Faraday Discuss. Chem. SOC., 1991, 91, 249-258 Coherence and Transients in Photodissociation with Short Pulses Horia Metiu Department of Chemistry and Physics, University of California, Santa Barbara, CA 93106, USA The time evolution of the product energy distribution and product state coherence following the one photon excitation of a dissociating state of a molecule with one or two short pulses is discussed qualitatively and illustrated with exact quantum calculations. In this paper the opportunity is taken to present some simple ideas in a very qualitative way. Two general questions are addressed: what can we do with short pulses that cannot be done with long ones? and which cw experiments most resemble those performed with short pulses? Fourier-transform theory shows that a short pulse is a linear superposition of cw beams.The probability that the pulse acting on a molecule induces a transition from a state i to a state f is proportional to Here cri -. is the cw cross-section for the transition i -+f, f(o) is the Fourier transform of the electric field of the pulse and ofi is the transition frequency from the initial state i to the final state f. In a cw experiment f(ofi) is replaced by S(o -of), where o is the laser frequency. Thus, eqn. (1) suggests that a short pulse is just a low-resolution laser, which excites more than one final state. Of course saying this misses a most important point: eqn. (1) is valid only at a sufficiently long time r,, after the pulse started acting on the molecule. Many of the things that make the use of short pulses worthwhile happen at t < T,,; moreover, the ability of simultaneously exciting more than one state allows us to enhance the information regarding the excited-state dynamics.1. Transient Final-state Distribution Let us consider a molecule that predissociates, after being excited by a laser pulse, to form the atom A and the molecule B. The state of the fragments is described by (k, n), where Ak is their relative momentum (in the centre-of-mass system) and n labels the internal states of B. What would one see if one measured the product distribution before r,,? We can solve the time-dependent Schrodinger equation and monitor the wavefunction in the asymptotic region (i.e. where the interaction between fragments is negligible) at discrete, regular and small time intervals t,, , A = 0, 1,2, .. . ; to is the time when the laser and the molecule begin interacting. At t l there is no product wavefunction in the asymptotic region; the pulse has created an excited-state wavefunction but the molecule did not have time to dissociate and it still lingers in the ‘transition state’ (to qualify as a topic of discussion at this meeting). At t2 a small wavepacket x ( R , r; t 2 ) appears in the asymptotic region; ( x ( R , r ; t,)12 is the probability that the molecule has dissociated at some time between t l and t Z . Here r represents the internal coordinates of B and R is 249250 Coherence and Transients in Photodissociation the distance between A and B. We can expand this packet in the basis set lk, n), which is appropriate for the asymptotic region, and obtain la,(k; t2)I2 is the probability that at t2 the fragments have relative momentum Zlk and B is in the internal state n.If we look at the product energy very early ( i e . at t 2 ) , the distance R between the fragments A and B is known fairly accurately and is close to the distance Ro, where the fragments stop interacting with a small uncertainty proportional to ( t2 - tl). There- fore, the momentum uncertainty in x ( R , r; t 2 ) is considerable. This means that, for every n, lar,(k; f2)I2 has a large width as a function of k. The width of Ia,(k; t2)I2 (at given n) can exceed all the energy scales in the system: the widths of the predissociation resonances (in the absorption cross-section) excited by the pulse, as well as the width of the pulse in the frequency domain.Energy conservation is ‘violated’. The energy distribution Ia,(k; t2)I2 is very different from eqn. ( l ) , which has peaks for all the predissociation resonances excited by the pulse. How does quantum mechanics go from the shapeless structure of the early measure- ment to the structure present when the measurement is late? Let us imagine again that we watch the asymptotic region as the time-dependent Schrodinger equation is being solved. At every time t A a new wavepacket x ( R , r; t A ) appears in the asymptotic region. It represents the probability amplitude that the molecule predissociated in the time interval between tA-* and t A . Since we refrain from performing any measurement until the time t > t,, we do not know when the predissociation took place.Because of this, to obtain the wavefunction at the time t when the measurement is made, we must add the amplitudes representing all possible predissociation times: The packet Ix, t A ) is the amplitude for the predissociation at tA and the operator U ( t - ?A) propagates the packet from tA to t. Since we call (R, r l x , t A ) = x ( R , r; tA) wavepackets, we call (R, r I $, t ) = $(R, r ; t) a wavetrain. If t exceeds the dissociation lifetime all wavepackets have reached the asymptotic region and the construction of the wavetrain, by the predissociation process, is completed. From now on the time dependence of the train is trivial: it shows that the fragments fly further and further apart from each other; since no forces act on them in the asymptotic region the wavetrain no longer changes its energy distribution; this means that (n, k I $, t) becomes independent of time.In fact one can show that (n, k I $, t ) 2 gives eqn. (1). The transition from a featureless energy distribution at early time to one rich in resonance peaks at later time takes place through the interference between the packets composing the train. To see how this comes about we use (4) U ( t - tA ) I k, n ) = exp { -i ( t - t, ) E ( k, n ) } 1 k, rz ) together with eqn. (2) and (3) to write (k, n l + , t)=Cexp{-i(t-fA)~(k, n ) } a , ( k ; t,)=exp{-itE(k, n)}a,(k; t ) ( 5 ) A with a,(k; t ) = C exp{+itAE(k, n ) } a , ( k ; tA). A Here E(k, n ) = [ ( h k ) 2 / 2 r n + ~ , ] / h and E,, is the internal energy of B.Eqn. (6) shows that the amplitude a,(k; t ) of the wavetrain in the energy representation at time t, is a coherent superposition of the wavepacket amplitudes ar,(k; t h ) , each weighted by aH. Metiu 25 1 c, .* P 4 -2 1200 fs 1300 fs I ( x 11 loo fs I 300 fs I cx 21 2.4 2.6 2.8 2.6 2.8 energy/ eV Fig. 1 The time evolution of the final-state distribution lank(t)12 at 100, 200, 300 and 1300fs, respectively, after the laser pulse started acting on the molecule. (-) n = 0, (- - -) n = 1. The curves corresponding to the vibrational states n = 0 and n = 1 are shown as a function of total energy E = fi2k2/2p+ en. Different scales are used, as indicated. The laser puke has an FWHM of 30 fs. The peak frequency 2.5523 eV is tuned to excite the (1,l) resonance in the absorption cross-section phase factor exp {+itAE(k, n ) } .The probability that at time t the fragments have the cnergy ~ ( k , n) is W k ; t) = I(k n I *, ?>I2 = l a n ( k 111' = C Ia,(k; tA)l*++C +C exp {-i(tp-tA)~(k, n ) ) a , ( k ; t,i>*a,(k, tp)+cc (7) A A P where cc means the complex conjugate of the preceding term. The single sum (the incoherent contribution) adds, as in classical physics, the probabilities that each wavepacket has the energy E ( k, n). Since each term la,( k ; ?,)I2 has a broad, featureless dependence on k, the incoherent contribution to the product energy distribution is broad and featureless. The structure in the distribution must be built by the double sums (the coherent contributions), which represent quantum interfer- ence.These terms must diminish the incoherent one at some values of k and add to it at others. The time t seems to have disappeared from eqn. (6), but that is not true: t determines the number of packets included in the wavetrain, since th must be less than t. The more packets the train has, the finer the interference pattern can be. En el and Metiu' have calculated the product state distribution for a two-dimensional model of the CH30N0 photodissociation into CH30 and NO. CH30 was treated as an atom and NO was allowed to vibrate, but not to rotate. The absorption cross-section for the model has sharp predissociation resonances* whose lifetimes are of the order of 1 ps. The molecule was excited by a Gaussian laser pulse with an FWHM of 30fs, whose peak frequency was tuned to excite the ( 1 , l ) resonance2 in the absorption cross-section.The pulse width, in the frequency domain, is sufficiently large to excite the ( 1 , O ) and the (2,O) resonances. Fig. 1 shows the product-state distribution I(k, n I $, t)12 = la,(k; ?)I2 at several times t after the pump pulse started acting on the molecule. When t is a hundred femtoseconds F252 Coherence and Transients in Photodissociation the product-state distribution is broad and featureless. After 200 fs the peaks are already defined and they are roughly at the energies corresponding to a long-time cw measure- ment;2 the peak heights at 300fs still differ from the final ones, which are established at 1300 fs. The distribution at 1300 fs is exactly that given by eqn.(1). 2. Coherence of the Product Wavefunction The most obvious way of detecting the transient final-state distribution is a pump-probe (PP) experiment that excites the product^.^ Let us assume first, for simplicity, that the probe pulse is extremely short and excites the fragment B to an electronic state B*. The states of the fragments after the excitation are denoted by Ik,, n,); hkf is the relative momentum and nf the internal state of B". Let us further assume that we measure how much light is emitted by a specific state n, of B". If the probe pulse was fired at the time t after the pump, the amplitude giving this signal is proportional to The matrix element (kf, nfl k, n ) satisfies the propensity rule (kf, nfl k, 4 = m - kf)(nfl n> (9) where (nfl n ) is the Franck-Condon factor for the transition from the internal state In) of B to the internal state Inf) of B".The physics behind the propensity rule is simple. The photon momentum being practically zero the excitation by the probe cannot change the relative momentum of the fragments, therefore, kf must be practically equal to k. Using eqn. (5) and (9) in eqn. (8) gives The probability that B" is in the state n f regardless of the final relative momentum kf is P b f ; t ) = dkfl(kf, %I+, t>12 n n m I =C I(nfln>12Pn(t)+CC (nfln>(mInf) exp{--it(~n -Ern))On,(t)+CC ( 1 1 ) The first term in eqn. (1 1) is the sum of the probabilities r that at time t the fragment B is in the state In), multiplied with the Franck-Condon factors I(n,l n)12. We must sum over all states n because we have used a delta function probe pulse; otherwise, the number of states n included in the sum will depend on the width of the probe pulse.The time dependence of this incoherent term is that of the product distributions Pn ( t), which we discussed earlier. The double sums in eqn. (1 1) represent coherent contributions. They oscillate as a function of the delay time of the probe, with the frequencies ( E, - E,) determined by the gaps between the internal states of B. One interesting feature of the coherent contribution to the signal is the presence of the overlap integral o n m ( t ) dkf an(kf; t)am(k,; t)* (13) I If a,( k,; t ) and a,( k,; t ) * do not overlap, as functions of k f , then the states In) and }m)H. Metiu 253 30 fs I- 0 200 400 t/fs Fig.2 The pump-probe LIF signal for the case when a &function probe excites the photodissoci- ation products. The pump FWHM is 10, 20 and 30 fs and the peak frequency is 2.5523 eV. The state populated by the pump is repulsive and the absorption cross-section is broad of B cannot contribute coherently to the LIF-PP signal; they cannot pair up to give an oscillating contribution (i.e. beats) to the signal. More precisely, if the dissociation dynamics is such that both ‘reactions’ AB -+ A+ B( n) and AB --* A+ B( rn) take place, the probe cannot excite coherently B(n) and B(m) unless there is a finite probability that A+ B( n) and A+ B( rn) have the same relative velocity. This translational overlap condition is a propensity rule because its derivation used eqn. (9).To test the rule we have performed4 exact calculations on a two-dimensional system that dissociates into A and NO. The NO molecule is allowed to vibrate but not to rotate. The excited state potential is repulsive. The probe excites NO to NO* and the total emission from one state of NO* is plotted as a function of the delay time between the pump and the probe (Fig. 2). The oscillations characteristic of coherence are seen for the shorter pulses only. In Fig. 3 we show the final-state distributions la,( kf, ?)I’ for the excitation by the three pulses. We note that all three pulses excite coherently the vibrational states n = 0, n = 1 and n = 2 of NO. However, coherent beats are not seen in the LIF signal of the longest pump pulse, because the overlap integral is zero. Furthermore, when beats are seen (for the shorter pump pulses), the frequency E ~ - E ~ is not present in the signal because the overlap integral 0,, is zero.This is why even though three internal degrees of freedom are excited coherently we observe only two level beats. Exact calculations4 with the two-dimensional model of CH,ONO predissociation lead to similar conclusions. We note that, as discussed earlier, the final state distribution varies in time and its dependence on momentum is broader at early times. Thus, we expect that the overlap integral On, ( t ) becomes smaller in time. Since On, ( t ) modulates the exponential time dependence in eqn. ( l l ) , the decay in the amplitude of the coherent signal in Fig. 3 reflects the decay of the overlap integral.The coherent signal due to the rotational states is simpler’ because of the selection rules for the excitation by the probe. If we monitor emission from the state n f = j of B* then this state can be excited by absorption from either the state j - 1 of B or from j + 1. This means that the emission from n, will show two level beats. Exact calculations with a simplified model of ICN,’show that this is the case. Because the ICN excited state is repulsive and the rotational energy is small, the translational requirement discussed above is always satisfied for rotations.254 Coherence and Transients in Photodissociation 1 I .5 2.0 2.5 3.0 translational energy/eV n = O 1 1.5 2.0 2.5 3.0 translational energy/eV n = O I I .5 2.0 2.5 3.0 translational energy/ eV Fig.3 The product energy distribution P,, ( k ; t ) = I(k, n 1 4, t)I2 = la, ( k ; ?)I2, for the model used in Fig. 2, as a function of the relative translational energy of the fragments, for n = 0, 1 and 2 and for the pump FWHM: ( a ) 30 fs; ( b ) 20 fs and (c) 10 fs It is often assumed that the laser-induced fluorescence signal in the pump-probe experiments is proportional to the product concentration. This is obviously not true when the working conditions are such that the coherent contribution to the signal [the double sums in eqn. ( 1 l)] is not negligible. One can hardly interpret the signal oscillations in Fig. 3 to mean that some of the fragments recombine and dissociate periodically. The coherence effects influence the total PP-LIF signal whether or not the total emissionH. Metiu 255 is frequency resolved.Several exact calculations596 have provided examples where the time evolution of the PP-LIF signal is not proportional to that of the population of the wave train created by the pump. Many research groups7-” have used a pump pulse to excite coherently more than one eigenstate of a molecule, and a probe pulse to induce beats in the dependence of the LIF-PP signal on the delay time between the pump and the probe. In these the pump excited coherently molecular bound states, while we are here with the transient coherence of the wavetrain describing the products of photodissoci- ation’ or prediss~ciation.~ The connection between coherence and photodissociation has also been discussed in the context of different experimental arrangements by Brumer and Shapiro.12 3.Transients caused by Laser Excitation The above discussion was confined to the transient properties of the wavetrains created by predissociation. The same concepts can be used to examine the wavetrain prepared by the excitation of a molecule by a laser. Instead of adding coherently the wavepackets representing the probability amplitude that the molecule dissociates at the time t A , we add the wavepackets representing the probability amplitude that the molecule has adsorbed the pump photon at the time t A . At the level of abstraction used here there is hardly any difference between these processes; they are both described by eqn. (3) and only the physical interpretation given to the wavepackets and the wavetrain differs.In the case of laser excitation I$, t) is the wavetrain describing the nuclear motion induced by photon absorption on the upper electronic energy surface. The states lk, n) used in the above equations are now the nuclear eigenstates of the excited electronic state. The excited state population created by the laser is given by eqn. (7). It is worthwhile to examine this analogy closely. To do this we calculate exactly the state created by one-photon absorption from a laser pulse and monitor its evolution at discrete, evenly spaced and ‘infinitesimally’ close times t A . The pulse makes contact with the molecule for the first time at to=O. At time t , the algorithm creates a small wavepacket Ix, t , ) , on the upper electronic state, representing the probability amplitude that the photon has been adsorbed in the time interval T = t l - t o .Ix, tl) is the ground state at time tl multiplied with the electric field at t , and the transition dipole between the ground and the excited electronic states. Let us examine what happens if we make an energy measurement at t , . Since we have initiated the absorption process at to = 0 and terminated it (by performing a measurement) at t , the molecule interacts with the pulse only for a time T ; the effect of this interaction is equivalent to that of a delta-function laser pulse, whose spectrum includes all frequencies; thus, at tl all the nuclear energy eigenstates that have a non-zero Franck-Condon factor with the initial state are excited. This includes all electronic states, all photoionization states, and, to dramatize a little, even those states in which the 1s orbitals of the atoms are ionized! This excitation process is independent of laser frequency or the temporal width of the laser pulse; both parameters are made irrelevant by the shortness of the time between the beginning of the laser-molecule interaction and the energy measurement.If we have a little more patience and allow the computer to go on for three time increments the algorithm will construct a wavetrain I$, t) composed of three wavepackets, created at the times t l , t2 and t , . An energy measurement at the time t = tl + f2 + t3 = 37 interrupts the excitation process and is equivalent to an excitation of the molecule by a pulse of temporal width 37.The spectrum of this equivalent pulse is narrower than that of the pulse equivalent to a measurement performed at t , ; we may still observe that several electronic states are excited, but the 1s orbitals of the atoms are no longer ionized. The peak frequency of the equivalent pulse does matter, since the states that are excited are grouped around the energy reached by the laser frequency.256 Coherence and Transients in Photodissociation Let us emphasize how intriguing this is: the train is composed of three wavepackets; each packet represents a state in which the 1s orbital is excited, but their sum is a state in which this excitation is absent! This behaviour has a simple explanation: the wavepackets have been created at different times t A with different phases exp (--hatA), where w is the peak pulse frequency; because of nuclear dynamics, their phases change during their brief lifetimes t - t h .These phase differences cause the destructive interfer- ence that makes the energy eigenstates which differ most from the laser frequency to disappear from the wavetrain, and the constructive interference that builds up the population of the states close to the laser peak frequency. As eqn. (7) indicates, without the coherent contribution made by the double sums the probability that the off-resonance states are excited will build up with each wavepacket. The state created by the pulse, at a long time after the pulse is extinguished, depends on how many molecular eigenstates Ef satisfy the equation f( Ef - ci) = 0. Here f( o) is the power spectrum of the pulse, E~ is the initial energy of the molecule and Ef represents the energy eigenvalues of the molecule.If the equation is satisfied for one eigenvalue Ef only, then we are performing a cw experiment which prepares the pure state I&f). If it is satisfied for several eigenvalues Ef, then a coherent superposition of these energies is prepared. If no eigenvalue satisfies the condition (Le. off-resonance excitation) then in the long-time limit the excited-state population is zero. In all cases, at the early times when the pulse acts on the molecule, many nuclear and even electronic states are excited. The lifetime of these transient excitations is the time needed by interference to destroy the states that do not resonate with the laser.This is of the order A/Ae, where Ae is the smallest difference between the laser peak frequency and the optically active eigen- states that satisfy f( Ef - ei) # 0. The off -resonance transients can be maintained indefinitely by exciting the molecule with a semi-infinite ‘pulse’. This happens because it takes a time A/Ae to destroy, by interference, the wavepackets forming the wavetrain. Thus, if we make a population measurement at a time t, the packets promoted on the upper surface at a short time prior to the measurement (i.e. between t - A/Ae and t ) are still in existence. The closer the laser is to resonance (i.e. the smaller is Ae), the longer is the survival time, and the transient wavetrain contains more packets. If the upper potential surface is repulsive, the off-resonance semi-infinite pulse creates a finite wavetrain whose length can be tuned by choosing the appropriate off-resonance parameter Ae.If the pulse is suddenly extinguished this train disppears on a timescale A/Ae. During its brief life on the upper electronic surface each wavepacket has a chance to either emit a photon and give a spontaneous Raman signal, or to absorb a photon from a second laser, which leads to two-photon absorption. Thus, the transients are the intermediate states from which off -resonance Raman and two-photon absorption spectroscopy proceeds. This observation, first made and extensively exploited by Heller,I2 indicates that by tuning the pump off resonance and performing Raman or two-photon spectroscopy we can study, with long pulses, the dynamics of short wavetrains on the excited potential-energy surface. These short wavetrains are similar to those excited by ultrashort or resonance pulses.Thus one can do femtosecond chemistry with a cw set up. Experiment~l~ and exact calculations on model systems14 justified these insights of Heller. 4. Interference between Wavetrains We have discussed so far how the interference of the wavepackets forming a wavetrain affects the results of the various measurements. We can also ask whether the interference between two wavetrains might lead to something interesting. Let us monitor one-photon absorption by a molecule exposed to two laser pulses; the photon will be absorbed from one of the pulses, but we do not perform any measurement to find out from which.AsH. Metiu I 257 600 1000 1400 delay time/fs Fig. 4 The total population [eqn. (14)] created by the two laser pulses, as a function of the delay time between them. Both pulses are Gaussian, with an FWHM of 1Ofs and a peak wavelength of 328 nm. The two curves correspond to a phase difference S between pulses of S = 0 and S = T a consequence, the amplitude I+, t ) = l + , t ; 1 ) t ; 2) ( 1 4 ) describing the state of the excited molecule at the time t, is a linear superposition of the amplitudes I+, t ; 1) and I+, t ; 2) describing the probability of absorption from the pulse 1 or from the pulse 2, respectively. The excited-state population is P ( t ) = ( + , t l + , t ) = ( + , t ; l l + , t ; l)+(+,t;21+,t;2)+2Re(+,t; 1 l + , t ; 2 ) ( 1 5 ) The first two terms represent the excited-state populations PI( t ) and P2( t ) created in two separate one-pulse experiments; P( t ) # PI( t ) + P2( t ) whenever the interference term PI2( t ) = 2 Re (+, t ; 11 +, t ; 2) # 0.Because the excited molecule dissociates, the reaction yield is modified by interference; in a two-pulse experiment in which we do not determine from which pulse the photon was absorbed the yield is different from the sum of the yields obtained in two separate one-pulse experiments. We have studied quantitatively the interference process discussed above by solving exactly4 the time-dependent Schrodinger equation for an NaI molecule interacting with two laser pulses. For simplicity we took the pulses to be identical except for an overall phase factor.The peak frequency was tuned to excite the molecule, through one-photon absorption, to a state from which predissociation occurs. The model and the method of calculation were described in ref. 1 5 . In Fig. 4 we show the excited-state population created by one-photon absorption after the molecule interacted with both pulses. The abscissa is the delay time between the pulses. For delay times <900 fs and > 1100 fs the total population is equal to the sum of the populations created if each pulse acted on the molecule alone. Interference takes place only for delay times between 900 and 1100 fs. This is the time it takes the wavetrain I+, t ; 1 ) to return to the Franck-Condon region of the upper potential-energy surface.The interference takes place only when the wavepackets being created by the second pulse overlap with the wavetrain created by the first pulse.258 Coherence and Transients in Photodissociation Such train-interference experiments complement the information generated by pump-probe experiments; furthermore, if performed on a molecule whose dynamics are well understood, they could be used to study the properties of laser pulses. Obviously other multiple pulse sequences could induce interesting behaviour, as demonstrated in numerous multiple-pulse NMR experiments. The general question of the relationship between pulse shape and excitation outcome has been studied extensively by Rabitz.16 The example used here is much simpler and in particular does not require strong pulses.This work was supported by NSF and AFOSR. I express my gratitude to V. Engel, R. Heather, K. 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