Abstract
1. Introduction
Funneling dynamics is a key mechanism in the framework of artificial lightharvesting processes, biological antennas, or organic photovoltaic devices^{[1–3]}. Transfer from an initial donor to a target acceptor proceeds through an excited state gradient, leading to a final localization from which the photonic energy may be captured. In complex molecular systems, such energy transport mechanisms are driven by ultrafast nonadiabatic transitions through conical intersections (CIs)^{[4–9]}. Investigation of the coherent preparation of the donor state from the initial ground state and the control of the subsequent dynamics using intense, ultrashort laser pulses are major issues for modifying the energy transport mechanisms^{[10–12]}. In this context, we have recently studied a rather complex molecular system, namely, polyphenylene ethynylene dendrimer, with a control objective aiming at the coherent preparation of two donor states involved in the dynamics, or their symmetric versus asymmetric superposition^{[13]}. In these systems, the acceptor state is radiatively coupled to the ground state. It is only by discarding this direct excitation that we could build control strategies to increase the coherence (asymmetry) lifetime to populate the acceptor through the vibrational baths of the donors.
In the present work, we address the coherent control of the pyrazine (${\mathrm{C}}_{4}{\mathrm{H}}_{4}{\mathrm{N}}_{2}$) photophysics and more precisely its ${S}_{2}\to {S}_{1}$ internal conversion leading to challenging issues, both with respect to its ultrafast nonadiabatic CI transfer (internal conversion time scale of 20 fs^{[14–18]}) and to its 24dimensional (24D) vibrational excitation dynamics. Some previous works in the literature are also concerned by the coherent control of population transfer from ${S}_{0}$ to ${S}_{2}$ and ${S}_{1}$ electronic states, putting the emphasis on a significant delay of the internal conversion (or, in general, radiationless transition), which is considered as an undesired phenomenon^{[19–21]}. This is in relation with the fact that a successfully populated excited state (${S}_{2}$) would be readily available for applications (such as the observation of desired photoproducts, mode selective chemistry, lighttriggered molecular rotors, separation of racemic mixtures) for only a brief period of time. In this respect, Refs. [20,21] focus specifically on an optimal control scheme to suppress radiationless transitions on nanosecond (ns) time scales after the external control is over. As for Ref. [19], the authors proceed to a detailed study of the ${S}_{0}\to {S}_{2}\to {S}_{1}$ process, putting the emphasis on maximal ${S}_{2}$ population during the control scheme (about 50 fs duration). Still other works^{[22,23]} refer to more general schemes by using overlapping resonances. The main computational limitation is identified as the reduced number of vibrational modes, usually four rather than 24. However, within some approximations, model extensions to 24 dimensions have been achieved, making use of a socalled QP (Q denotes the small subspace of interest and P the rest) partitioning^{[24]}. Motivated by lightharvesting systems, we pursue an opposite goal. Our purpose is to maximize the population of the energetically lower and thus more stable excited state ${S}_{1}$. The basic idea is to deposit the exciton energy from the bright donor state ${S}_{2}$ to the dark acceptor state ${S}_{1}$, through a funneling mechanism triggered by a CI and nonadiabatic internal coupling, taking advantage of a decreasing electronic gradient from ${S}_{2}$ to ${S}_{1}$.
Control objectives being opposite, the strategies we are developing also differ completely from the ones referred to in previous works. More precisely, when maximizing the ${S}_{2}$ population, the emphasis is put on stable, electronically localized eigenstates among the very dense manifold of vibronically coupled levels. References [20,21] are concerned by an optically coherent superposition of a specific eigenstate localized in the ${S}_{2}$ potential well. Reference [19] also deals with highlying ${S}_{2}$ vibrational states with significantly slower decay rates into ${S}_{1}$. A complete study of excitation frequencies is undertaken to resonantly reach the superposition of such excited vibrational states, using rather long duration pulses (180 fs for Refs. [20,21] and 40 to 100 fs for Ref. [19]). Some other works deal either with the dynamic Stark effect used as a basic mechanism to shift the CI away from the Franck–Condon (FC) region^{[25–27]}, or with optimal control theory^{[28]}, but they are limited to three or four modes of vibronic couplings. We are adopting different control strategies based on two mechanisms: either interference or sudden excitation kicks, as extended to highdimensional complex molecular systems. This is conducted by using ultrashort broadband pulses, without a specific resonant excitation frequency. We note that such strategies remain in the spirit of what has been suggested in the preliminary work of Ref. [29]. Our control schemes are built so as to decrease the ground state ${S}_{0}$ population and simultaneously reach the optimal contrast favoring ${S}_{1}$ population as compared to the one of ${S}_{2}$. As a signature of a stable deposit of electronic energy, we are interested in the postpulse ${S}_{1}$ population, taking into account the dephasing processes of the full 24mode dynamical description of pyrazine. On the experimental side, it is worth mentioning that several works are addressing the control of the internal conversion to the energy transfer ratio, using in particular, open learning loop setups and evolutionary algorithms^{[30–33]}. In particular, the multipulse excitation (a degenerate fourwavemixing sequence) has its counterpart in the present work, in terms of ultrashort pulse trains to propel the wave packet towards the CI.
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The manuscript is organized as follows. Section 2 is devoted to the Frenkel vibronic Hamiltonian involving the 24mode model of pyrazine, with all relevant parameters taken from Ref. [34]. The methodology adopted to treat this model is the multiconfigurational timedependent Hartree (MCTDH)^{[34–39]}, which is shortly reviewed with respect to its numerical convergence criteria in the CI branching space. The postpulse fieldfree dynamics and its recursive patterns in terms of wavepacket vibrational periods on both ${S}_{2}$ and ${S}_{1}$ potential energy curves are analyzed. A detailed interpretation is provided when going from 2D to fourdimensional (4D) and 24D models. Control strategies are presented in Section 3, putting the emphasis on control observations and parameters on the one hand and interference and kick mechanisms from weak to strong field regimes on the other hand. Results are presented in Section 4 for both the mechanisms with their interpretation. Conclusions and some perspectives are given in Section 5.
2. Methodology: Model Hamiltonian and Numerical Techniques
As in many previous works, we are considering three electronic states of pyrazine, namely, the ground state ${S}_{0}({{}^{1}A}_{g})$, together with the two lowest excited electronic states, ${S}_{1}({{}^{1}B}_{3u})$ and ${S}_{2}({{}^{1}B}_{2u})$. These states are wellseparated from others, close in energy, and vibronically coupled to each other^{[15]}. Figure 1 displays a schematic view of the corresponding states and their couplings.
Figure 1.Schematic view of the threestate model used for pyrazine. The ground state S_{0} is indicated in thick solid blue line. The bright donor state S_{2} is a thick solid black line, and the dark acceptor state S_{2} is a thick solid red line. The thin horizontal lines are for the corresponding vibrational states. The interstate coupling W_{12} in the dotted blue line and the transition dipole moments µ_{01} and µ_{02} are also indicated.
Following the usual model of displaced harmonic oscillators in open quantum systems, we work with a linear vibronic Hamiltonian expressed in a diabatic representation. In this section, we examine the fieldfree vibronic Frenkel Hamiltonian, the MCTDH method retained for solving it, and the generic fieldfree dynamics after launching part of the ground state wavepacket on the excited states. That is more precisely conducted with successive 2D, 4D, and 24D models for a clear understanding of the role of dimensionality.
2.1. Vibronic Hamiltonian
We briefly recall the generic fieldfree Frenkel Hamiltonian, which reads^{[40]}
It is to be noted that the two normal modes that span the CI branching space, using the symmetry group notations, are ${Q}_{6a}$ for state ${S}_{1}$ and ${Q}_{10a}$ for state ${S}_{2}$^{[40]}. As mentioned before, the three models will be referred to as 2D (${N}_{\alpha}=2$) involving ${Q}_{6a}$, ${Q}_{10a}$, 4D (${N}_{\alpha}=4$) when enlarging the branching space to ${Q}_{9a}$ and ${Q}_{1}$, and the full 24D (${N}_{\alpha}=24$) description involving all normal modes.
2.2. MCTDH survey
The MCTDH approach for multielectronic states and multimode nuclear dynamics has been presented in detail in the literature. For pyrazine, with its three electronic states in consideration, the multistate vibrational wave function is taken as a threedimensional column vector ${({\mathrm{\Psi}}_{1},{\mathrm{\Psi}}_{2},{\mathrm{\Psi}}_{3})}^{T}$, of singlestate nuclear wave functions ${\mathrm{\Psi}}_{n}(n=1,2,3)$, $T$ being the transpose. The total electronuclear eigenvector is then written as
$n$ labels the electronic states, and the unknown nuclear wave functions are solutions of the following closecoupled equations:
A standard multiconfiguration approach consists of expanding these functions on a timeindependent basis set ${\mathrm{\Phi}}_{J,n}$, with timedependent coefficients ${A}_{J,n}$, as
The major improvement brought by MCTDH is that not only the coefficients but also the basis functions are taken as timedependent. The challenge is that, by adapting the basis set functions to the temporal evolution, one must reduce the total number of these functions (i.e., ${N}_{J}$) for a given convergence criterion. To proceed along this line, the now timedependent basis set functions ${\tilde{\mathrm{\Phi}}}_{J,n}(Q,t)$ are given as a tensorial product of timedependent singleparticle functions (SPFs) ${\phi}_{{j}_{\alpha}}^{(\alpha )}({Q}_{\alpha},t)$ describing a given nuclear degree of freedom $\alpha $. The complete expansion reads as
Interestingly enough, there is no prescription for SPFs to depend on a single coordinate, and they may depend on several coordinates. To further reduce the memory and the numerical effort, several physical coordinates can be regrouped into a socalled combined mode^{[34]}:
Here, we use $p=8$ combined modes ${\tilde{Q}}_{j}$ ($j=\mathrm{1,\dots 8}$). More precisely, ${\tilde{Q}}_{1}$ results from a combination of ${Q}_{6a}$ and ${Q}_{10a}$. For the others, the combination scheme is indicated in parenthesis, i.e., ${\tilde{Q}}_{2}(1,9a,8a)$, ${\tilde{Q}}_{3}(2,4,5)$, ${\tilde{Q}}_{4}(6b,3,8b)$, ${\tilde{Q}}_{5}(7b,16a,17a)$, ${\tilde{Q}}_{6}(12,18a)$, ${\tilde{Q}}_{7}(19a,13,18b,14)$, and ${\tilde{Q}}_{8}(19\text{\hspace{0.17em}}b,20b,16b,11)$.
As already said, we use the normal coordinates of pyrazine in its electronic ground state for the dynamics ($f=24$). For the primitive basis set [${\chi}^{(\kappa ,j)}({Q}_{\kappa ,j})$ correspond to the primitive basis functions for the nuclear coordinate ${Q}_{\kappa ,j}$ and ${M}_{\kappa j}$ their number], we have used 22 harmonic oscillator discrete variable representation (DVR) functions for the nuclear (normal) coordinates ${Q}_{6a}$ and ${Q}_{10a}$, whereas only 12 and 11 such functions are retained for the normal coordinates ${Q}_{9a}$ and ${Q}_{1}$ normal modes, respectively. All other nuclear coordinates, not playing a crucial part in the dynamics, are merely described by four DVR functions. Numerical convergence is obtained by a much smaller basis set (${n}_{1}\cdots {n}_{8}$) of singleparticle functions, which are no more onedimensional (${\phi}_{j}^{(\kappa )}({\tilde{Q}}_{\kappa},t)$), as given in Table 1.

Table 1. Number of SingleParticle Functions (
The equations of motion for the coefficients $A$ of Eq. (8) result from the timedependent evolution equation involving the total Hamiltonian, i.e., the one of the system, together with the molecule field coupling. The closecoupled system of differential equations [Eq. (6)] is solved by projecting on the basis function of the combined modes. The initial condition, at time $t=0$, being taken as one for the vibrationless ground state $n=1$, and zero for all other states, the nuclear wave functions are built following Eq. (8). Finally, the timedependent population in each electronic state is given as
2.3. Fieldfree dynamics
In order to have a typical generic overview of the ${S}_{2}\to {S}_{1}$ population transfer, we now proceed to three dynamical calculations extending over a period of 500 fs. More precisely, we successively analyze 2D and 4D approximate models and relate them with the full 24D model. The minimal 2D model merely involves the two normal modes $\alpha =6a$ and $\alpha =10a$ building up the CI branching space. The population transfer dynamics are induced by some Franck–Condon vertical launching of the initial vibrationless (${v}_{\alpha}=0$ for all $\alpha $) ${S}_{0}$ wavepacket on the excited states. To fix the ideas, this is practically done referring to a low intensity ($5\times {10}^{12}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$), short duration (14 fs), resonant laser pulse (carrierwave frequency of 4.8 eV), which is symbolized by the vertical blue arrow of Fig. 2 positioned at time $t=0$. Note that several timeresolved photoelectron spectroscopy experimental studies have been reported using pump pulses around 4.7 eV: they do not report any dipoleallowed resonant (1 + 1) transition to a higher lying electronic state but a slow decay (of around 20 ps) to the electronic ground state; see the introduction of Ref. [42].
Figure 2.Excited states populations as a function of propagation time in a minimal 2D model involving the CI branching space. P_{2} and P_{1} are, respectively, indicated by thin black and thick red solid lines. The blue vertical arrow at t = 0 symbolizes the vertical launching of the v = 0 wavepacket from the initial state S_{0}. T_{1} and T_{2} are, respectively, the vibrational periods of the excited states S_{1} and S_{2}. W_{12}^{−1} is a notation for the CI characteristic transfer time.
It is worthwhile noting that the precise parameters of this excitation are an illustrative example of the kind of sudden excitation we refer to initiate the transfer dynamics. During the 14 fs ${S}_{0}\to {S}_{2}$ resonant excitation process, it is basically the ${S}_{2}$ state that is populated up to ${P}_{2}=0.37$, whereas ${P}_{1}$ does not exceed 0.03. The short pulse here is just to initiate a typical population transfer from the initial ground ${S}_{0}$ to the bright donor state ${S}_{2}$, without any control purpose. Actually, what we call freefield dynamics is to be understood as the wave packet dynamics following the switchoff of this pulse.
The postpulse fieldfree dynamics starts with this configuration and proceeds towards the descending energy gradient from ${S}_{2}$ to ${S}_{1}$ electronic potential energy surfaces coupled by ${W}_{12}$. In particular, due to the CI, an important amount of population is transferred to ${S}_{1}$. After a delay corresponding to the ${S}_{1}$ vibrational period (${T}_{1}=56\text{\hspace{0.17em}}\mathrm{fs}$ with the parameters of our model), the wavepacket reflected on the outer right turning point returns back to the CI region with a partial back transfer to ${S}_{2}$ and then evolves with another oscillation with the vibrational period (${T}_{2}=44\text{\hspace{0.17em}}\mathrm{fs}$) of the state ${S}_{2}$. The combination of these two oscillation modes in the vibrational baths of ${S}_{1}$ and ${S}_{2}$ gives rise to a recurrent pattern in the population dynamics that is then periodically repeated. Figure 2 provides a complete illustration of such recursive patterns. Calculations based on longer propagation times show that the revival structures last for more than 1 ps. They are progressively attenuated, and their periodicity is slightly altered due to wavepacket broadening. We now proceed by including higher degrees of freedom through additional normal modes. Figure 3 displays such calculations over 500 fs of propagation time, with the same excitation initiating the population transfer from ${S}_{0}$. It is to be noted that the additional degrees of freedom in multidimensional calculations dramatically increase the density of vibrational states. As a consequence, the specificity of the ${T}_{1}$ and ${T}_{2}$ vibrational periods of the $\alpha =6a$ and $\alpha =10a$ normal modes, building up the 2D model, is partially lost. The wavepacket is spreading over all the additional modes through all the vibrational states that form a quasicontinuum. Typically after 250 fs, the revival structures are completely suppressed. The ${P}_{1}$ and ${P}_{2}$ populations are moderately oscillating and progressively stabilizing in time.
Figure 3.Excited states populations as a function of propagation time in 4D (upper panel) and full 24D models (lower panel). Same notations as for Fig.
3. Coherent Control
The pyrazine molecule is studied in planar geometry and assumed to be oriented in a plane Oyz orthogonal to the propagation direction Ox of the electromagnetic field $E(t)$. The laser is supposed to be linearly polarized along the Oz axis. The timedependent total Hamiltonian is written in the length gauge and within the dipole approximation as
The transition dipole matrix elements between the ground ${S}_{0}$ and excited ${S}_{1}$, ${S}_{2}$ states are, respectively, noted as ${\mu}_{01}$ and ${\mu}_{02}$. Their explicit spatial expansions in terms of the normal mode coordinates are given by^{[26]}
It is worthwhile noting that for symmetry arguments, ${\mu}_{02}$ is much larger than ${\mu}_{01}$ due to its permanent dipole component ${\mu}_{02}(0)$. Numerical values we are using are extracted from Ref. [27]. It is also to be noted that we are working in such field conditions where referring to the polarizability is not necessary, as opposed to Ref. [27], concerned by very strong intensities and excitation conditions far from resonance. We will now examine the relevance of some control observables, together with two strategies to reach them in an optimal way, by exploiting two different mechanisms; namely, interference and kicks.
3.1. Control observable
The primary objective of this study is to optimize the population ${P}_{1}$ of the acceptor state ${S}_{1}$. But for pyrazine, the acceptor state ${S}_{1}$ is a quasidark one, as is clear from Eq. (17), practically without the possibility of a direct radiative excitation from the ground state. The population transfer process first proceeds with the radiative excitation of the ground ${S}_{0}$ state to the bright excited ${S}_{2}$, which then acts as a donor. The second step is monitored by the interstate coupling ${W}_{12}$, leading to a CIinduced strong nonadiabatic transfer from ${S}_{2}$ to ${S}_{1}$. A rather intuitive control objective could be taken as the maximization of the ratio $r(t)={P}_{1}(t)/{P}_{2}(t)$. But, this goes with the difficulty of getting very high ratio $r$, between actually very small excited states populations, that is without any practical interest. To properly take into account the two steps, excitation and funneling, we built a population contrast type control observable combining two requirements: (i) maximization of the excited states populations, which actually amounts to the minimization of the residual population ${P}_{0}$ of the ground state ${S}_{0}$ and (ii), among the two excited states, bring the maximum of population on the dark acceptor state ${S}_{1}$ or, in other words, increase the contrast between excited states population. Such a timedependent contrast is given by
It is worth noting that, in terms of $r$, Eq. (19) can be written as
Maximizing $\tilde{p}$ requires simultaneously maximizing the numerator, that is $r>1$, and minimizing the (always positive) denominator, which is achieved with ${P}_{2}(t)<1/[1+r(t)]$. Finally, combining these two requirements, in particular limiting the ${P}_{2}$ increase, our control objective goes much beyond the passive observation of the acceptor state population ${P}_{1}$ increase (i.e., increasing the donor state population ${P}_{2}$ and waiting for its CImediated transfer towards the acceptor).
On numerical grounds, in weak fields, ${P}_{0}$ would be considered not less than 0.5, meaning that no more than half of the initial population is assumed to be transferred to the excited states. As a consequence, even in the case of the best contrast that could ideally be reached between the donor and acceptor states (that is ${P}_{1}=0.5$ and ${P}_{2}=0$), the maximum possible value for $\tilde{p}$ should not exceed one. In other words, it is only in the strong field regime that we could expect a contrast $\tilde{p}>1$.
Finally, instead of the timedependent contrast given by Eq. (19), we choose a scalar control observable by taking time averages of $\tilde{p}(t)$:
This involves the definition of time windows over which the average is performed. A local contrast can be defined over a time window displaying the maximum of contrast, which happens during the first vibrational pattern, lasting over 45 fs, as illustrated in Fig. 2, and leading to the choice $[{t}_{\mathrm{min}},{t}_{\mathrm{max}}]=[45\text{\hspace{0.17em}}\mathrm{fs},90\text{\hspace{0.17em}}\mathrm{fs}]$. Although, during this 45 fs time window, some experimental observables are reachable, more interestingly, an asymptotic contrast is adopted in the following defining a semifinite time interval above 300 fs. Incidentally, it is to be noticed that 300 fs corresponds to typical electronic coherence times for such molecular systems [the dissipation comes from the intramolecular vibrational redistribution (IVR)] and an upper limit of 500 fs is taken as our final propagation time, leading to $[{t}_{\mathrm{min}},{t}_{\mathrm{max}}]=[300\text{\hspace{0.17em}}\mathrm{fs},500\text{\hspace{0.17em}}\mathrm{fs}]$.
3.2. Control fields
In the following, we refer to two laser control strategies based on two mechanisms, namely, interference and kicks. In both cases, the control field is made up of a train of $N$ timedelayed individual pulses. Two such pulses are illustrated in Fig. 4. The general expression for the field involved in Eq. (16) is given by
Figure 4.Sinesquare laser electric field envelopes for two identical pulses as a function of time. The illustrated pulse duration corresponds to T = 14 fs, whereas the interpulse time delay is τ = 6 fs.
More precisely, we compare pulse trains with the same total energy, namely, the one provided by the single pulse. To fix the ideas, for two identical pulses ${E}_{1}={E}_{2}=\u03f5$, $\mathcal{A}$ can be written as
Referring to Fig. 4, three cases are examined: (i) $\tau \ge T$, no overlapping pulses, resulting in $a=2$; (ii) $\tau =0$, full overlapping, with $a=4$; and (iii) $0<\tau <T$, partial overlapping, leading to a delaydependent $a$ ($2<a<4$), to be calculated according to Eqs. (24) and (25). Our control fields amplitudes are normalized so as to provide the same total electromagnetic energy. This is practically done by dividing the nominal intensities $I$ by $a$, or equivalently, the electric field amplitudes by $\sqrt{a}$.
We also wish to emphasize that when developing these strategies, we always have in mind the experimental feasibility with respect to pulse duration, intensities, and maximum number of pulses in the train. For the interference scenario, the laser leading frequency $\omega $ is taken to be approximately resonant with the excited states vibrational levels at the Franck–Condon vertical region of the ground state. The optimal value that is adopted is $\omega =4.77\text{\hspace{0.17em}}\mathrm{eV}$, i.e., a wavelength of $\lambda =260\text{\hspace{0.17em}}\mathrm{nm}$ in the UV region. The pulse duration $T$ is calculated such that the corresponding energy band broadening covers enough vibrational levels taking part in the funneling process. For the kicks scenario, the resonance condition is no more a relevant requirement, since the ultrashort pulses involved in this strategy would lead to excitation with even broader energy bands. All ultrashort pulses are taken as identical and typical values, which are retained ($T=14\text{\hspace{0.17em}}\mathrm{fs}$ or even $T=10\text{\hspace{0.17em}}\mathrm{fs}$) and correspond to band broadening ranging from $2382\text{\hspace{0.17em}}{\mathrm{cm}}^{1}$ up to $3335\text{\hspace{0.17em}}{\mathrm{cm}}^{1}$. Following a few attempts to roughly optimize the frequency $\omega $ and the duration $T$, we are finally left with two main control parameters, namely, the field intensity $I$ and the time delay $\tau $.
3.3. Coherent control strategies
The first strategy we are referring to relies on vibrational wave packets interference as a basic mechanism, envisioned either in a pumpprobe or a pumppump process. More precisely, as illustrated in Fig. 1, the pumpprobe process involves two routes to reach ${S}_{1}$, starting from the ground state ${S}_{0}$. Route 1 (pump pulse) proceeds through the intermediate state ${S}_{2}$ with the lowestorder transition amplitude given by^{[43]}
Finally, the interference scheme we are considering through Eq. (30) is between the transition amplitudes ${\mathcal{T}}_{1}$, with a controlled delay $\tau $. Such interference mechanisms have already been recently exploited in the control of rotational anisotropy^{[44]}. A physical understanding of the mechanism can be obtained by considering the vibrational wavepacket back and forth oscillations in the excited states potentials. Following a first laser pulse leading to a vertical Franck–Condon launching from ${S}_{0}$ to ${S}_{2}$, the vibrational wavepacket oscillates during its early dynamics $t<70\text{\hspace{0.17em}}\mathrm{fs}$ in the excited ${S}_{2}$ and ${S}_{1}$ states harmonic potentials. A second pulse is then applied with a controlled delay $\tau $ launching a second wavepacket from the ground state ${S}_{0}$, which can interfere with the first one, as they may overlap if the delay is adequately chosen. A constructive interference would produce an amplitude enhancement close to the CI, and thus an efficient population transfer from ${S}_{2}$ to ${S}_{1}$. Later on ($t>70\text{\hspace{0.17em}}\mathrm{fs}$), due to wavepacket spatial broadening and also its dispatching over additional degrees of freedom, the control efficiency is expected to be lost, as a consequence of less important successive wavepacket overlapping.
The interference mechanism can be extended and complemented by a kick mechanism, in the spirit of the one already discussed previously in the context of molecular alignment or orientation processes^{[45]}. At this respect, we are considering a train of $N=5$ identical individual pulses in Eq. (22). The kick mechanism assumes sudden excitation of the system. We are therefore addressing even shorter duration $T=7\text{\hspace{0.17em}}\mathrm{fs}$ to 10 fs pulses without the specific need for a resonance condition. The molecule can be kicked every time when ${P}_{1}$ starts decreasing.
As has previously been stated, the control observable given by Eq. (19) highly depends on the field intensity. In particular, it is only for strong field regimes that it can take the values exceeding 0.5. In order to fix the intensity regimes for the present system described within a 4D model, we proceed with dynamical calculations using both a single pulse or two (equal fluence) pulses with a typical delay of 13 fs and for a series of increasing intensities. Figure 5 plots the bright donor state population ${P}_{2}$ at the final time ($t=50\text{\hspace{0.17em}}\mathrm{fs}$) as a function of the field leading intensity. As is clear from the figure, for the single excitation, the weak field regime extends up to intensities $I{\text{\hspace{0.17em}}=\text{\hspace{0.17em}}10}^{13}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$, for which ${P}_{2}$ increases rather linearly. The strong field regime shows nonlinear behaviors: first, a saturation in ${S}_{2}$ population, for about $I=2\times {10}^{13}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$, followed by a decrease down to $I\text{\hspace{0.17em}}=\text{\hspace{0.17em}}7\times {10}^{13}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$, corresponding to partial population trapping in the ground state ${S}_{0}$, and then again an increase. We checked that the limit between weak (linear behavior) and strong fields (nonlinear behaviors) remains practically unchanged when applying two pulses of equal fluence with a typical delay of $\tau =13\text{\hspace{0.17em}}\mathrm{fs}$, as will basically be the case for the following control issues.
Figure 5.Excited state population P_{2} as a function of the leading intensity. Red dashed thin line for a single pulse, black thick solid line for two equal fluence pulses, with a delay of τ = 13 fs.
4. Results and Discussion
We will successively examine the efficiency and robustness of the interference and kick mechanisms as implemented in their respective control strategies, when going from a reduced 4D to a full 24D model. The guiding principle is to define the parameters of the control field in a 4D realistic and tractable model. This field, once obtained, is later used in a fulldimensional dynamic calculation including all 24 degrees of freedom. The challenge is to discover the efficiency and robustness of such a control field that persists when confronted with the presence of the other numerous degrees of freedom of the molecule.
4.1. Interference mechanism
Having in mind the general postpulse evolution of the excited states populations as illustrated by Fig. 2, two possible interference schemes could be envisioned. One concerns the nuclear wavepacket early dynamics for times typically less than $t<70\text{\hspace{0.17em}}\mathrm{fs}$ within the first vibrational periods, the second corresponding to longer times, typically $t>70\text{\hspace{0.17em}}\mathrm{fs}$ when the wavepacket revisits the FC region, following the revival patterns that are observed in Fig. 2. The asymptotic contrast, as defined by Eq. (21), is calculated in a 4D model as a function of the delay $\tau $ between the two pulses of fixed duration $T=14\text{\hspace{0.17em}}\mathrm{fs}$. All calculations are conducted within the appropriate field amplitude renormalization condition [cf. Eq. (24)] leading to the same fluence as the one of the single pulse. The results are displayed in Fig. 6. Following the analysis of Fig. 5, we consider three laser intensities: two pertaining to the weak field regime $I=5\times {10}^{12}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$ and $I{\text{\hspace{0.17em}}=\text{\hspace{0.17em}}10}^{13}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$ and one to the strong field regime $I=8.8\times {10}^{13}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$. Weak fields, apart from being easier to realize experimentally, have the advantage of better supporting the firstorder perturbation approach of Eq. (28). However, the rather low contrast obtained for $I=5\times {10}^{12}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$ makes us favor the choice of $I{\text{\hspace{0.17em}}=\text{\hspace{0.17em}}10}^{13}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$ for a typical illustration of the weak field regime. In this regime, the optimal contrast $p=0.8$ is obtained for a delay of $\tau =25\text{\hspace{0.17em}}\mathrm{fs}$, which, as discussed before, denotes a rather good efficiency, taking into account that less than 20% of the initial state population is transferred to the donor state ${S}_{2}$, as seen from Fig. 5.
Figure 6.Asymptotic contrast (4D) as a function of the interpulse delay τ for two pulses, each with a duration of T = 14 fs, equal fluence pulses of leading intensity I. The weak field regime is illustrated by the intensities I = 5 × 10^{12} W/cm^{2} in the blue dasheddotted line, and by I = 10^{13} W/cm^{2} in the red dashed line. I = 8.8 × 10^{13} W/cm^{2} corresponds to the strong field regime, represented by the black solid curve.
As a clear indication of the efficiency of the optimal control strategy, we compare the two population transfer dynamics resulting from either a single pulse or two pulses of equal fluence with the optimal delay of $\tau =25\text{\hspace{0.17em}}\mathrm{fs}$ (Fig. 7). The single pulse depletes the ground state up to ${P}_{0}=0.65$. The second pulse in constructive interference, producing a depletion up to ${P}_{0}=0.38$. The contrast is increased by a factor 3.3 from $p=0.24$ up to $p=0.8$. The final population, which is deposited on state ${S}_{1}$, is ${P}_{1}=0.5$, meaning that about 50% of the ground state population is in the acceptor state, while only 16% remains in the donor.
Figure 7.Excited states populations as a function of the propagation time for two pulses of intensity I = 10^{13} W/cm^{2} delayed by τ = 25 fs. P_{0} is indicated in the dasheddotted black line, P_{1} in the thick red solid line, and P_{2} in the thin solid black line.
The strong field regime offers much better contrasts by depleting more of the ground state population ${P}_{0}$. The results displayed in Fig. 6 show that the maximum is reached for a delay of $\tau =6\text{\hspace{0.17em}}\mathrm{fs}$ and leads to $p=1.55$ as a contrast. Moreover, fast oscillations as a function of the delay show the sensitivity of the observable to this delay, which is another signature of interference effects. Four characteristic delays are considered in the 4D reduced model for an intensity $I=8.8\times {10}^{13}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$. The single pulse (corresponding to $\tau =0\text{\hspace{0.17em}}\mathrm{fs}$, after equal fluence renormalization) and $\tau =9\text{\hspace{0.17em}}\mathrm{fs}$ (corresponding to a local minimum in Fig. 6) result in comparable dynamics leading to small asymptotic contrasts (typically $p<0.3$). The third choice for the delay $\tau =6\text{\hspace{0.17em}}\mathrm{fs}$ corresponds to the optimal value reached during the early dynamics $p=1.55$, whereas the fourth choice $\tau =75\text{\hspace{0.17em}}\mathrm{fs}$ stands for the optimal value $p=0.9$ obtained during the late dynamics. In an a posteriori way, we also checked the sensitivity and optimality with respect to the resonance condition. The first rows of Table 2 for the interference scenario collect the asymptotic contrasts for two offresonance conditions. The frequency $\omega =4.56\text{\hspace{0.17em}}\mathrm{eV}$ below the resonance is better fitted for a population transfer towards state ${S}_{1}$. But, as such transfers cannot operate in a direct way, the efficiency in terms of contrast is not satisfactory. The frequency $\omega =5.15\text{\hspace{0.17em}}\mathrm{eV}$ above the resonance, although opening the possibility for some interesting superposition of vibrational levels mixing the two excited states, turns out to be less efficient. This study validates the choice made for the resonance frequency, as the presumably optimal one.

Table 2. Contrasts for Several Excitation Frequencies for Both Mechanisms Discussed in the Text
Finally, we introduce the optimal control parameter values obtained from the reduced 4D dynamics into a full 24D model describing population evolution. The results are displayed in Fig. 8 for the early dynamics and Fig. 9 for the late dynamics. It is worthwhile noting that such fulldimensional calculations are done once the laser parameters are optimized within the context of the reduced 4D model. As for the 24D calculations, they necessitate a central processing unit (CPU) time of about 32 h using the OpenMP parallelization scheme with 16 processors on an Intel E52665 computer. The most striking observation is that the population evolution in the 24D full dynamics closely follows the one of the 4D calculations. Both end with very comparable asymptotic contrast exceeding $p\simeq 1.6$ for $\tau =6\text{\hspace{0.17em}}\mathrm{fs}$ and $p\simeq 0.9$ for $\tau =75\text{\hspace{0.17em}}\mathrm{fs}$. Following this observation, late dynamics control turns out to be less efficient. Actually, better interference schemes are operating for early dynamics where the vibrational wavepacket dynamics is more accurately periodic, with better marked revival structures, better adapted for the interference mechanism. The asymptotic contrasts obtained in the 24D model are comparable with the ones of the reduced 4D model, even very slightly better. This is presumably due to the fact that the wavepacket is more efficiently and rapidly dispatching towards the additional degrees of freedom, resulting in a faster population stabilization between the excited states. As a conclusion, the optimal result obtained for an early dynamics control scheme gives an acceptor dark state population of about ${P}_{1}=0.6$, which means that 60% of the ground state initial population is transferred to the acceptor ${S}_{1}$, while only 16% remains on the donor ${S}_{2}$.
Figure 8.Excited states populations as a function of the propagation time for two pulses of intensity I = 8.8 × 10^{13} W/cm^{2} delayed by τ = 6 fs from a full 24D model. P_{0} is indicated in the dasheddotted black line, P_{1} in the thick red solid line, and P_{2} in the thin solid black line.
Figure 9.Excited states populations as a function of the propagation time for two pulses of intensity I = 8.8 × 10^{13} W/cm^{2} delayed by τ = 75 fs from a full 24D model. P_{0} is indicated in the dasheddotted black line, P_{1} in the thick red solid line, and P_{2} in the thin solid black line.
4.2. Kick mechanism
The kick mechanism operates with ultrashort (broadband) pulses, leading to successive sudden, nonresonant momentum transfer to the molecule^{[45]}. Figure 10 displays the asymptotic contrast as a function of a constant single delay $\tau $ between the kicks imparted to the system from a train of five identical pulses, with fixed intensity $I=5\times {10}^{12}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$. Here, we deal with two control parameters, namely, the pulse duration $T$ and the delay $\tau $. It is important to note that a precise value for the excitation frequency is not very relevant, since the broadband character of the individual pulses renders the strategy essentially nonresonant. As with the previously discussed mechanism, once the optimal pulse parameters are determined on the reduced more tractable 4D model, they are transposed to the full 24D dynamics. More precisely, we obtain from Fig. 10 an optimal asymptotic contrast $p=1.71$, for individual pulse duration $T=10\text{\hspace{0.17em}}\mathrm{fs}$ and delay $\tau =13\text{\hspace{0.17em}}\mathrm{fs}$. Here, also we checked the sensitivity of the results with respect to the excitation frequency $\omega $. The last rows of Table 2, corresponding to the kick scenario, collect the results for three offresonant frequencies. As expected, the contrast is not very sensitive to frequencies within a reasonable window covering $\omega =4.77\text{\hspace{0.17em}}\mathrm{eV}$ to $\omega =4.91\text{\hspace{0.17em}}\mathrm{eV}$, leading to almost the same $p=1.7$. It is only far from these values that the results differ but with a less efficient control.
Figure 10.Asymptotic contrast (4D) as a function of the interpulse delay
Figure 11 displays the corresponding time evolution of the electronic states populations. As can be seen from the stepbystep decreasing evolution of ${P}_{0}$, at each kick, the momentum imparted to the system induces an increasing population towards the excited states. Our primary aim was to increase regularly the population ${P}_{1}$. Such a control could presumably be possible with different delays between the kicks. But, for easier experimental requirements, we assume in this control scheme a maximum of equally delayed, five identical pulses pertaining to a weak field regime $I=5\times {10}^{12}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$. With such restrictions, the population ${P}_{1}$ is not regularly increasing, even during the time when the kicks are applied. Another strategy could be to apply kicks every time ${P}_{1}$ starts to decrease, as has been previously done in Ref. [45]. Targeting experimental feasibility, within this restrictive parameters sampling space, we get very encouraging results, with contrasts up to $p=1.75$. We also observe from the fulldimensional calculation of Fig. 11 that 60% of the population of the ground state is transferred to the dark acceptor, while only 16% remains on the donor. About 54 CPU hours are needed for this calculation with the same computer as mentioned before. It is worthwhile noting that short duration and low intensity fivepulse trains can do as well, even better than the two interfering long duration (resonant) intense pulses, as can be observed from Figs. 8 and 11.
Figure 11.Excited states populations as a function of the propagation time for five ultrashort T = 10 fs pulses of intensity I = 5 × 10^{12} W/cm^{2} delayed by τ = 13 fs for a full 24D model. P_{0} is indicated in the dasheddotted black line, P_{1} in the thick red solid line, and P_{2} in the thin solid black line.
5. Conclusion
In this work, we consider the intense laser control of funneling dynamics between electronic states with decreasing energy gradient and passing through a CI. The pyrazine molecule offers such a framework involving two excited electronic states, presenting a strong nonadiabatic coupling in relation with their CI. Moreover, the initial ground state is radiatively coupled to the highest energy excited state, which is considered as a bright donor (D). We are concerned by the population transfer dynamics from this state, to the lowest energy dark acceptor (A) state, motivated by the possibility of long term deposit in (A), like in lightharvesting systems. We propose some efficient and robust external control schemes to achieve stable optimal final population in (A). Facing an indirect population transfer process, our control observable should incorporate both a maximum depletion of the ground state and a maximum final population in (A), when sharing between (D) and (A). This is done by defining an asymptotic contrast in terms of the ratio of the population difference between (A) and (D) to the remaining ground state population. In order to maximize this contrast, and inspired by a thorough understanding of the 2D (CI branching space) postpulse fieldfree population evolution, we refer to two basic coherent control mechanisms, namely, pumppump interference and kicks. The search for the optimal control parameters for an electromagnetic field in terms of a train of ultrashort laser pulses is conducted on a reduced but still realistic 4D model of pyrazine. More precisely, we optimize the pulse leading frequency and intensity, duration, and the interpulse delay. Once obtained and fixed, these parameters are used in the full 24D dynamics to calculate the time evolution of the electronic states populations. It is very gratifying to see that this extension of control mechanisms, often established for over simplified systems involving only a small number of levels, can survive and be effective when applied to systems with a large number of degrees of freedom. This is probably the most relevant message of this study.
Having in mind the limitations of realistically achievable experimental conditions, the excited states population contrasts we obtained can be considered as efficient ones for pyrazine CImediated funneling dynamics. Referring to either interference or kick mechanisms, about 60% of the ground state population is deposited in the acceptor state, while about 16% remains in the donor state. As a perspective, we can mention other control strategies that could be checked in terms of their possible extension to multidimensional systems. One is the strong field pumpprobe interference we already mentioned [cf. Eq. (27)]; the other being the frequency chirp. The density of levels for the vibrational baths being important, addressing in an optimal way some specific levels playing a major role in the funneling dynamics control may deserve interest. Even more importantly, pyrazine should be considered as an example of a modestly complex system. As a conclusion and consequence of our findings, we anticipate the possibility of potential transposition of such control mechanisms to other larger biological systems. This would certainly be conditioned by having a small number of active normal modes among all the others, which are spectators in the dynamics we wish to control. Then, similar to what we have observed for pyrazine, external control fields once defined and fixed in lowdimensional models should be effective at higher dimensions.
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