Molecules in strong laser fields: vibrational inversion and harmonic generation

Transcription

1 Molecules in strong laser fields: vibrational inversion and harmonic generation ERIC E. AUBANEL AND AND& D. BANDRAUK' Laboratoire de chimie thiorique, Faculte' des sciences, Universite' de Sherbrooke, Sherbrooke, QC JIK 2R1, Carlada Received July 2, 1993 This paper is dedicated to Professor John C. Polanyi on the occasiorl of his 65th birthday ERIC E. AUBANEL and AND& D. BANDRAUK. Can. J. Chem. 72,673 (1994). We examine two consequences of the unique behaviour of molecules in strong fields. First, by time gating of laser-induced avoided crossings with femtosecond laser pulses, one can obtain'efficient vibrational inversion into a narrow distribution of vibrational levels of a molecular ion. We demonstrate this by numerical solution of the time-dependent Schrodinger equation for Hz. Second, we show results of numerical calculation with vibrationally excited Hz of harmonic generation up to the 11 th order of an intense 1064-nm laser. We predict that competition of photodissociation can be minimized by trapping the molecule in high-field-induced potential wells, thus enhancing the high-order harmonic generation process. Furthermore, the harmonic spectrum can serve as a measure of the structure of these laser-induced potentials. ERIC E. AUBANEL et AND& D. BANDRAUK. Can. J. Chem. 72,673 (1994). On examine deux conscquences du comportement particulier des molicules dans un champ fort. En premier lieu, la provocation, ii l'aide de pulsations laser de femtoseconde, d'un declenchement intermittent des croisements CvitCs par une induction au laser permet d'obtenir une inversion vibrationnelle efficace dans une distribution itroite des niveaux vibrationnels d'un ion moleculaire. On demontre cette affirmation par une solution numerique de I'iquation de Schrodinger en fonction du temps pour le Hz. En deuxikme lieu, on presente les resultats de calculs numeriques avec du Hz vibrationnellement excite d'une generation harmonique jusqu'au 1 le ordre d'un laser intense ii 1064 nm. On fait la prediction qu'il est possible de minimiser la compctition de la dissociation en piegeant la molecule dans des puits de potentiel de champ eleve et que I'on peut ainsi amiliorer le processus de generation harmonique d'ordre Clevt. De plus, le spectre harmonique peut servir B la mesure de la structure de ces potentiels induits au laser. [Traduit par la redaction] 1. Introduction The study of molecules in strong fields is establishing itself as an important part of chemical physics, particularly because of its role in the control of photodissociation dynamics (see, e.g., ref. 1). Strong laser fields induce high-order multiphoton absorption and emission that can lead to the creation of field induced potentials (2). In this paper we will show how one can exploit field-induced potentials to obtain efficient vibrational inversion, and multiphoton absorptions to generate high-order harmonics of a 1-pm laser. High vibrational excitation in a molecule is desirable either as a means to achieve selective dissociation of chemical bonds or to prepare selectively excited reactants. Over 30 years ago, J.C. Polanyi proposed an infrared laser based on vibrational excitation achieved by chemical reaction in the gas phase (3). Other ways to achieve high vibrational excitation include the climbing of the vibrational ladder using a frequency-chirped infrared laser pulse (4) and excitation via an electronically excited state with visible-ultraviolet lasers (stimulated emission pumping, SEP) (5). In the case of infrared-inactive molecules, the SEP method is required. This method is complicated, however, if the lifetime of the excited state is shorter than the interaction time with the laser needed to achieve complete inversion. If the electronically excited state is bound, one can use the STIRAP (stimulated Raman adiabatic passage) or counterintuitive method (6), where the order of the pump and Stokes lasers is reversed from the usual intuitive order. If the excited electronic state is repulsive, a special high-field method is necessary, as was recently examined in detail by us (7), so that stimulated emission becomes more rapid than dissociation. In this paper we will summarize our method, which we call vibrational inversion via time gating of laser-induced avoided crossings, and present new results of high harmonic generation '~uthor to whom correspondence may be addressed. FIG. 1. Inversion scheme (inside dotted box) for H;, starting from v = 2 and ending at v = 14, using two lasers of wavelengths X, = 2 13 nm and X2 = 320 nm. Also shown are two one-photon nonresonant processes that can compete. from vibrationally excited H;. The latter will show interesting features of vibrational trapping via the laser-induced avoidedcrossing mechanism. 2. Vibrational inversion Consider the problem of inverting the vibrational population of H; in its bound 2 ~i state. One must do so via the excited 22: state. Figure 1 (inside dotted lines) illustrates the case for inver-

2 674 CAN. I. CHEM. VOL. 72,1994 FIG. 2. Three-dressed-state representation of the excitation scheme of Fig. 1, showing crossing I due to laser 1 and crossing I1 due to laser 2. sion from vibrational level v = 2 to v = 14 using lasers of frequencies hl = 213 nm and X2 = 320 nm. If one simply applies laser 1 followed by laser 2, most of the excitation is lost as photodissociation. The same thing happens if one applies the lasers in the reverse order, unlike the situation in Na2 (6) where the excited electronic state is bound, unless the laser intensities are high enough. The reason for this is evident when one considers the dressed states of the molecule-field system (Fig. 2) (1): (i) Vg + nlfiwl + n2?iu2, where population initially resides; (ii) V, + (nl - 1) fiol + n2fio2; and (iii) Vg + (nl - 1) fiol + (n2 + 1) ho2. The second state corresponds to absorption of one photon w and the third state to subsequent emission of one photon 02. Curve crossings I and I1 due to lasers 1 and 2, respectively, become avoided crossings as each laser is turned on, through the field-molecule coupling: where fli is the Rabi frequency of laser i (i = 1, 2), 1 is the laser intensity, and p, = erl2 is the electronic transition moment (8). Note that we do not take into account rotational excitation, so we place the molecule parallel to the laser polarization. One can see from Fig. 2 how the counterintuitive method might work: first turn on laser 2 so that crossing I1 is adiabatic (1) (i.e., the crossing probability between the two field-molecule states is high), then turn on laser 1 to release the initial vibrational wave function through crossing I, and finally turn off both lasers before the wave packet returns to the initial state after hitting turning point b. This leaves Hz in the ground state at v = 14. For crossing I1 to be adiabatic, laser 2 must be very intense (>1013 w/cm2). Laser 1 must be intense as well, to dissociate all of the initial wave function in a time shorter than the return time from turning point b. The discussion in the previous paragraph has only taken into account resonant transitions (inside the dotted line, Fig. 1). At the high intensity required to make crossing I1 adiabatic, the initial v = 2 state can be nonresonantly photodissociated with laser 2. The solution that we propose to avoid this problem is to apply two simultaneous pulses. Competition with other multiphoton nonresonant processes leads to possible dissociative loss into two channels: a nonresonant low kinetic energy channel corre- FIG. 3. Final vibrational populations of 'CgC state from exact calculations of rotationless H: initially in v = 2, for two simultaneous pulses of wavelengths X1 = 213 nm and X2 = 320 nm, having a 10-fs rise to a 13-fs plateau at 2 x 1013 w/cm2 followed by a 10-fs fall. The total final ground state population is sponding to net absorption of one photon of frequency o2 from v = 2, which dominates at high intensity, and a resonant high kinetic energy channel corresponding to net absorption of one photon of frequency ol from v = 2, which dominates at low intensity. An optimum intensity can be found that minimizes dissociative loss in both channels (7). Figure 3 shows the final bound state population after application of two simultaneous pulses of wavelengths X1 = 213 nm and X2 = 320 nm, having a 10-fs rise to a 13-fs plateau at 2 x 1013 w/cm2 followed by a 10-fs fall, to Hz in v = 2, from numerical solution of the time-dependent Schrodinger equation assuming the molecule to be aligned with the field polarization and neglecting rotational excitation (7). Less than 20% dissociation and almost complete

3 AUBANEL AND BANDRAUK 67 5 inversion of the remaining bound population occurred into a distribution centered around v = 1 1. Inversion into v = 14 alone did not occur, owing to the large dynamic Stark shifts resulting from the high intensities used. We have also shown in ref. 7 that rotational excitation increases slightly the dissociative loss and broadens the distribution of excited vibrational levels. Thus we have exploited the concept of laser-induced avoided crossings (Fig. 2), necessary to interpret high-field effects, to control population inversion. We expect that heavier diatomic molecular ions, with less steep repulsive potentials and slower nuclei, would allow even better control. The increased electronic complexity of heavier molecules would not pose a problem if resonant transitions were used, and if the two desired electronic potentials were linked by a particularly large transition moment, as is the case for charge-transfer transitions (9, 10). 3. Harmonic generation One of the features of strong laser fields is their ability to induce a nonlinear response in atoms and molecules: where kind is the induced dipole, E(t) is the applied field, and a, p, and y are the first-, second-, and third-order polarizabilities of the atom or molecule. For centrosymmetric systems such as H;, one expects only odd-order polarizabilities (a,y,...) to contribute. Even-order polarizabilities (P,...) can only occur in symmetry broken systems (1 1). At laser intensities of the order 1013 w/cm2 and higher, the response of the system becomes highly nonlinear, i.e., many terms in eq. [2] become large. For sinusoidal time-dependent fields, one can see that the second term in eq. [2] will produce oscillations of kind at twice the laser frequency (second harmonic), the third term will produce oscillations at three times the laser frequency (third harmonic), and so on (1 1). The study of harmonic generation in atoms in intense fields has been actively studied in recent years (see, e.g., ref. 12), and now attention is being focused on molecules. Molecular ions in particular have been the focus of attention (9, 10) because of their charge resonance states, which couple very strongly to electromagnetic fields (transition moments proportional to R/2 (8); see eq. [I]). The induced dipole in G can be evaluated from a time series of the nuclear wave functions xp and X, obtained from numerical solution of the time-dependent Schrodinger equation. The induced dipole is obtained by evaluating the expectation value of the electronic coordinate along the laser polarization (2); here we use a double electronic state (2Ci and 2C:) representation of G and ignore rotation: [31 (11 = (xp(r.t) Og(r,R) + xu(r.t) +,,(r,r) zi +g(r,r) x xg(r,t) + +,(r,r) xu(r,f)j = 2Re(xg(R,t) +g(r,r) z +,(r,r) xu(r,t)) = Re(xp(R,t) ~1 xu(r,t)) where r and R are the electronic and internuclear coordinates, respectively. Before we proceed we should address three possible processes that could compete with harmonic generation in molecules: photoionization, photodissociation, and rotational excitation. The first is partly circumvented by the choice of molec- FIG. 4. Photodissociation probability from initial level v = 13, h = 1064 nm, and 30-cycle pulse with 5-cycle linear ramp. ular ions, which have higher ionization potentials than their parent neutral molecules. Calculations show that the ionization rate for Hz exceeds 1013 s-i for intensities greater than 1014 w/crn2 (13). The competition between ionization, photodissociation, and harmonic generation is the subject of ongoing study by our group. We will address here only the competition of photodissociation with harmonic generation. The effect of rotation will be the subject of an upcoming publication (14). Numerical calculations have shown that molecular photodissociation can be stabilized in the presence of strong laser fields, owing to trapping in laser-induced bound states (I, 2, 15-18). An example of this is shown in Fig. 4, where the photodissociation probability Pd for Hl starting in v = 13 and a laser wavelength of 1064 nm is shown as a function of peak intensity. The pulses used consisted of a 5-cycle linear ramp to a 25-cycle plateau (total time 106 fs). For details of the calculation, which employed a split-operator algorithm, refer to refs After a rise to complete dissociation, Pd goes through a series of minima, which can be attributed to trapping in laser-induced potential wells (14-16). For our harmonic generation spectra we use a longer pulse of 200 cycles with the same 5-cycle linear ramp. The power spectra are calculated by Fourier transforming the dipole moment: where v is the harmonic order and wo is the laser frequency; the function ~ (t) is equal to one between 6 and 200 cycles, and zero otherwise; this reduces the noise in the spectra due to the turn-one of the pulse (10, 19). In Figs. 5 and 6 we show harmonic generation spectra at intensities of 7.4 x 1013 and 1.6 x 1014 w/cm2, corresponding to a minimum and maximum in Pd (Fig. 4), respectively. In Fig. 5 one can see that split peaks at even and odd harmonics are produced up to 1 lth order. The even harmonics can be interpreted as odd harmonics that are split (Rabi splitting) by the strong laser field (9, 10). Alterna-

4 676 CAN. L CHEM. VOL harmonic order FIG. 5. Harmonic generation spectrum at 7.4 x 1013 w/cm2, v = 13, and A = 1064, and with 200-cycle pulse with a 5-cycle linear ramp, obtained from induced dipole moment (eq. [4]) from 6 cycles to end of pulse. tively, one can think of the field creating a symmetry-broken electronic state, lag + la, rn i.e., the electron becomes localized on one atom on a femtosecond time scale; this will result in generation of even harmonics. This spectrum correlates very well with a simple two-level calculation with the internuclear distance fixed at 6 a.u. (the right turning point of v = 13) (10, 14); i.e., the relative efficiencies of the harmonics are well reproduced. Additional structure results from the nuclear motion that allows harmonic generation into several adiabatic vibrational levels, i.e., levels of the field- induced potential well (10, 14). From exact calculations (i.e., electronic motion treated exactly) on Hl fixed at R = 6 au., it is observed that the molecular contribution, reproduced by the two-level model and the two-electronic state results shown here (minus structure due to vibrational motion), is separate from the atomic contribution that occurs at higher harmonic order (10). The harmonic generation spectrum at 1.6 x 1014 w/cm2 (Fig. 6, solid line) is nearly obscured by the background. The peaks that do appear are all less intense than those in Fig. 5. This contrasts with calculations based on H l with fixed nuclei, and with the two-level model, where higher order harmonic generation is obtained at higher intensity (10). The explanation lies in the competition between harmonic generation and photodissociation. We found that harmonic generation occurs only at short (R < 5 A) internuclear distances, where the electronic states are nondegenerate. This is not surprising, since both from the point of view of perturbation theory, and from the high-field analytic solution of the two-level model, no harmonic generation can occur in a system of degenerate states (10). With this in mind, one can make the link between photodissociation and harmonic generation, namely, that more of the former means less of the latter. All that the photodissociating fragments contribute is a linear trend in pz! (t) that gives rise to the large background in the solid line inbfig. 6. The dotted line in the same figure shows the result of subtraction of this linear trend from the dipole moment before Fourier transforming to obtain the spec- harmonic order FIG. 6. Same as Fig. 5, except at 1.6 x l0i4 w/cm2 (solid line). Also shown is the result of subtracting the linear trend in the dipole moment before Fourier transforming to obtain the spectrum (dotted line). trum; the background is greatly reduced and a similar structure as in Fig. 5 is revealed, i.e., split peaks around even and odd harmonics, but much weaker in intensity. The interplay between photodissociation and harmonic generation should now be evident. At intensities where photodissociation is minimized, owing to trapping in laser-induced potentials, harmonic generation will be maximized and vice versa. 4. Conclusions In this note we have shown that laser-induced avoided crossings, necessary to explain laser-molecule interactions in strong laser fields, can be exploited to achieve vibrational inversion into a narrow distribution of vibrational levels in diatomic molecules via a dissociative excited state. We then showed that vibrationally excited Hl can generate harmonics up to 1 lth order of an intense, but less than 1014 w/cm2, 1064-nm laser. Of course, harmonic generation could be produced from vibrationally relaxed Hl, but less efficiently, owing to a smaller transition moment (rn R/2) and a larger number of photons needed to make the transition from the ground to the excited state. Harmonic generation from vibrationally relaxed Hi with a higher frequency laser would be less efficient as well, since the intensity of a given harmonic depends inversely on the laser frequency (lo), and competition from ionization would be more important. Finally, for harmonic generation to compete with photodissociation, laser-induced avoided crossings must be exploited to achieve stabilization of photodissociation. One sees that high-order (nonperturbative) harmonic generation can serve as an excellent diagnostic tool to probe intense fieldmolecule interactions and dynamics. 1. A.D. Bandrauk. Molecules in laser fields. Marcel Dekker, New York A.D. Bandrauk and M.L. Sink. Chem. Phys. Lett. 57,569 (1978); J. Chem. Phys. 74, (1981); J. Yuan andt.f. George. J. Chem. Phys. 68, 3040 (1978).

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