as the laser is on. Each piece is "labeled" by a phase factor contained in E ( t ) , added to the already existing and moving l$U(t)), and propagated by Hex. Note that, as each piece arrives, it can constructively or destructively add to I$=(f)). This is the mechanism by which the laser projects out of I $ ( t ) ) one frequency component. To obtain the Raman spectrum, the overlaps between the excited-state wave function and all the ground-state eigenstates are required; the time-dependent Raman amplitude, afi,into final state I$f), is then given by afi(t)= ($d$ex(t))
(10)
Since the excited-state wave function is time dependent, the Raman amplitudes will also be time dependent. The time evolution is on the order of a fraction of a vibrational period; thus to be able to observe it requires a pulse much shorter than a vibrational period. The total Raman intensity into a final state I&) is {if a Jm~afi(t)l2
dt
(1 1)
According to eq 10, the Raman spectrum at any time, t', is a projection of the wavepacket l$ex(t)) onto ground-state eigenfunctions. A short pulse will create a localized wavepacket whose dynamics can be followed by obtaining Raman spectra at different delay times between the excitation and acquisition of the spectrum. The shorter the exciting pulse, the more localized Iqcx(t))is. Obviously for very short pulses, the Raman spectrum will be broadened according to the uncertainty principle, and we would lose the information we are after. Instead we can devise a scheme by which both time and frequency resolution are retained. Consider a pump pulse on the order of 2 fs. This pulse is used to prepare I$cx(t)). A second pulse of comparable width delayed by time T can then be used to transfer the molecule back to the ground state. If that pulse is a &function in time, it will transfer the complete I$=(t)) to the ground state. A third high-resolution long-pulse laser (probe laser) can now be used to map out the population in the ground-state vibrational levels by laser-induced fluorescence. I$ex(t)) can be written in the form IP$ex(t>)
=
CCfl4Jf) f
(12)
This is always true since the state form a complete set. As long as IqU(f))is on the excited state, the coefficients are time dependent. Once the second pulse transferred I&x(t)) back to the ground state, the coefficients ICf( become time independent. The population in each state I&) is proportional to IC# which is the Raman intensity for the peak corresponding to the state I&). By recording these types of spectra at different delays, a direct map of the time evolution of Iw$ex(r)) on the excited state can be obtained. In the following sections, the Raman spectra are calculated assuming such an experiment was performed. To simulate the pulsed excitation, we solve the time-dependent Schrodinger equation (eq 5 ) using modified FlT grid methods.*q9 Briefly, the wave functions and the potentials are discretized on (8) Kosloff, D.; Kosloff, R. J . Compur. Phys. 1983, 52, 35. (9) Kosloff, R.; Kosloff, D. J . Chem. Phys. 1983, 79, 1823.
Williams and Imre
6638 The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 TABLE I: Parameters Used in Calculations (in atomic units)
1
Laser Pulse P = I x 105 A = I x 105 10
1
1
= 200 (-5 fs) disp
disp
disp
diap
disp
Potential Surfaces harmonic oscillator repulsive excited state Morse excited state
t
O.O189(x - xo)2
ground state 1.34 excited state
xo = 0 xo =
0.038e4.529(”) 0.04556[1 - e0.644(x4.7) I
a grid. The propagation depends on the grid size and the time step. To check for convergence, we halve the grid size and time step. For these calculations, the transition moment, p, is assumed to be constant. Note, since the results are obtained by integrating the Schrijdinger equation, the results are accurate to infinite order in perturbation theory. 111. Long-Pulse Excitation: A ( t ) Turned On at t = 0 and
Remains On In an effort to make a somewhat smooth transition from the C W limit, we will first present results for a laser source that is turned “on” at t = 0, rises smoothly to a maximum in 5 fs, and stays “on” for all times. The parameters for A ( t ) are shown in Table I (a). a. Repulsive Excited State. For these calculations, a harmonic oscillator ground state with wo = 1000 cm-l is used, and the excited state is a simple exponential of the form e-n(x-xo). As the laser is turned on, “pieces” of the initial wavepacket are transferred to the excited-state surface where they evolve in time. To study how the wavepacket I$ex(t))moves on the excited state, we take “snapshots” of l$ex(t)).These are shown in Figure 1. In Figure l a we present results for laser excitation on resonance with the excited-state potential, wL = ( $olHexl$o). Figure lb-d presents results for detuning by Aw = 4500, 9000, and 13 500 cm-I, respectively. The first four of the five frames shown are for snapshots taken at 8-fs intervals. The fifth frame shows I$,(t)) on the excited state after long times ( t > 100 fs); no change in $, is observed after this time. Figure l a (laser central frequency on resonance) shows that, as the laser is turned on, the ground-state wave function is transferred to the excited state, since p is constant, without any change. It is continuously arriving at the same position (zero displacement). The pieces which have arrived early are propagated toward large displacement while new pieces keep arriving. Eventually I$ex(t)) covers the entire region from zero displacement to infinity. When the laser is tuned away from resonance (Figure lb-d), a very different behavior is observed. For short times, aside from a decrease in amplitude, the shape of the excited-state wave function is very much the same as that in Figure la. For longer times, we find that only a small fraction of the wave function manages to escape from the Franckcondon region. There is small transient effect induced by turning on the laser. At long times, the majority of l$ex(r)) is localized in the F-C region. What appears to be happening is that, as the initial pieces arrive, they are propagated by Hex. Since at t = 0, $ex = 0, these pieces cannot destructively interfere with the excited-state wave function, and thus they manage to escape. However, the later pieces arrive out of phase with already existing ones, causing destructive interference, and return to the ground state before they manage to escape from the F-C region. The larger the detuning frequency, the faster the destructive interference occurs. This phenomenon will be further illustrated in the bound case. In the limit that the laser is on for a long time, we should recover the C W limit results.’ Figure 2 shows a comparison between the Raman wave function obtained assuming a C W laser, eq 3 and
-1
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Figure 1. Evolving wavepacket on the repulsive excited state for (a) resonant excitation, and for detuning by (b) 4500, (c) 9000, and (d) 13 500 cm-’, respectively. These results are for a pulse that is turned on at t = 0 and remains on for all times. The first four frames were taken at 8-fs intervals, while the last frame represents I h ( t ) ) after very long times ( t >> 100 fs).
the last frames in Figure 1, demonstrating that indeed the two are identical. In Figure 3 we follow the time evolution of the emission spectrum. The snapshots were taken at the same time as shown in Figure 1. Again, results for an on-resonance excitation are shown in Figure 3a, with detuning by 4500,9000, and 13 500 cm-’ presented in Figure 3b, c and d, respectively. When the laser is tuned on-resonance with the excited state, the spectrum develops in a simple sequential manner, with higher overtones developing intensity later in time. Again, as we detune from resonance, we see remnants of the transient effect showing up in the emission spectrum. The further off-resonance we tune the laser, the less converges faster to the visible the transient effect, since Raman wave function.’ A summary of the results on the repulsive excited state is presented in Figure 4 which shows how the population develops on the excited state and also shows how the intensity into the ground-state vibrational eigenstates u” = 1 (-), u” = 2 (- - -), and u” 3 changes with time. Figure 4a illustrates that for resonant excitation population is constantly transferred to the excited state, so that there is a steady, continuous increase in population with time. As we detune from resonance, a smaller population is transferred to the excited surface (e.g., a factor of 30 less for detuning by 4500 cm-‘). When the laser is off-resonance, the population initially follows the laser pulse and shows an increase in intensity up to approximately 5 fs (the maximum pulse power). Since the “piece” transferred at time t is out of phase with the “piece” transferred earlier at time t - At, the population decreases. The oscillations in the population are due to the oscillatory nature of elAwt;that is, the population oscillates at the detuning frequency Aw. The greater the detuning frequency, the higher the frequency of oscillations in the population. While these oscillations do not represent the normal saturation Rabi frequency since we are in the weak-field limit, they can be understood by examining the formalism leading up to the Rabi frequency.’&l2 In this formalism there are two contributions to (.-a)
Time Evolution of Single- and Two-Photon Processes
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6639
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Figure 3. Time development of the emission spectrum for (a) resonant excitation, and for detuning by (b) 4500, (e) 9000, and (d) 13 500 cm-', respectively. Note here that each frame shows a normalized emission spectrum. The pulse shape and times for each frame are the same as in Figure 1. Note that intensity into v"= 0 (Rayleigh line) is not shown in these plots.
why is there population on the excited-state surface when the laser is off-resonance? We can address this question by writing eq 6 as a simple expansion dLap
diap
>> 100 fs calculated by using eq 6 and (ii) the Raman wave function, I$R), calculated by using eq 3 for (a) resonant Figure 2. (i) lt/ex(t))for t
excitation, and for detuning by (b) 4500, (e) 9000, and (d) 13 500 em-'.
oscillation in population-a laser power dependent term, which in our case is very small (weak field), and a term dependent on the amount of detuning, Aw. It is this second term- which determines the oscillations in the population, in Figure 4. We can get further insight into the time development of the emission spectrum by studying the time development of the overlaps with the ground-state vibrational eigenstates. The second set of plots in Figure 4 shows that emission into u" = 1 develops before emission into u" = 2. In fact, the intensity develops in a simple sequential order with intensity into higher overtones developing later in time. Note also that after approximately 20 fs, the overlaps and indeed the emission spectrum has converged as any remnants of the transient effect has disappeared. If we take a cut at any time across these overlaps, we can determine the emission spectrum at that time, Clearly, the long-time results show that the ratio of intensity into u" = 1 to u" = 2 increases as we detune from resonance. Therefore, not only will the signal obtained in the emission spectrum decrease as we detune further off-resonance, but there would be fewer overtones in the emission spectrum as we tune further off-resonance. We have shown how the population and spectrum develop with time, One of the intriguing questions we have yet to address is (10) Sargeant, M.; Scully, M. 0.;Lambo, W. E. Laser Physics; Addison-Wesley: Reading, MA, 1974. (1 1) Allen, L.; Eberly, J. H.Optical Resononce and Two-Leuel Atoms; Wiley: New York, 1975. (12) Loudon, R. The Quantum Theory ofLight; Oxford University Press: London, 1973.
where I$(O)) = L L I + ~ ~ ( O )and ) 14(t)) = e - i H + # J ( 0 )Also, ). w = wi + WL, where q is the initial ground-state energy and wL is the laser so that = wL - (Ic$IHcxIIc$) represents the amount Of detuning* According to eq 13, even for large detuning frequencies, Po, there would be amplitude on the excited-state potential surface as long as A ( t ) is not zero. It also says that the larger the detuning frequency, Aw, the shorter time (teff),it takes before e ' b f a b m e s a rapidly oscillating function, and hence the shorter effective time + e x ( t ) spends On the excited-state surface. That is, for times t t e f f there would be no net increase in contribution to +eAt), since population transferred to the excited state at teff undergo phase cancellation. The expansion in eq 6 makes it clear what we mean by pieces of the ground-state wave function; each term in the sum is one Of these pieces. Note each term involves the ground-state wavefunction labeled by a phase factor determined by Aw, the detuning frequency. Thus what we mean by a piece Of the wave function does not refer to Some part Of +gr in coordinate space* For a constant transition moment, $8r is transferred without any change. b. Bound Excited State. ( i ) Harmonic Excited State. In this case, both the ground- and excited-state potential surfaces were harmonic oscillators with wg = 1000 cm-l. The excited state was from the ground state by 0.67 A so that the F-c excitation is to v'= 7 .
'
'
6640
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988
POPULRTIONlon
e x , etatol
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1
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Figure 4. Time development of the population on the excited state for excitation (a) on-resonance and for detuning by (b) 4500, (c) 9000, and (d) 13 500 cm-I. The second plot (to the right) shows how the overlap with ground-state eigenstatesd’=1 (-), u ” = 2 (---),and u ” = 3 develop in time for the same frequencies mentioned above. (-a*)
In Figure 5a, we show snapshots of for a resonant excitation to u’ = 7. These snapshots were taken at 7.5-fs intervals. Unlike the repulsive excited state, the pieces transferred at early times now have the opportunity to revisit the F-C region, and interact with newly arriving pieces. As the laser pulse increases in intensity, larger pieces of the ground-state wave function are transferred to the bound excited state. These pieces develop momentum on the excited state and move away from the F-C region. The laser acts as a filter and with time projects out u’ = 7 from the evolving wavepackets on the excited-state surface. Indeed the figure shows that at long times, the time-dependent wave function converges to u’ = 7. When the laser is detuned from resonance with a discrete excited state, very different results are obtained. Results for excitation to u’ = 6.5 (Le., halfway between u’ = 7 and u’ = 6) are shown in Figure Sb. At early times, the behavior of I$&)) is very similar to the resonant case presented in Figure 5a. Note, however, that in this case a smaller amplitude is transferred to the excited state and after approximately one period I$ex(t)) looks very much like a linear combination of u‘ = 7 and u‘ = 6 . After one period (at about the fifth frame), the amplitude of I$,(t)) begins to decrease until the population on the excited state totally disappears. At that point, the laser is still on, and thus it will transfer population to the excited state. Since there is no amplitude on the excited state to interfere with the newly arriving pieces, population will remain on the excited state and a new cycle will begin. This perfect periodic cancellation of IqeX(t))can occur only in harmonic potentials where the dynamics is perfectly periodic. A similar set of results for excitation to u’ = 0 and for excitation ‘ / 2 h w below u ’ = 0 (Le., to bottom of well) are presented in Figure 5 , c and d, respectively. We expect I$,,(t)) to converge to u ‘ = 0 when the laser is tuned to on-resonance with u’ = 0. Note, however, that since the F-C factor for u ’ = 0 is small compared to, say, u’ = 7, it will take a much longer time to produce v’ =
Williams and Imre 0 than it has for u’ = 7. Therefore, for the time shown (approximately three vibrational periods), does not converge to u’ = 0. When the laser is 1/2 hw below u’ = 0 (Figure 5d), we see two distinct components in I $ e x ( t ) ) . One component is localized at zero displacement and represents the piece of the initial ground-state wave function transferred at that instant. The second component at larger displacement represents pieces of the initial ground state that were transferred to the excited state at earlier times and propagated by the excited-state Hamiltonian. Note that this second component is totally dephased after every vibrational period. Figure 6 illustrates how the emission spectra evolve for the system described above. In Figure 6a, the results are for excitation resonant to u’ = 7, while Figure 6b shows results for an excitation between u’ = 6 and u’ = 7. The frames shown here were taken at identical times with those presented in Figure 5 (7.5 fs). When the laser is tuned to u’ = 7, the emission spectrum grows in intensity with time, evolving from a spectrum which at short times is of the form of a typical Raman spectrum, to a fluorescence spectrum. The transition from Raman to fluorescence can be seen much clearer in Figure 6b; however, here the laser is not onresonance with any excited eigenstate and as a result both l$ex(t)) and the spectrum begins to dephase after each vibrational period. Thus not only is there less amplitude transferred in this case, but the spectrum oscillates from Raman to fluorescence, back to Raman, etc. The development of the emission spectrum for excitation to u’ = 0 and for excitation hw below u ’ = 0 are shown in Figure 6, c and d, respectively. As with results for Ifim(t)),for excitation to v’ = 0, Figure 6c shows two contributions to the emission spectrum. The smooth envelope at higher overtones is due to the laser projecting out v’ = 0 on the excited state, while most of the intensity into u’’ = 1, 2, and 3 is due to the pieces of the ground state transferred at that instant (i.e., the short-time dynamics). Note that, as time increases, Ifiex(t))will converge to u’ = 0 and the effect of the short-time dynamics (i.e., intensity into u” = 1 and 2) will become negligible. When the laser is tuned to the bottom of the potential well (Figure 6d), the spectrum displays the oscillatory nature depicted by i$cx(t)) (Figure 5d). In this case, there is always a large Raman component in the emission spectrum which says that the emission spectrum is dominated by short-time dynamics. Figure 7 provides a summary for the bound excited-state case. It shows the steady increase in the population for the resonant excitation to u’ = 7 (Figure 7a) along with the oscillatory nature in the total population on the excited state for excitation to u’ = 6.5 (Figure 7b). Note here that the population dephases completely to zero because the laser is exactly halfway between u’ = 6 and u’ = 7. For excitation to, say, u‘ = 6.7, the population would not dephase completely to zero. The second plot shows how the Raman intensity into ground-state wave functions u” = 1 (-), v ” = 2 (---), and u ” = 3 changes with time. For the case resonant to u’ = 7, the overlaps show a rather steep increase after every vibrational period; that is, the intensity deposited into u” = 2 or u”= 3 is greatest after each period. Since the excited-state potential is bound, the overlaps, and indeed the emission spectrum, will be very sensitive to laser frequency.’~’~As expected, for excitation to v’ = 6.5, the overlaps with the ground-state wave functions also show the oscillatory behavior depicted by the total population. The overlaps show a steady increase as the laser power increases and larger pieces of the ground state are transferred to the excited state. As these pieces move away from the F-C region, they do not add any appreciable intensity to u“ = 1, 2, and 3 and hence the flat portion in plots. After they are reflected off the outer wall, they proceed toward the F-C region where they again develop overlap with the ground-state wave functions. This is shown by the rapid increase in u” = 1 and 2 (or rapid decrease in u” = 3). After exactly one period, the new pieces transferred to the excited state by the laser are out of phase with those already there and begin to interfere destructively such that, after the second (.-a)
(13) Rousseau, D.L.;Williams, P. F. J . Chem. Phys. 1976, 64, 3519.
Time Evolution of Single- and Two-Photon Processes
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The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6641
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frame represents 7.5 fs. period, I q e X ( t ) has ) totally dephased. Since there is nothing left on the excited state to interfere with newly promoted pieces, the cycle starts all over again. Parts c and d of Figure 7 show how the population develops on the excited state and also show how intensity into ground-state hw wave functions evolve for laser excitation to u’= 0 and to below u’ = 0, respectively. For excitation to u’ = 0, the population profile shows a relatively steady increase with time. There is a
small amount of dephasing after each vibrational period since the wavepacket on the excited state returning to the F-C region is slightly out of phase with that transferred to the excited state at that instant. Another way of explaining this is to realize that the only time the laser is truly on-resonance is when it is tuned to u’ = 7, since the F-C excitation (or the energy on the excited state at zero displacement with respect to the ground-state potential minimum) coincides with the energy to u’ = 7. When we tune
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off-resonance to, say, the bottom of the excited-state potential well (Figure 7d), we observe a very rapid dephasing in the population after each vibrational period and in fact the population does not show an overall increase with time but is oscillatory. (ii) Morse Excited State. In an effort to investigate the new features introduced by an anharmonic potential, we perform similar studies on a Morse excited-state potential with a dissociation limit of 10000 cm-l and with 20 bound state (wo = 1000 cm-I). The Morse potential was displaced from the ground-state potential by -0.35 8, so that the F-C excitation was to u ’ = 2.
Figure 8 shows changes in the population on the excited state for laser excitation resonant to v’ = 2 and laser excitation to v’ = 7.5. When the laser is resonant to v’ = 2, the population on the excited state continues to increase as a large component of the initial ground-state wave function transferred to the excited state remains in phase and, hence, interfere constructively with the population transferred at earlier times, and the state v’ = 2 is created. When the laser is tuned between vibrational levels, much less population is transferred to the excited state. In Figure 8b, where the excitation is to v’= 7.5, the amplitude on the excited
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The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6643
Time Evolution of Single- and Two-Photon Processes
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The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6647
VI. Conclusion We have utilized a time-dependent approach to show how both the population on the excited state and the emission spectrum evolve in time for two different laser pulses. This study was designed to complement a previous study’ which presented results for C W excitation source. Here we have demonstrated that l $ e x ( t ) ) converges to the Raman wave function for a long pulse (in time). We have also been able to show much more explicitly that large detuning from resonance is identical with observing short-time dynamics. For a bound excited state, the emission spectrum grows from one that would typically be described as Raman to a fluorescence spectrum. The experimental consequences of using pulsed excitations are significant. Clearly, with a sufficiently narrow gate, one can experimentally obtain very different results in emission by placing the gate at different points after the excitation pulse. However, some of the apparent drawbacks from using short-pulse excitation (mainly a consequence of the large uncertainty in energy) are also evident from these calculations. Recently, numerous experiments using femtosecond and picosecond laser pulses to observe real time dynamics in ICN and NaI have been reported by Zewail et al.l5 The limitations involved in time-resolving molecular vibrations and other motions are obvious. With a subfemtosecond pulse, the uncertainty broadening is such that one cannot readily excite and monitor the “interaction” in a well-defined way. Even though we are limited by the time-energy uncertainty principle, we were able to numerically show discrete Raman intensities into final ground-state wave functions. This is not a contradiction. As explained earlier, the results provided in this study can be obtained experimentally by introducing a second laser (CW). We can use this second laser to verify the population of the ground-state vibrational levels by projecting these levels onto another (bound) electronic surface. In effect, the second laser will project the population of each of the ground-state vibrational levels onto eigenstates (as was done numerically here). As short laser pulses become more readily available, and since we are always searching for better ways to characterize the motion of atomic nuclei and study reaction dynamics, this time-dependent approach will continue to be a growing area of research. In a future paper,I6 we will present in depth calculations that simulate recent results15 for ICN where the first account of experimentally observing real time dynamics in molecules was reported. Acknowledgment. This work has been supported by NSF Grant CHE-8507168 and by the donors of the Petroleum Research Fund, administered by the American Chemical Society. We thank Jinzhong Zhang, Sandra Tang, Professor Carl Salter, and Professor Eric Heller and his group for helpful suggestions, and David Tannor for introducing us to the grid method used in these calculations. (15) (a) Scherer, N. F.; Knee, J. L.; Smith, D. D.; Zewail, A. H. J . Phys. Chem. 1985,89,5141. (b) Dantus, M.; Rosker, M. J.; Zewail, A. H. J . Chem. Phys. 1987,87,2395. (c) Rosker, M. J.; Rose, T.; Zewail, A. H. Chem. Phys. Lett. 1988, 146, 175. Rose, T.; Rosker, M. J.; Zewail, A. H. J. Chem. Phys. 1988, 88, 6672. (16) Williams, S. 0.; Imre, D. G. J. Phys. Chem., following paper in this issue.
