Open Access Article

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A.
Serrano-Jiménez
^{a},
L.
Bañares
^{b} and
A.
García-Vela
*^{a}
^{a}Instituto de Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain. E-mail: garciavela@iff.csic.es
^{b}Departamento de Química Física, Facultad de Ciencias Químicas (Unidad Asociada I+D+i al CSIC), Universidad Complutense de Madrid, 28040 Madrid, Spain

Received
1st March 2019
, Accepted 18th March 2019

First published on 18th March 2019

A coherent control scheme is suggested to modify the output of photodissociation in a polyatomic system. The performance of the scheme is illustrated by applying it to the ultrafast photodissociation of CH_{3}I in the A-band. The control scheme uses a pump laser weak field that combines two pulses of a few femtoseconds delayed in time. By varying the time delay between the pulses, the shape of the laser field spectral profile is modulated, which causes a change in the initial relative populations excited by the pump laser to the different electronic states involved in the photodissociation. Such a change in the relative populations produces different photodissociation outputs, which is the basis of the control achieved. The degree of control obtained over different photodissociation observables, like the branching ratio between the two dissociation channels of CH_{3}I yielding I(^{2}P_{3/2}) and I*(^{2}P_{1/2}) and the fragment angular distributions associated with each channel, is investigated. These magnitudes are found to oscillate strongly with the time delay, with the branching ratio changing by factors between two and three. Substantial variations of the angular distributions also indicate that the scheme provides a high degree of control. Experimental application of the scheme to general polyatomic photodissociation processes should be straightforward.

It has been shown experimentally^{8,29–31} and theoretically^{32} for atomic and diatomic systems that when a weak electric field combining two pulses delayed in time is applied, by varying the time delay between the pulses it is possible to modify the shape of the spectral profile associated with the field, which implies changing the relative populations of the different states excited within a superposition. The shape of the profile changes as the delay between the pulses is varied due to interference between the pulses. Such a combination of pulses produces a spectral profile with a fixed bandwidth that is independent of the delay between the pulses. Thus, the spectral bandwidth of the laser field applied is not modified when the time delay between the pulses is varied.

Based on the above interference effect, a weak-field coherent control scheme has been recently proposed for the first time to modify the asymptotic photofragment state distribution of a polyatomic system, Ne-Br_{2}(B,ν′), produced upon resonance decay.^{33,34} The scheme was applied to control the long-time, final vibrational distribution of the Br_{2}(B,ν_{f} < ν′) fragment produced upon dissociation from an initial superposition of either intermolecular overlapping or isolated resonances,^{33} and also from initial single isolated resonances.^{34} The control scheme applied an electric field consisting of two pulses that overlap spectrally with a time delay between them. When the time delay is varied, the shape of the corresponding fixed bandwidth spectral profile changes, causing modulation of the initial population of the different resonances in the superposition, leading to variation of the asymptotic Br_{2}(B,ν_{f} < ν′) fragment vibrational distribution. Thus, control of the final fragment distribution is possible just by varying the time delay between the two pulses.

In the previous applications of the control scheme,^{33,34} the specific interest was to control the fragment state distribution produced in a resonance-mediated photodissociation process on a single excited electronic state. Thus, the spectral bandwidths associated with the laser fields applied were correspondingly narrow, of a few wavenumbers (≤6 cm^{−1}). While it has been shown that for overlapping resonances the fragment state distribution produced upon resonance decay can vary significantly within those narrow energy ranges,^{35} using a narrow spectral bandwidth limits remarkably the potential performance of the control scheme. It is expected that the scheme will display its largest potential when a broad spectral bandwidth (associated with pulses on the femtosecond or even the attosecond time scale) is used to excite the several electronic states that are usually involved in general polyatomic photodissociation processes. This work pursues investigation of the application of the control scheme under these general conditions of photodissociation. In this sense, the photodissociation process associated with the A-band of the CH_{3}I molecule, which involves excitation to three different electronic states, is an ideal case in order to test the performance of the control scheme.

A great deal of effort has been devoted to investigating the A-band photodissociation of methyl iodide, both experimentally^{27,36–46} and theoretically.^{43,44,47–59} In brief, the absorption spectrum associated with this band involves excitation from the ground electronic state ^{1}A_{1} to the three excited states ^{3}Q_{0}, ^{1}Q_{1}, and ^{3}Q_{1} (in Mulliken's notation^{60}). The parallel transition to the ^{3}Q_{0} state dominates at most of the excitation wavelengths, and particularly at the maximum of the band at 260 nm. Weak perpendicular transitions to ^{1}Q_{1} and ^{3}Q_{1} take place at the blue and red edges of the band, respectively. In addition, a conical intersection couples the ^{3}Q_{0} and ^{1}Q_{1} states. The ^{3}Q_{0} state correlates adiabatically with the CH_{3}(^{2}A_{2}′′) + I*(^{2}P_{1/2}) products, while both ^{1}Q_{1} and ^{3}Q_{1} correlate with CH^{3}(^{2}A_{2}′′) + I(^{2}P_{3/2}). Photodissociation through each excited electronic state produces a different output, and thus varying the initial relative population of the three states implies controlling the final global photodissociation yield. As schematically depicted in Fig. 1, the goal of the present work is to investigate to what extent such control can be achieved for different product fragment distributions by applying the above-mentioned control scheme based on two time-delayed excitation pulses.

In order to describe the CH_{3}I system the reduced-dimensionality model of Guo^{48} has been used. In this model the CH_{3}I molecule is considered as a CXI pseudotriatomic molecule where the pseudoatom X (X = H_{3}) is located at the center-of-mass (CM) of the three H atoms. The model considers three degrees of freedom (DOF) represented by the (R,r,θ) Jacobi coordinates. The dissociation coordinate R is the distance between I and the CH_{3} (or C–X) CM, r is the C–X distance, which represents the umbrella bend of the C–H_{3} group (ν_{2}), and θ is the angle between the vectors associated with R and r, and represents the X–C–I bend (ν_{6}). The specific form of the nuclear kinetic energy operator within this model has been given in detail in previous work.^{43} Since the purpose is to calculate angular distributions of the photofragments, an additional DOF, θ_{R}, which is the angle between the laser electric field polarization direction and the CH_{3}–I molecular axis R, is also included.

In the present model four electronic states are considered, namely the ground state ^{1}A_{1} and the three excited states ^{3}Q_{0}, ^{1}Q_{1}, and ^{3}Q_{1}. The potential-energy surfaces used to represent these four states, as well as the transition dipole moment functions that couple them, have been described in detail earlier.^{43,44} Thus, the total wave packet of the system can be expressed as

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

Thus, the spectral profile can be expressed as

(7) |

The phase dependence term 2A(ω,ω_{1})B(ω,ω_{2})cos(ωΔt) is actually an interference term that arises when is squared. The requirement for this term to be nonzero is that the product A(ω,ω_{1})B(ω,ω_{2}) must be nonzero, that is, the bandwidths A(ω,ω_{1}) and B(ω,ω_{2}) of the two pulses of eqn (3) must overlap in a certain range of frequencies ω. When this happens, this term will modulate the spectral profile in that range of frequencies. It is important to notice that the maximum value of cos(ωΔt) = 1 (which occurs at Δt = 0) leads to the maximum intensity of . For all other values cos(ωΔt) < 1, the contribution of the interference term will produce a profile with a smaller intensity, reaching a minimum when cos(ωΔt) = −1.

In the present simulations, it is assumed that the two Gaussian pulses of eqn (3) are identical, i.e., ω_{1} = ω_{2}. In this case A(ω,ω_{1}) = B(ω,ω_{2}), and . The implication of this condition is that for those values of Δt leading to cos(ωΔt) = −1, the intensity is suppressed. The specific ω_{1} = ω_{2} frequencies used correspond to the energy of the maximum of the calculated absorption spectrum of the CH_{3}I A band, namely 38680 cm^{−1}, associated with the excitation wavelength λ = 258.6 nm from the ground electronic state. The temporal full width at half-maximum (FWHM) of both pulses of eqn (3) is

Fig. 2 Calculated absorption spectrum of the A-band of CH_{3}I, along with the sub-bands associated with the transitions ^{3}Q_{0} ← ^{1}A_{1}, ^{1}Q_{1} ← ^{1}A_{1}, and ^{3}Q_{1} ← ^{1}A_{1}. Also shown are the spectral profiles (eqn (7)) of the laser field of eqn (3) applied with three different time delays, (a) Δt = 0, (b) 1.98, and (c) 4.80 fs. |

As discussed above, the interference term governed by cos(ωΔt) (see eqn (7)) allows one to modify the shape of the laser field spectral profile when Δt > 0. By choosing appropriately the delay time between the two pulses of eqn (3), can be shaped so that population of a given region (and therefore of a given sub-band) of the absorption spectrum is favored. This is shown in Fig. 2, where the profiles associated with three different time delays, namely Δt = 0, 1.98, and 4.80 fs, are displayed. For Δt = 0 the spectral profile nearly covers all the absorption spectrum, thus populating the three sub-bands associated with ^{3}Q_{0}, ^{1}Q_{1}, and ^{3}Q_{1}. However, for Δt = 1.98 fs the corresponding profile populates mainly the ^{3}Q_{0} and ^{3}Q_{1} states, minimizing the population of ^{1}Q_{1}, while for Δt = 4.80 fs population of the ^{1}Q_{1} state is favored. The photodissociation yield in each case is expected to be different.

A deeper physical insight into how interference between the two pulses operates can be gained by analyzing the behavior of the laser field for different values of Δt. This is done in Fig. 3, where the behavior of both and is shown for three time delays which correspond to the situations of maximum (Δt = 0), nearly minimum (Δt = 0.43 fs), and intermediate (Δt = 1.98 fs) intensity of , as displayed in Fig. 3(a). Indeed, for Δt = 0.43 fs we have that cos(ωΔt) ≈ −1, and . For simplicity, let us rewrite eqn (3) in a more compact form

(t) = _{1}(t) + _{2}(t), | (8) |

Fig. 3 (a) Spectral profile (eqn (7)) of the laser field applied with three different time delays between the two Gaussian pulses, Δt = 0, 0.43, and 1.98 fs, and temporal profiles of the two Gaussian pulses _{1}(t) and _{2}(t) along with their sum (t) for the time delays (b) Δt = 0, (c) Δt = 0.43 fs, and (d) Δt = 1.98 fs. |

Several quantities have been analyzed in order to assess the degree of control achieved over the output of the CH_{3}I photodissociation process when the time delay Δt between the two excitation pulses is varied. Such quantities include the final populations (when t → ∞) in the CH_{3}(ν′) + I* and CH_{3}(ν′) + I dissociation channels (from now on referred to as the I* and I channels), the branching ratio between these two final populations, and the anisotropy parameter β associated with the angular distributions produced through both channels. The analysis is carried out for the final states ν′ = 0, 1, and 2 of the umbrella mode of the CH_{3} fragment, and the results are shown in Fig. 4 for ν′ = 0 and in Fig. S1 and S2 (ESI†) for ν′ = 1 and ν′ = 2, respectively (reported as ESI†).

Fig. 4 Behavior of different observables with the time delay Δt between the two excitation pulses of the laser field of eqn (3) for the vibrational state ν′ = 0 of the CH_{3} fragment. (a) Final populations in the I* and I dissociation channels. (b) Branching ratio I*/I between the final populations in the I* and I dissociation channels. (c) Anisotropy parameter β associated with the angular distribution produced by dissociation through the I* channel. (d) Same as panel (c) for the I channel. |

The most interesting result of Fig. 4 and Fig. S1 and S2 (ESI†) is that all quantities investigated display a strongly oscillating behavior with Δt. This behavior is caused by the oscillating nature between 1 and −1 of the cos(ωΔt) term in . The oscillations are qualitatively similar for the three vibrational states of the CH_{3} fragment. The main difference found with the final vibrational quantum number ν′ is the relative population of the I* and I channels (and therefore the branching ratio), determined by the CH_{3} product vibrational distribution associated with each channel. Around the maximum of the CH_{3}I A-band spectrum, the CH_{3} vibrational distribution associated with the I* channel is very cold, with most of the population located at ν′ = 0 and little population for ν′ > 0, while the distribution of the I channel is hotter, peaking at ν′ = 1.^{43}

Fig. 4(a) shows the final populations of the CH_{3}(ν′ = 0) fragment produced through the I* and I channels in a range of time delays of nearly 10 fs. As mentioned above, the maximum population occurs in both channels for Δt = 0, which implies that cos(ωΔt) = 1. Then cos(ωΔt) ≈ −1 for Δt = 0.43 fs, and the population of both channels nearly vanishes. A second maximum of the populations occurs at Δt = 0.99 fs, and a second minimum is found at Δt = 1.29 fs. Successive maxima and minima occur in the populations for longer Δt. Since the maxima and minima are determined by cos(ωΔt) = 1 and −1, respectively, the corresponding Δt values can be determined from the conditions

(9) |

(10) |

The variation of the branching ratio between the populations of the two dissociation channels (denoted by I*/I) with Δt is probably even more interesting than that of the populations themselves. The branching ratios corresponding to ν′ = 0, 1, and 2 are shown in Fig. 4(b) and Fig. S1(b) and S2(b) (ESI†), respectively. The three branching ratios display a qualitatively similar structure of pronounced oscillations, related to the oscillations of the populations themselves. The I*/I ratio varies between 7 and 12 for ν′ = 0, between 0.7 and 1.4 for ν′ = 1, and between 0.1 and 0.3 for ν′ = 2. This means a variation of the ratio by about a factor of two for ν′ = 0 and ν′ = 1, and by a factor of three for ν′ = 2 when Δt is changed. In addition, while the maxima and minima of the I* and I populations always coincide, they do not necessarily coincide with the maxima and minima of the corresponding ratios. The interesting implication is that despite the qualitatively similar, in phase oscillations of the I* and I populations with Δt, the quantitative variation of the two populations is significantly different, giving rise to strong variations of the I*/I ratio by factors of two or three that allow for a high degree of control.

Three regions can be distinguished in the behavior of the I*/I ratios of Fig. 4 and Fig. S1 and S2 (ESI†). There is a central region 2–3 fs < Δt < 7 fs where the oscillations of the ratios are more pronounced, while in the side regions Δt < 2–3 fs and Δt > 7 fs the intensity of the oscillations is somewhat weaker. A trivial explanation would be that this effect is due to a lack of resolution in the grid of points of Δt used that would prevent finding the correct position and value of the maxima and minima, which would have a similar value across the whole range of Δt. It is noted, however, that for Δt = 0 (where a possible lack of resolution has no effect) the I* and I populations reach the maximum value possible, but still the value of the corresponding ratio is smaller than the values of the maxima at 2–3 fs < Δt < 7 fs.

While a lack of resolution could contribute to some extent to the effect, an additional explanation is that the variation of the intensity of the oscillations in the ratios as Δt increases is also due to the different behavior of the effect of interference between the two pulses when Δt changes, and how it affects the population of the electronic states correlating with the I* and I channels. For relatively short delays Δt < 2–3 fs the spectral profile displays few ocillations due to the cos(ωΔt) term. This may favor population of the ^{1}Q_{1} and ^{3}Q_{1} states, which lead to the I channel, causing lower maximum and higher minimum values of the I*/I ratio in this region of Δt. In the intermediate region 2–3 fs < Δt < 7 fs the number of oscillations in gradually increases, which may alternately favor population of either the ^{3}Q_{0} or the ^{1}Q_{1} and ^{3}Q_{1} states, producing higher differences between the maxima and the minima of the ratio oscillations. Finally, for Δt > 7 fs the effect of the cos(ωΔt) term is vanishing, as shown by the behavior of the individual I* and I populations, and then . This spectral profile would populate the whole CH_{3}I absorption spectrum, approaching a situation very similar to that of Δt = 0, which would explain the similar behavior of the ratios for Δt > 7 fs and for Δt = 0. While the present behavior of the I*/I ratios is due to the specific structure of the sub-bands of the CH_{3}I absorption spectrum, the behavior of with increasing Δt described above is general for any photodissociation process, and will affect it correspondingly.

Angular distributions were calculated for photodissociation into the CH_{3} + I* and CH_{3} + I product channels in order to investigate the behavior of these distributions when Δt is varied. In the absence of any conical intersection in between excited states, the only contribution to the I* channel angular distribution would come from the ^{3}Q_{0} state, while in the case of the I channel the angular distribution would be the sum of the contributions produced upon dissociation through both the ^{1}Q_{1} and ^{3}Q_{1} electronic states. The angular distributions obtained for both dissociation channels were fitted to the familiar expression^{61}

(11) |

The β parameters associated with the angular distributions of the I* and I channels are shown vs. Δt in the (c) and (d) panels of Fig. 4 and Fig. S1, S2 (ESI†) for ν′ = 0–2, respectively. Not surprisingly, the anisotropy parameters for both dissociation channels display a similar structure of oscillations to those found in the upper panels for the populations and the branching ratios. In the case of the I* channel, the oscillating values of β are very close to 2, reflecting the parallel character of the ^{3}Q_{0} ← ^{1}A_{1} transition, and the fact that little population is transferred from the ^{1}Q_{1} state through the conical intersection across all the range of Δt investigated. Consistently with this small contribution of the ^{1}Q_{1} population the variation of the I* channel β with Δt is rather small, the largest variations being from 1.84 to 1.94 for ν = 0, from ∼1.965 to ∼1.985 for ν = 1, and from ∼1.865 to ∼1.92 for ν = 2. Such small variations imply very little (or negligible) control of the I* channel angular distribution.

The situation, however, is quite different for the I dissociation channel. Indeed the β parameter for this channel shows remarkably larger variations with Δt. The reason is that in the CH_{3}I excitation, the ^{3}Q_{0} state is the most populated one (as shown by the absorption spectrum of Fig. 2), and thus the fraction of this population that is transferred to the ^{1}Q_{1} state through the conical intersection is large enough to cause substantial changes in the angular distribution of the I channel. The largest variations found are from −0.4 to 0.3 for ν′ = 0, from ≈−0.4 to ≈0.2 for ν′ = 1, and from ≈−0.4 to ≈0.1 for ν′ = 2. Those variations imply large changes in the exit angle of the photoproducts, and thus a large degree of control of the angular distribution of the I channel.

It is interesting to note that for the three vibrational states of the CH_{3} fragment, the shape of the β dependence on Δt is very similar for the two dissociation channels, with coinciding positions of the maxima and minima. Moreover, the shape of the two β curves is also very similar to the shape of the corresponding I*/I ratio as a function of Δt, showing three distinct regions, with the most intense oscillations appearing at intermediate Δt values. The coincidence in the shape of the branching ratio and β curves seems to support that their trend is determined by the behavior of the interference cos(ωΔt) term as Δt increases, as discussed above. A close or even exact coincidence (for most of the points) in the position of the maxima and minima of the β curves with those of the branching ratio is found. Such a correlation is not very surprising since when the I*/I ratio reaches a maximum (involving an increase of the I* population), it is expected that the β value of the I* and I channels will also reach a maximum (i.e., becoming closer to 2 and more positive for the I* and I channels, respectively), while when the I*/I ratio reaches a minimum (decreasing the I* population) the expected result is also a minimum of the two β parameters (i.e., becoming farther away from 2 and less positive for the I* and I channels, respectively). The important implication is that by controlling the initial relative population of the different excited electronic states, it is possible to control the photodissociation output of all the observables affected by those states.

For simplicity the present work was restricted to the situation where the two pulses of eqn (3) overlap completely in the frequency domain (ω_{1} = ω_{2}). It is interesting to comment briefly on what it is expected in the more general scenario of only partial overlap of both pulses (ω_{1} ≠ ω_{2}). In this case the interference term cos(ωΔt) operates only in the region of frequency overlap (see eqn (3)), and thus control by varying Δt could only be exerted in the region of the absorption spectrum that coincides with that overlapping region. In some cases it may be desirable to restrict the action of control to a limited region of the absorption spectrum, and this can be done by varying the size of the pulse overlap region, changing the carrier frequencies ω_{1} and ω_{2} and the spectral width of the two pulses of eqn (3).

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## Footnote |

† Electronic supplementary information (ESI) available: Trends of different observables with the time delay between two excitation laser pulses for the product vibrational states ν′ = 1 and 2 of the CH_{3} fragment: final populations in the I* and I dissociation channels, branching ratios between the final populations in the I* and I dissociation channels and anisotropy parameters. See DOI: 10.1039/c9cp01214a |

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