Coherent Control of Photofragment Distributions Using Laser Phase

Feb 16, 2015 - Department of Chemistry, Building 207, Technical University of Denmark, ... Weak-field laser phase modulation coherent control of asymp...
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Coherent Control of Photofragment Distributions Using Laser Phase Modulation in the Weak-Field Limit Alberto García-Vela*,† and Niels E. Henriksen*,‡ †

Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain Department of Chemistry, Building 207, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

‡

ABSTRACT: The possibility of quantum interference control of the final state distributions of photodissociation fragments by means of pure phase modulation of the pump laser pulse in the weak-field regime is demonstrated theoretically for the first time. The specific application involves realistic wave packet calculations of the transient vibrational populations of the Br2(B, vf) fragment produced upon predissociation of the Ne−Br2(B) complex, which is excited to a superposition of resonance states using pulses with different linear chirps. Transient phase effects on the fragment populations are found to persist for long times (about 200 ps) after the pulse is over due to interference between overlapping resonances in Ne−Br2(B).

no phase control of the final state distributions of the fragments is possible.14,15 Thus, after the excitation pulse ,(t ) has vanished and in the long-time limit where dissociation is completed, the probability of observing fragments in an eigenstate |E, n⟩ of a given arrangement channel of the products, can be written in the form (see ref 16 for details)

E

nergy is required in order to drive chemical reactions and it is normally supplied in the form of heat or as photons in photochemistry. Photochemistry can lead to isomerization or fragmentation, that is, bonds breaking as well as bonds forming. In parallel with the continued development of laser technology a new branch of photochemistry has emerged. The use of optimized laser fields to guide the dynamics of an atom or molecule from a given initial state into a desired final state is a topic of much current interest.1−12 The time-dependent phase-coherent electric field of a laser pulse can be represented by

t →∞

(1)

where A(ω) is the real-valued distribution of frequencies and ϕ(ω) is the real-valued frequency-dependent phase. The energy in the field is completely determined by the frequency distribution A(ω) and the amplitude , 0 , that is, it is independent of the phases.13 Besides frequency and intensity control provided by lasers, an intriguing question concerns pure phase control and the control of chemical reactivity via quantum mechanical interference effects. Genuine active coherent laser control is defined as control which depends on the phases ϕ(ω). To be specific, we consider in the following a molecule which is bound in its electronic ground state and where fragmentation can take place in an excited electronic state. In the weak-field (one-photon) limit, amplitude is exclusively transferred from the electronic ground state to excited state surfaces. However, when excitation out of a single eigenstate (or an incoherent superposition of eigenstates) of a molecule is considered, and products are observed in the long-time limit when dissociation is completed, a seminal proof suggested that © XXXX American Chemical Society

(2)

where E is the total energy of the fragments and degenerate states are labeled by a global quantum number n, |ϕX⟩ is the initial eigenstate (e.g., the vibrational ground state) at energy ϵ1 times the projection of the transition dipole moment along the polarization direction of the field, ωE = (E − ϵ1)/ℏ is the excitation frequency, HB is the nuclear Hamiltonian of the excited electronic (B) state, and * is a constant. In the asymptotic time limit t → ∞, according to eq 2, only the frequency distribution of the laser field is reflected in the probability, that is, via |A(ωE)|2, and any dependence on the phases of eq 1 is absent, which implies that asymptotic phase effects on the fragment distributions are not possible. For indirect photodissociation, where products may continuously “leak out” over an extended period of time, the wave packet exp(−iHBt/ℏ)|ϕX⟩ behaves as sketched in Figure 1a. In the related area of photoisomerization, phase dependence of isomerization yields has been reported recently for weak-field excitation out of a single stationary state.17−20 It was shown that the phase dependence can persist over long times when the few

∞

∫−∞ A(ω)eiϕ(ω)e−iωtdω]

,(t ) = , 0Re[

P(E , n) = *|A(ωE)|2 lim |⟨E , n|exp( −iHBt /ℏ)|ϕX ⟩|2

Received: January 20, 2015 Accepted: February 16, 2015

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in much detail,30−34 and it has been shown that the dynamics in some energy ranges encompass overlapping resonances.32−34 The effects of interference between these overlapping resonances of Ne−Br2(B) have been used to design schemes to control the lifetime of specific resonance states.35−38 Recent work on coherent control has discussed the significant role of interference between overlapping resonances in relation to the population dynamics of electronically excited molecular states.39,40 As we will show in the following, pure phase modulation of the laser pulse can lead to different postpulse distributions of the final vibrational states vf when the dynamics involve interfering overlapping resonances. Upon laser excitation, Ne−Br2(X, v″ = 0) + hν → Ne−Br2(B, v′), a vdW resonance or a superposition of resonances of Ne− Br2(B, v′), is populated (hereafter, the labels v″ and v′ will be used to denote the vibrational states of Br2 in the X and B electronic states, respectively). Then the resonance or resonances excited decay to the fragmentation continuum through vibrational predissociation, Ne−Br2(B, v′) → Ne + Br2(B, vf < v′).30−33 The process of Ne−Br2(B, v′) excitation with a laser pulse and the subsequent predissociation of the complex was simulated with a full three-dimensional wave packet method (assuming J = 0) described in detail elsewhere.32,35 We consider a pump pulse with a fixed Gaussian frequency distribution centered around ω0 and a quadratic phase function with chirp parameter β0. The electric field applied in the Ne− Br2(X, v″ = 0) + hν → Ne−Br2(B, v′) excitation takes the form

Figure 1. (a) Schematic picture showing the photofragmentation of a van der Waals complex. A laser pulse creates a wave packet in an excited electronic (B) state, the wave packet oscillates corresponding to vibrational motion in Br2 and at the same time there is a slow dissociation into the final product states Ne + Br2(B, vf). (b) Temporal profiles of the pump pulse of eq 3 for different values of the chirp parameter β0.

⎡ ⎛ (t − t )2 τ2 iβ(t − t0)2 0 − ,(t ) = , 0Re⎢ 2 0 exp⎜ − 2 ⎢⎣ τ0 − iβ0 2 2τ ⎝ ⎞⎤ − iω0t ⎟⎥ ⎠⎥⎦

modes that are active in the isomerization are coupled intramolecularly to the many vibrational modes in a large molecule or, in general, to an external environment. This coupling allows for dissipation of energy from the isomerization coordinates and effectively constrains the observables to finite times of the system dynamics. It has even been predicted that cis/trans branching ratios can be phase-controlled without a requirement for an open system.21 As discussed above, for weak-field photofragmentation out of a single eigenstate genuine active laser control is not possible in the asymptotic long-time limit. However, we considered recently the fragmentation of diatomic molecules and showed that the total dissociation probability22 as well as the branching ratio between different electronic states can be modified within a certain time window13 before dissociation is completed. In the long-time limit, a phase dependence associated with a fixed bandwidth phase modulated pulse can be observed2,23−26 when excitation out of a coherent superposition of stationary states is considered. In the strong-field limit, genuine active laser control is possible even for photofragmentation out of a single eigenstate.27,28 The use of strong fields creates, however, potential problems with the population of unwanted channels, for example, related to ionization.29 In this Letter, we explore in more detail the question of weak-field phase control of final product distributions. To that end, we consider the fragmentation of a van der Waals (vdW) complex Ne−Br2(X, v″ = 0) + hν → Ne−Br2(B) →Ne + Br2(B, vf). The dissociation dynamics of this complex has been studied

(3)

with a linear chirp rate β and a pulse duration τ related to τ0 and β0 via β=

β0 (τ04

+ β02)

(4)

and ⎛ β2⎞ τ 2 = τ02⎜⎜1 + 04 ⎟⎟ τ0 ⎠ ⎝

(5)

The value τ0 = 2.12 ps is used in the above equations. The spectral full width at half-maximum (fwhm) associated with the pulse is 6 cm−1, and the full bandwidth of the pulse covers ∼16 cm−1. For a transform-limited pulse (β0 = β = 0) the above value of τ0 corresponds to a temporal fwhm of the pulse of τfwhm = τ(8 ln 2)1/2 = τ0(8 ln 2)1/2 = 5 ps. The laser pulse is assumed to be linearly polarized. The transition dipole moment function for the X → B transition is unknown, so it was assumed to be constant (1 atomic unit). The maximum pulse amplitude used is , 0 = 1.0 × 10−6 a.u., which corresponds to a maximum pulse intensity of about 3.5 × 104 W/cm2, well within the weak-field regime. The effect of the linear chirp of the pulse on the vibrational state distribution of the Br2(B, vf < v′) fragment produced upon photodissociation is analyzed by varying the value of the spectral chirp β0 in eqs 3−5. Specifically, the values β0 = 0, 12.0, 24.0, 50.0, and 75.0 ps2 have been used. In Figure 1b the 825

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associated with v′ − 1 orbiting resonances. The second most intense feature in the spectrum located at −60.63 cm−1 corresponds to the v′ − 1 orbiting resonance separated from the v′ ground resonance by 1.2 cm−1. Thus, Ne−Br2(B, v′ = 27) is used to investigate the case where a superposition of overlapping resonances (essentially those located at −61.80 and −60.63 cm−1) is prepared by tuning the pulse frequency ω0 to excite the v′ ground vdW resonance. Figure 3 shows the normalized time-dependent vibrational populations of the Br2(B, vf) fragment in the vf = v′ − 1 and v′

Gaussian envelopes of the pulse are shown for the different values of β0, showing the temporal range covered by the pulse as the chirp rate increases. We investigated the effect of the pulse chirp on the Br2(B, vf < v′) fragment vibrational distribution by preparing a superposition of Ne−Br2(B) resonances with the ,(t ) laser field, that decay to the fragmentation continuum. As previously shown,39,40 the effect of phase on time-dependent observables can be remarkably different depending upon whether the resonances populated in the superposition overlap or not. This is due to interference effects between the overlapping resonances, which are absent when the resonances do not overlap. Thus, these two different situations have been studied in the present work. The Ne−Br2(B, v′ = 16) system supports intermolecular vdW resonances that do not overlap, and it is used to illustrate the situation where a superposition of nonoverlapping resonances is prepared. More specifically, in this case the frequency of the pulse ω0 is tuned to excite the n′ = 8 vdW resonance of Ne−Br2(B, v′ = 16), located at the energy −24.95 cm−1, relative to the Ne + Br2(B, v′ = 16, j′ = 0) dissociation threshold. The n′ index labels the energy position of the resonance, with n′ = 0 corresponding to the ground one. Taking into account the bandwidth of the pulse of eq 3, a superposition of about eight nonoverlapping resonances is created. The specific resonances populated are n′ = 5−12, located at the energies −31.89, −30.07, −26.96, −24.95, −22.39, −20.02, −18.08, and −17.78 cm−1, respectively. On the other hand, it has been shown that the ground intermolecular resonance of Ne−Br2(B, v′ = 27) overlaps with some vdW orbiting resonances corresponding to the lower v′ − 1 vibrational manifold of Br2(B).33,34 These orbiting resonances lie above the Ne + Br2(B, v′ − 1 = 26, j′ = 0) dissociation threshold and are supported by centrifugal barriers. In particular, the v′ ground resonance overlaps mainly with a v′ − 1 orbiting resonance located ∼1.2 cm−1 above in energy. The excitation spectrum of the Ne−Br2(B, v′ = 27) ground resonance is shown in Figure 2. The spectrum displays a main peak located at −61.80 cm−1, associated with the v′ ground vdW resonance, and several other overlapping peaks

Figure 3. Time-dependent normalized vibrational populations of the Br2(B, vf) fragment produced in the vf = v′ − 1 and v′ − 2 final vibrational states upon predissociation of Ne−Br2(B, v′ = 16), calculated for different values of β0.

− 2 final vibrational states for v′ = 16, calculated for different values of the chirp parameter β0. The vf = v′ −3 population is already very small (0.005) and it is not shown. The normalized vibrational populations are calculated as (t ) = Pvnorm f

Pvf (t ) ∑v Pvf (t ) f

(6)

with vf = v′ − 1, v′ − 2, .... The vibrational populations are shown only for times t ≥ 0 because at early (negative) times the curves display uninteresting oscillations due to the fact that at those times the vibrational populations are still small, and the denominator of eq 6 becomes very small. The vibrational population curves display changes as the β0 value increases, with respect to the case when the unchirped pulse (β0 = 0) is used. However, overall, only small effects of the chirp are observed during the interaction with the laser pulse. When no chirp is present, all the resonances of the superposition created are populated simultaneously by the pulse. Thus, the Pnorm (t) populations are the result of an vf average of the vibrational populations produced by the decay of all the resonances populated in the superposition. The chirp alters the temporal profile of the pulse. Therefore, when a positive chirp (β0 > 0) is present in the pulse, some of the superposition created, those at lower energies, are populated at earlier times than the resonances at higher energies. As a consequence, both the nature and amount of the resonances populated in the superposition change with time, causing a corresponding change in the average Pnorm (t) populations with vf respect to the case of an unchirped pulse, as shown in Figure 3.

Figure 2. Calculated excitation spectrum associated with the ground vdW resonance of Ne−Br2(B, v′ = 27). The energy axis is relative to the Ne + Br2(B, v′ = 27, j′ = 0) dissociation threshold. 826

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The Journal of Physical Chemistry Letters As expected, these changes increase in magnitude as the chirp rate (or the β0 parameter) increases. The above result is well known, and has been previously found in time-dependent observables in different systems and processes.20,39−41 When a coherent superpositon of orthogonal bound states (e.g., nonoverlapping resonances like in the present case of Ne−Br2(B, v′ = 16)) is prepared, the effects of the pulse chirp on time-dependent observables are expected to disappear once the pulse is over.40 The reason is that when the pulse is over all the states of the superposition have been populated, so all of them have contributed to the time-dependent observables, and no further changes of these observables can be expected at longer times. Thus, for a superposition of orthogonal states, a chirped pulse can induce changes in time-dependent observables only within the period of pulse duration, or at most for slightly longer times,13,22 when the states populated latest by the pulse are resonances with a finite lifetime that delays to some extent their contribution to the time-dependent observables. Indeed, this is the result found for the vibrational populations of Ne−Br2(B, v′ = 16) shown in Figure 3, where all appreciable changes in the population occur within the time range t < 100 ps, which is the maximum time duration of the pulse with the highest chirp rate, β0 = 75 ps2. A different situation may arise when a coherent superposition of nonorthogonal states, like overlapping resonances, is prepared due to the possibility of interference between them. As mentioned above, the Ne−Br2(B, v′ = 27) system provides the chance to prepare such a superposition because its ground intermolecular resonance overlaps with some v′ − 1 vdW orbiting resonances (and particularly with one of them, see Figure 2). This superposition is created with the pulse of eq 3, and Figure 4a displays the vf = v′ − 1 and v′ − 2 normalized vibrational populations of the Br2(B, vf) fragment produced upon predissociation of Ne−Br2(B, v′ = 27) using pulses with different values of the chirp parameter β0. In Figure 4b the vf = v′ − 1 normalized populations are shown with a different scale in order to display more clearly the details. It is noted that the calculations have been carried out up to a final time t = 840 ps, where the Br2(B, vf) fragment populations are asymptotically converged. However, in order to show more clearly the details of the phase effects in the population curves, the time scale of Figure 4 is limited to t = 600 ps. The slower convergence of the populations of Figure 4 as compared to those of Figure 3 appears to be an effect of the overlapping resonances. The populations show a clear effect of the pulse chirp in the range t < 100 ps where the different pulses operate. For the same β0 value, the effect during pulse operation is stronger for v′ = 27 than for v′ = 16. The most interesting finding, however, is that both the vf = v′ − 1 and v′ − 2 curves show clear changes in the population with respect to the unchirped pulse result at times well after all the pulses are over, especially around t = 150 ps, and to a smaller extent around t = 200 ps, t = 300 ps, and longer times. It is noted that the changes in the populations for t > 100 ps occur at the same times regardless of the time duration of the pulse applied. In addition, the times at which the changes in the populations take place are very long in comparison with the lifetimes of the two overlapping resonances mainly populated by the pump pulse (about 25 and 13.5 ps, respectively34,38), particularly for the shorter pulses with β0 = 12.0 and 24.0 ps2. The above results indicate that the postpulse times at which the effects on the populations occur are not due to a delay associated with the resonance lifetimes. We also note that the period of rotation of the Ne−Br2(B)

Figure 4. (a) Time-dependent normalized vibrational populations of the Br2(B, vf) fragment produced in the vf = v′ − 1 and v′ − 2 final vibrational states upon predissociation of Ne−Br2(B, v′ = 27), calculated for different values of the parameter β0. (b) Same as in panel a but showing only the Br2(B, vf = v′ − 1) vibrational populations for positive β0 values and for β0 = −75.0 ps2.

complex is shorter than the postpulse time window for which the phase effects persist. The postpulse effects on the populations for t > 100 ps are unambiguously sensitive to the magnitude of the chirp rate, increasing their intensity with increasing β0 value. Calculations using negative values of β0 have also been carried out, and the v′ − 1 population curve obtained with β0 = −75.0 ps2 is shown in Figure 4b. The effects on the populations are found to be of similar magnitude but opposite sign to those obtained for positive chirps. The only difference between the resonances populated in the Ne−Br2(B, v′ = 16) superposition and the v′ and v′ − 1 resonances populated in the superposition leading to the populations of Figure 4 is that in the latter case, the resonances overlap and can interfere between themselves. This interference is responsible for the pulse-free effects at t > 100 ps of Figure 4, which are absent in the populations of Figure 3. Indeed, because the effect of the chirp is essentially to modify the time at which each resonance of the superposition is populated, it also modifies correspondingly the way that the different resonances interfere between themselves along time, causing changes in the Br2(B, vf) fragment populations at long times after the pulse applied is over. The effect of the chirp on the resonance interference also takes place during pulse operation, and this is probably the reason why the effects on 827

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The Journal of Physical Chemistry Letters the populations of Figure 4 at times t < 100 ps are stronger than in the case of Ne−Br2(B, v′ = 16), where no interference is present. Figure 4b presents the vf = v′ − 1 populations in more detail. The largest postpulse effects of the chirp take place at around t = 150 ps. Actually, at t = 150 ps the populations for β0 = 50.0 and 75.0 ps2 increase by 4.5% and 7.6%, respectively, with respect to the β0 = 0 population. This variation of the population can be even larger if the phase effects caused on the population by the negative chirps are also considered (see Figure 4b). It is noted that the magnitude of the phase modulation effects found experimentally in the photoisomerization of retinal were about 5−7%.17 Figure 4b shows that phase modulation effects on the populations persist after the pulse is over for times longer than t = 300 ps. Interestingly, the phase effects on the populations exhibit an oscillating behavior, in the sense that the β0 > 0 populations alternately increase and decrease along time with respect to the β0 = 0 population. The magnitude of the phase effects is gradually damped with time and finally vanishes asymptotically in agreement with eq 2. It should be very interesting to investigate whether other types of chirps (nonlinear ones) could produce stronger and more persistant in time phase modulation effects on the fragment state populations. Similarly, it would also be interesting to study the scaling of the phase effects on the fragment populations with field coupling strength when using both positive and negative chirps, as done in a recent work.42 The experimental verification of the predicted phase effect should be relatively straightforward. Control effects due to phase modulation of the pulse in the weak-field limit have previously been found computationally for the time-dependent populations of some electronic states of different systems. For example, for the S0 → S2/S1 excitation followed by S2 ↔ S1 internal conversion in pyrazine and βcarotene,40 and the cis−trans isomerization in retinal.20 However, the present results are the first computational demonstration of the possibility of postpulse coherent control of the final state distributions of photodissociation fragments using phase modulation of a fixed bandwidth pulse in the weakfield regime. It has been shown21 that phase control of an observable Ô in a closed system (i.e., a system not coupled to an environment) like Ne−Br2(B) is possible in the weak-field limit if [Ĥ , Ô ] ≠ 0, being Ĥ the system Hamiltonian. The operator Ô associated with the Ne + Br2(B, vf) fragment state distributions is the projector onto the product fragment states. The fragment states are eigenstates of the asymptotic Hamiltonian Ĥ (i.e., the Hamiltonian in the product asymptotic region, a subspace of the full configuration space), but they are not eigenstates of the full Hamiltonian (including the Ne− Br2(B) interaction region). The implication is that the condition [Ĥ , Ô ] ≠ 0 is fulfilled, making possible phase coherent control of the Br2(B, vf) fragment vibrational populations, as shown in Figure 4. In summary, this work reports the first computational demonstration of coherent interference control of the final state distributions of photodissociation fragments based on pure phase modulation of the pump pulse in the weak-field regime. The phase effects on the fragment populations are found to persist for long times after the pulse is over, as a result of interference between nonorthogonal states, like overlapping resonances as in the present case, populated in a superposition. Because the phase modulation applied in this study is by no means optimized, the range of possible control both in

magnitude and in terms of the time window for observation of such phase effects, is unknown. The possible application of this phase modulation coherent control to internal (rovibrational) cooling of molecular species by applying a second laser pulse at the times when the phase effects on the populations are more intense, is envisioned. Furthermore, similar phase effects might occur in the branching ratio control into different chemical product channels.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: garciavela@iff.csic.es. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was funded by the Ministerio de Ciencia e Innovación (Spain), Grant No. FIS2011-29596-C02-01, and the COST Action, Grant No. CM1002. The Centro de Supercomputación de Galicia (CESGA, Spain) is acknowledged for the use of its resources.

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REFERENCES

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DOI: 10.1021/acs.jpclett.5b00129 J. Phys. Chem. Lett. 2015, 6, 824−829

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(38) García-Vela, A. Resonant Detection of the Signature of Control of a Resonance State Lifetime Using a Pump−Probe Scheme. RSC Adv. 2014, 4, 52174−52180. (39) Grinev, T.; Shapiro, M.; Brumer, P. Coherent Control of Radiationless Transitions: Simultaneous Excitation and Decay of Overlapping Resonances. J. Chem. Phys. 2013, 138, 044306−1−12. (40) Grinev, T.; Brumer, P. Theory of Perturbative Pulse Train Based Coherent Control. J. Chem. Phys. 2014, 140, 124307−1−7. (41) Mishima, K.; Yamashita, K. A Theoretical Study on Laser Control of a Molecular Nonadiabatic Process by Ultrashort Chirped Laser Pulses. J. Chem. Phys. 1998, 109, 1801−1809. (42) Am-Shallem, M.; Kosloff, R. The Scaling of Weak Field PhaseOnly Control in Markovian Dynamics. J. Chem. Phys. 2014, 141, 044121−1−10.

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DOI: 10.1021/acs.jpclett.5b00129 J. Phys. Chem. Lett. 2015, 6, 824−829