Quantum Control of Population Transfer and Vibrational States via Chirped Pulses in Four Level Density Matrix Equations

Abstract

We investigate the effect of chirped excitation and the excitation detuning on the coherent control of population transfer and vibrational states in a four-level system. Density matrix equations are studied for optimally enhanced processes by considering specific parameters typical of oxazine systems. Our simulations show a strong dependence on the interplay between chirp and excitation detuning and predict enhancement factors up to 3.2 for population transfer and up to 38.5 for vibrational coherences of electronic excited states. The study of the dynamics of the populations and vibrational coherences involved in the four-level system allows an interpretation of the different enhancement/suppression processes observed.

1. Introduction

Tailoring of the optical phase of femtosecond pulses has been used in several applications in photochemistry and time-resolved spectroscopy [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. In general, tailored pulses can be applied using two different approaches: Coherent Control (CC) or Quantum Control Spectroscopy (QCS). Closed-loop learning algorithms are exploited in CC to generate complex optical phases in order to optimize specific reaction pathways. CC has been used as a tool in the control of dynamic processes [17,18,19,20,21], and it explains how a specific state can be selectively populated, for example, to attain 100% population transfer [22,23,24,25], and recent studies have shown the possibility of having enhancements even higher than 100% [26,27,28,29,30,31,32,33,34,35]. QCS also exploits tailored excitation but differs from CC in its applications [30,36]. Its goal is to simplify complex molecular dynamics by tailoring excitation pulses to enhance or suppress specific molecular signal features. A central aspect in both approaches, CC and QCS, is the control of population transfer between the ground and excited states, as well as the generation of vibrational coherence in both potential surfaces. By controlling the population transfer or by suppressing specific molecular vibrational coherences, a photochemical reaction channel can be selectively chosen [27,29,32,37,38,39,40,41,42,43,44,45,46,47,48,49], or a certain mode in a multidimensional time-resolved signal can be suppressed [37,50,51,52,53,54,55].

The control of population transfer and vibrational coherences in molecules in general is challenging since several molecular and optical parameters play a role. Indeed, the molecular system under study dictates the displacement of the addressed levels and the vibrational and electronic dephasing rates. While the electronic dephasing rates can in some cases be influenced by the choice of the solvent [56,57,58,59], the displacement between the ground and the excited states is generally fixed. Pulse characteristics can be more easily tailored than molecular properties. Crucial parameters are the spectral detuning of the excitation with respect to the molecular absorption region as well as the temporal shape of the excitation, which is tailored by using phase and/or amplitude shaping. In the present paper we address the coherent control of population transfer and vibrational states in laser dye oxazine systems. We have chosen oxazine systems since these systems are being extensively studied in spectrally and time-resolved transient absorption spectroscopy experiments, in the femtosecond time regime, to elucidate the fastest fundamental processes in photochemistry and photobiology, like isomerization and electron and photon transfer [60]. In particular, we analyze the response of a four-level system, consisting of two electronic states both including two vibrational sub-states (see Figure 1), to the excitation of chirped and detuned femtosecond laser pulses. We will show that the basic processes producing enhancement/suppression factors on the population transfer and vibrational coherences can be understood from this relatively simple molecular model by studying the dynamics of the density matrix elements, and will compare our results with previous theoretical studies and experimental measurements.

Indeed, an asymmetry in the molecular response to chirped pulses was first predicted theoretically by Ruhman and Kosloff [61] and observed experimentally by Bardeen et al. [62]. They explained how the excitation of a vibration in the ground state of a molecular system can be accomplished by negatively chirped pulses in an intrapulse pump-dump process. A similar scheme was implemented by Cao et al. [63] for quantum coherent control of population transfer. They demonstrated theoretically that electronic population inversion probabilities of up to 99% in molecular systems can be achieved by using intense positively chirped broadband laser pulses. Their results are robust with respect to thermal and condensed phase conditions and are supported by experimental evidence. For instance, solvent environment was recently shown to be sensitive to positively chirped pulses, and a dependence of the optimal chirp with respect to the displacement between excited and ground state’s molecular potentials was also found. On the way to find the optimal chirp for coherent manipulations, the electronic dephasing rate is theoretically studied to take both effects into account. It is found that for increasing dephasing times, the coherence is enhanced especially if negative chirp is added [64].

In this work we study in detail the influence of chirp, detuning and dephasing on the enhancement in population transfer and vibrational coherence in a four level system (Figure 1). We first describe the density matrix model and give the parameters used in Section 2. As mentioned above, the parameters used to solve our system are specific for the case of oxazine; however, due to the simplicity of the mathematical model that we consider, the physical mechanisms elucidated in our study are of a fundamental type and can therefore be extended to other systems. In Section 3 we show the main results obtained from the simulations. Maps showing the enhancement in the populations and the vibrational coherences have been produced by varying the incident spectral phase chirp from −10³ fs² to 10³ fs² and the excitation spectral detuning from −5 × 10³ cm⁻¹ to 5 × 10³ cm⁻¹. The most relevant regions in these maps are then studied in detail by plotting the evolution in time of the respective density matrix elements for the important specific cases (enhancement/suppression), and an interpretation of the observed effects is given. In Section 4 we present a study of the influence of the duration of the laser pulse on the observed enhancement effects. The influence of the laser peak intensity in some specific cases is shown in Section 5, and in Section 6 we investigate the important effect of electronic dephasing. A discussion and the conclusions are given in Section 7.

2. Density Matrix Simulations

A non-perturbative, time-dependent density matrix approach is used to simulate the interaction of a femtosecond laser pulse with a four level system consisting of 2 electronic states—ground and excited state, each containing two vibrational levels (Figure 1).

All simulations were carried out under the slowly-varying envelope (SVEA) and rotating-wave (RWA) approximations, by using a 4–5 order Runge-Kutta algorithm (see also [65]). The laser temporal pulse shape is a direct input to the code, and the system allows the laser pulse to interact with all electronic transitions. The model can be written as follows:

$$\begin{array}{l}{\stackrel{•}{\mathsf{\rho }}}_{22}=-{\mathsf{\gamma }}_{22}{\mathsf{\rho }}_{22}+{\mathsf{\gamma }}_{32}{\mathsf{\rho }}_{33}+{\mathsf{\gamma }}_{42}{\mathsf{\rho }}_{44}+i\{{\mathsf{\mu }}_{23}({\mathsf{\rho }}_{32}{\Omega }^{*}-{\mathsf{\rho }}_{23}\Omega )+{\mathsf{\mu }}_{24}({\mathsf{\rho }}_{42}{\Omega }^{*}-{\mathsf{\rho }}_{24}\Omega )\},\\ {\stackrel{•}{\mathsf{\rho }}}_{33}=-{\mathsf{\gamma }}_{33}{\mathsf{\rho }}_{33}+{\mathsf{\gamma }}_{43}{\mathsf{\rho }}_{44}+i\{{\mathsf{\mu }}_{13}({\mathsf{\rho }}_{13}\Omega -{\mathsf{\rho }}_{31}{\Omega }^{*})+{\mathsf{\mu }}_{23}({\mathsf{\rho }}_{23}\Omega -{\mathsf{\rho }}_{32}{\Omega }^{*})\},\\ {\stackrel{•}{\mathsf{\rho }}}_{44}=-{\mathsf{\gamma }}_{44}{\mathsf{\rho }}_{44}+i\{{\mathsf{\mu }}_{14}({\mathsf{\rho }}_{14}\Omega -{\mathsf{\rho }}_{41}{\Omega }^{*})+{\mathsf{\mu }}_{24}({\mathsf{\rho }}_{24}\Omega -{\mathsf{\rho }}_{42}{\Omega }^{*})\},\\ {\stackrel{•}{\mathsf{\rho }}}_{11}=-{\stackrel{•}{\mathsf{\rho }}}_{22}-{\stackrel{•}{\mathsf{\rho }}}_{33}-{\stackrel{•}{\mathsf{\rho }}}_{44},\\ {\stackrel{•}{\mathsf{\rho }}}_{12}=-{\overline{\Gamma }}_{12}{\mathsf{\rho }}_{12}+i{\mathsf{\omega }}_{21}{\mathsf{\rho }}_{12}+i\{{\mathsf{\mu }}_{13}{\mathsf{\rho }}_{32}{\Omega }^{*}-{\mathsf{\mu }}_{23}{\mathsf{\rho }}_{13}\Omega +{\mathsf{\mu }}_{14}{\mathsf{\rho }}_{42}{\Omega }^{*}-{\mathsf{\mu }}_{24}{\mathsf{\rho }}_{14}\Omega \},\\ {\stackrel{•}{\mathsf{\rho }}}_{13}=-{\overline{\Gamma }}_{13}{\mathsf{\rho }}_{13}+i({\mathsf{\omega }}_{31}-{\mathsf{\omega }}_1){\mathsf{\rho }}_{13}+i{\Omega }^{*}\{{\mathsf{\mu }}_{13}({\mathsf{\rho }}_{33}-{\mathsf{\rho }}_{11})+{\mathsf{\mu }}_{14}{\mathsf{\rho }}_{43}-{\mathsf{\mu }}_{23}{\mathsf{\rho }}_{12}\},\\ {\stackrel{•}{\mathsf{\rho }}}_{14}=-{\overline{\Gamma }}_{14}{\mathsf{\rho }}_{14}+i({\mathsf{\omega }}_{41}-{\mathsf{\omega }}_1){\mathsf{\rho }}_{14}+i{\Omega }^{*}\{{\mathsf{\mu }}_{14}({\mathsf{\rho }}_{44}-{\mathsf{\rho }}_{11})+{\mathsf{\mu }}_{13}{\mathsf{\rho }}_{34}-{\mathsf{\mu }}_{24}{\mathsf{\rho }}_{12}\},\\ {\stackrel{•}{\mathsf{\rho }}}_{23}=-{\overline{\Gamma }}_{23}{\mathsf{\rho }}_{23}+i({\mathsf{\omega }}_{32}-{\mathsf{\omega }}_1){\mathsf{\rho }}_{23}+i{\Omega }^{*}\{{\mathsf{\mu }}_{23}({\mathsf{\rho }}_{33}-{\mathsf{\rho }}_{22})+{\mathsf{\mu }}_{24}{\mathsf{\rho }}_{43}-{\mathsf{\mu }}_{13}{\mathsf{\rho }}_{21}\},\\ {\stackrel{•}{\mathsf{\rho }}}_{24}=-{\overline{\Gamma }}_{24}{\mathsf{\rho }}_{24}+i({\mathsf{\omega }}_{42}-{\mathsf{\omega }}_1){\mathsf{\rho }}_{24}+i{\Omega }^{*}\{{\mathsf{\mu }}_{24}({\mathsf{\rho }}_{44}-{\mathsf{\rho }}_{22})+{\mathsf{\mu }}_{23}{\mathsf{\rho }}_{34}-{\mathsf{\mu }}_{14}{\mathsf{\rho }}_{21}\},\\ {\stackrel{•}{\mathsf{\rho }}}_{34}=-{\overline{\Gamma }}_{34}{\mathsf{\rho }}_{34}+i{\mathsf{\omega }}_{43}{\mathsf{\rho }}_{34}+i\{{\mathsf{\mu }}_{13}{\mathsf{\rho }}_{14}\Omega -{\mathsf{\mu }}_{14}{\mathsf{\rho }}_{31}{\Omega }^{*}{+\mathsf{\mu }}_{23}{\mathsf{\rho }}_{24}\Omega -{\mathsf{\mu }}_{24}{\mathsf{\rho }}_{32}{\Omega }^{*}\},\end{array}$$

(1)

where ${\mathsf{\rho }}_{ii}$ are the populations of the levels, the off-diagonal elements ${\mathsf{\rho }}_{ij}$ represent the coherences between levels, and ${\mathsf{\mu }}_{ij}$ are the dipole coupling coefficients of the electronic transitions. ${\mathsf{\omega }}_{ij}=|E_j-E_i|/\hslash$ are the angular frequencies of the transitions $i-j$, with $E_{i,j}$ being the energies of the quantum states $i$ and $j$ (see Figure 1). $\Omega (t)=E(t)\text{exp}[i\mathsf{\phi }(t)]/2\hslash$ represents the input slowly-varying complex amplitude of the electric field. E(t) is the slowly varying real field envelope, and $\mathsf{\phi }(t)$ is the slowly varying phase, which for transform limited pulses is chosen constant. The decay rates of the transitions $i-j$ are given by ${\mathsf{\gamma }}_{ij}$, and the relaxation times of the levels satisfy ${\mathsf{\gamma }}_{22}{=\mathsf{\gamma }}_{21}$, ${\mathsf{\gamma }}_{33}{=\mathsf{\gamma }}_{31}{+\mathsf{\gamma }}_{32}$ and ${\mathsf{\gamma }}_{44}{=\mathsf{\gamma }}_{41}{+\mathsf{\gamma }}_{42}{+\mathsf{\gamma }}_{43}$. The decay rates of the coherences are defined by ${\overline{\Gamma }}_{12}=({\mathsf{\gamma }}_{22}/2+{\Gamma }_{12})$, ${\overline{\Gamma }}_{13}=({\mathsf{\gamma }}_{33}/2+{\Gamma }_{13})$, ${\overline{\Gamma }}_{14}=({\mathsf{\gamma }}_{44}/2+{\Gamma }_{14})$, ${\overline{\Gamma }}_{23}=(({\mathsf{\gamma }}_{22}+{\mathsf{\gamma }}_{33})/2+{\Gamma }_{23})$, ${\overline{\Gamma }}_{24}=(({\mathsf{\gamma }}_{22}+{\mathsf{\gamma }}_{44})/2+{\Gamma }_{24})$ and ${\overline{\Gamma }}_{34}=(({\mathsf{\gamma }}_{33}+{\mathsf{\gamma }}_{44})/2+{\Gamma }_{34})$, which include the relaxation of the coherences and the purely dephasing rates ${\Gamma }_{ij}$. The parameters used are summarized in

Table 1. Parameters used in the simulations.

Parameters		Value
Transitional dipole coupling coefficient (μij)		$4.22\times {10}^{-29} \text{Cm}$
Peak amplitude of field, E		$8.7\times {10}^7 \mathrm{V}/\mathrm{m}$
Vibrational Frequency
(${\tilde{\mathsf{\upsilon }}}_{12}$ and ${\tilde{\mathsf{\upsilon }}}_{34}$)${\tilde{\mathsf{\upsilon }}}_{12}={\tilde{\mathsf{\upsilon }}}_{34}=600 {\text{cm}}^{-1}$		600 cm−1
Wave numbers of the different transitions.
${\tilde{\upsilon }}_{13}$	$|1\rangle$ → $|3\rangle$	19,400 cm−1
${\tilde{\upsilon }}_{14}$	$|1\rangle$ → $|4\rangle$	20,000 cm−1
${\tilde{\upsilon }}_{23}$	$|2\rangle$ → $|3\rangle$	18,800 cm−1
Spectral chirp φʺ		−1000 fs2 to 1000 fs2
Purely dephasing rates (${\Gamma }_{\text{ij}}$)		${\Gamma }_{12}^{-1}={\Gamma }_{34}^{-1}=1.7 \text{ps}$
		${\Gamma }_{13}^{-1}={\Gamma }_{14}^{-1}={\Gamma }_{23}^{-1}={\Gamma }_{24}^{-1}=100 \text{fs}$
Decay rates of the transitions, i − j (${\mathsf{\gamma }}_{ij}$)		${\mathsf{\gamma }}_{ij}^{-1}=1 \text{ns}$
Duration of the Gaussian transform-limited (TL) pulses		30 fs, 17 fs

.

The non-diagonal terms ${\mathsf{\rho }}_{13}$,${\mathsf{\rho }}_{14}$, ${\mathsf{\rho }}_{23}$ and ${\mathsf{\rho }}_{24}$ define the electronic coherences and include dispersive and absorptive properties in the optical transitions, while ${\mathsf{\rho }}_{12}$ and ${\mathsf{\rho }}_{34}$ represent the vibrational coherences of the ground and excited states, respectively.

The enhancement factors are defined by the ratio between the values calculated using a chirped pulse (PS) and the ones calculated with a transform limited (TL) Gaussian pulse as: enhancement of population in the ground states ${({\mathsf{\rho }}_{11}+{\mathsf{\rho }}_{22})}_{\text{PS}}/{({\mathsf{\rho }}_{11}+{\mathsf{\rho }}_{22})}_{\text{TL}}$, of vibrational coherence in the ground states ${|{\mathsf{\rho }}_{12}|}_{\text{PS}}/{|{\mathsf{\rho }}_{12}|}_{\text{TL}}$, of population in the excited electronic state ${({\mathsf{\rho }}_{33}+{\mathsf{\rho }}_{44})}_{\text{PS}}/{({\mathsf{\rho }}_{33}+{\mathsf{\rho }}_{44})}_{\text{TL}}$, and of vibrational coherence in the excited electronic state ${|{\mathsf{\rho }}_{34}|}_{\text{PS}}/{|{\mathsf{\rho }}_{34}|}_{\text{TL}}$. The initial population is in level |1〉 in all the simulations.

3. Main Results Obtained from the Simulations

In order to resolve the dependence on the central input pulse frequency and the amount of chirp on the transfer of population and the creation of vibrational coherences in the system, the central wavenumber of the input pulse is considered from $\tilde{\nu }=$ 15,000 cm⁻¹ to 25,000 cm⁻¹ (see Figure 1), in steps of 100 cm⁻¹, and the values of the spectral phase chirp have been taken from φʺ = −10³ fs² to 10³ fs², in steps of 10 fs². The spectral chirp parameter φ'' has been implemented in the Fourier domain and is defined as φ(ω) = 0.5 φʺ ω², with φ(ω) being the group delay dispersion (GDD) added to the spectral phase of the pulse. Therefore, since only the spectral phase of the laser pulse is varied in our study, the frequency content and the power remains as in the TL pulse for any value of the spectral chirp φʺ, as it is usually the case in the experiments. The rephasing of the spectral components produces a time broadening of the pulse given by $t_{p_c}=\sqrt{t_p^4+16{(ln2)}^2{\mathsf{\varphi }}^{″2}}/t_p$, where $t_{p_c}$ is the full width at half maximun (FWHM) of the chirped pulse intensity profile and $t_p$ is the duration corresponding to the TL pulse [66].

Figure 2 shows the results for the enhancement/suppression factor of the population in the ground state (Figure 2a), the vibrational coherence in the ground state (Figure 2b), the population in the upper state (Figure 2c) and the vibrational coherence in the upper state (Figure 2d), sufficiently long after the interaction of the system with the laser pulse.

As is clear from Figure 2, the interplay between chirp and excitation detuning is crucial for the coherent control of the population transfer and the enhancement or suppression of the vibrational coherences. We can observe regions with more than a 2% enhancement/suppression in the case of the population in the excited state (Figure 2c), and enhancement factors as high as 4 for the vibrations in this state (Figure 2d). In order to understand the origin of the processes involved in the calculations of Figure 2, the different time-scales involved in the interaction need to be considered. On the one hand, the period of the vibration both in the ground and the excited states, which in our simulations is T_vib = 1/(c$\tilde{\nu }$) ≃ 60 fs. On the other hand, we note that the duration of the 30 fs laser pulse with φʺ = ±10³ fs² is ≃ 94 fs. For the values of the peak field amplitude and dipole coupling that we have considered, the period associated to the Rabi oscillations is T_Rabi = h/(μE) ≃ 180 fs. Also important is the spectral width of the 30 fs Gaussian laser pulse, which is 0.441/(ct_p) ≃ 490 cm⁻¹, which is to be compared with the 600 cm⁻¹ of the vibrations considered in our simulations. The fastest dephasing rate in the simulations shown in Figure 2 is 100 fs, which therefore allows in this case for purely coherent processes, since all the interaction is basically finished before that time. We study the influence of the dephasing rates on the enhancement effects in the last part of the paper. We next describe in detail and provide an interpretation of the main results shown in Figure 2.

(1) Population in the ground state (ρ11 + ρ22):

The peak enhancement of the population in the ground state (Figure 2a) is ≃ 1.073 at a chirp value of φʺ = −10³ fs² and for an excitation wave number of $\tilde{\nu }$ ≃ 19,520 cm⁻¹, which is red-detuned and close to the resonance of the first upper vibrational state at $\tilde{\nu }$ = 19,400 cm⁻¹. The highest suppression is obtained at a chirp of 10³ fs² with a laser central wavenumber of $\tilde{\nu }$ ≃ 20,320 cm⁻¹, which is blue-detuned to the resonance of the second upper vibrational state at $\tilde{\nu }$ = 20,000 cm⁻¹. The asymmetry of the enhancement with respect to the sign of the chirp can be understood as follows. The first half of the interacting laser pulse promotes population inversion, which can be available for stimulated emission depending on the shape of the laser pulse and its spectral components. In the case of the highest enhancement of the population ρ₁₁ + ρ₂₂, for instance, if we consider that the upper state is first excited at a wave number at $\tilde{\nu }$ ≃ 19,765 cm⁻¹, which corresponds to the beginning of the laser pulse, a vibrational wave packet will be created in the upper state, which will initiate its oscillation between the upper state vibrational sublevels at this particular frequency. The round trip of the wave packet to the highest vibrational level at $\tilde{\nu }$ = 20,000 cm⁻¹ and back to $\tilde{\nu }$ ≃ 19,765 cm⁻¹ will take about 23.5 fs. From that time, the resonant transitions of the excited atom or molecule available for stimulated emission will be smaller than 19,765 cm⁻¹, and therefore the red shifted frequencies of the negatively chirped laser pulse will meet them and stimulated transfer back to the ground state will be promoted. This is therefore an adiabatic intra-pulse pump-dump scenario that explains the enhancement observed for negatively chirped pulses. For a laser pulse positively chirped, e.g., at this same central pulse frequency of 19,520 cm⁻¹, however, the later higher frequencies of the pulse cannot promote the system down and will instead continue exciting the system to the upper states. Consequently, the population of the ground state when the laser pulse is gone is not enhanced for positively chirped pulses, as it can be seen in Figure 2a. The results that we obtain with our simple four-level system are therefore fully corroborated by the previous theoretical studies and experimental measurements [61,62,63]. In Figure 3 we have plotted the evolution in time of the populations at some specific values of the central laser pulse wavenumber and chirp. Figure 3a shows how the maximum enhancement obtained for negatively chirped pulses at 18,800 cm⁻¹ presents an oscillatory behavior of ρ₁₁ + ρ₂₂, while the TL pulse at this wavenumber induces a pump-dump scenario. Note that the transferred population is small at this detuning (≃5%). This behavior is qualitatively different from the more resonant cases in Figure 3b,c, where the pump-dump processes are not that clear and the population transfers become ≃40%.

(2) Coherence in the ground state |ρ₁₂|:

The highest enhancement of the coherence in the ground state is ≃1.35 and is obtained at a negative chirp of φʺ ≃ −430 fs² and at the excitation wavenumbers of $\tilde{\nu }$ ≃ 19,660 cm⁻¹ and $\tilde{\nu }$ ≃ 19,150 cm⁻¹ (see Figure 2b). The enhancement of the vibrational coherence in the ground states therefore responds basically to the same phenomena that the one for the ground state population, i.e., the pump-dump intra-pulse mechanisms are promoted for negatively chirped pulses and suppressed for positive chirp. The details of the precise values of the two central laser pulse wavenumbers that produce the maximum enhancement are intriguing intra-pulse mechanisms which have not been clarified completely. Our result is in accordance with the studies made in [60], where it is observed for oxazine 1 systems in ethanol that ground-state vibrational coherences are enhanced predominantly by negatively chirped pulses. Figure 4 shows the evolution in time of the coherences at some specific values of the central laser pulse wavenumber and chirp. In Figure 4a the time evolution of the vibrational coherence in the ground states for the two different wave numbers that produce enhancement at negative chirp can be seen, and also the reduction of the coherence |ρ₁₂| after the interaction of the first half of the pulse in the case of positively chirped pulses.

(3) Population in the upper states (ρ₃₃ + ρ₄₄):

Figure 2c shows that the peak enhancement in population transfer to the upper states is ≃1.23 at the chirp values φʺ = ±10³ fs² and for an excitation wave number of $\tilde{\nu }$ ≃ 20,550 cm⁻¹. A qualitatively different smaller enhancement of ≃1.1 is also obtained at the positive chirp φʺ = 260 fs² and for an excitation wave number of $\tilde{\nu }$ ≃ 18,840 cm⁻¹. The highest suppression is obtained at a close wave number of $\tilde{\nu }$ ≃ 18,920 cm⁻¹ for a negative chirp of φʺ = −10³ fs². On the one hand, the symmetric enhancement with respect to chirp at $\tilde{\nu }$ ≃ 20,550 cm⁻¹ can be understood by the evolution in time of the population in the upper states for the TL and the chirped pulses at this particular central laser pulse wavenumber, in Figure 3f. The behavior can be seen as the opposite to the effect in the ground state and for red detuned pulses (Figure 3a). In this case the oscillatory behavior corresponds to the positively chirped pulse. On the other hand, the enhancement obtained for red detuned pulses at $\tilde{\nu }$ ≃ 18,800 cm⁻¹ is the consequence of an optimized pump-dump transition, as shown in Figure 3d, which as opposed to the enhancement of the population in the ground state, it is favored by positively chirped pulses, as was also observed in [57,58,59].

(4) Coherence in the upper states |ρ₃₄|:

Figure 2d shows two narrow peaks of the upper vibrational coherence that are obtained at a pulse central frequency of $\tilde{\nu }$ ≃ 19,110 cm⁻¹ and for spectral phase chirps as φʺ ≃ −180 fs² and φʺ ≃ 170 fs². The corresponding values are ≃3.56 and ≃4.34, respectively, and therefore a slightly higher enhancement is obtained with positively chirped pulses. Of relevance is the narrow parameters window observed in this case. The nearly symmetric values of the upper vibrational coherence observed in our simulations, however, shows that the vibrational states in the upper levels can be enhanced by relatively small positive or negative chirp values, with a central laser pulse wavenumber red shifted with respect to the resonance of the upper vibrational levels. This result is partially in accordance with the studies made in [60], where it is shown that excited state vibrational coherences are enhanced predominantly by positively chirped pulses. Figure 4b shows the evolution in time of the coherence |ρ₃₄| at the relevant parameter values. It is clear that the two chirped pulses after the interaction of the first half of the pulse are able to retain the |ρ₃₄| coherence as compared to the TL pulse.

4. Influence of the Duration of the Laser Pulse

Although a systematic study of all the parameters involved is far from the scope of the present work, we have evaluated the effect of the TL laser pulse duration by repeating some of our simulations using a shorter laser pulse durations (17 fs), and then comparing the results to the ones obtained with the TL 30 fs pulse. Figure 5 shows the output obtained for five specific different laser pulse excitation wavenumbers (18,800 cm⁻¹, 19,400 cm⁻¹, 19,700 cm⁻¹, 20,000 cm⁻¹ and 20,600 cm⁻¹). As it is indicated, in the left-hand part of the figure (Figure 5a,c,e,g) the simulations have been performed with a laser pulse of 30 fs, as in Figure 2. The right hand panels (Figure 5b,d,f,h) show the results obtained by keeping all the same parameters but using the shorter laser pulse.

The results in the left hand side of Figure 5 are cuts of the density maps shown in Figure 2 at specific excitation wavenumbers. The right hand side panels obtained for 17 fs show basically the same qualitative behavior as the ones obtained for 30 fs, except that we observe some undulations of the enhancement factors as the chirp is varied. An overall increase in the value of their maximum is also observed. The observed undulation may occur due to the larger effect of chirp on the shorter 17 fs pulse, since its duration for φʺ = ±10³ fs² results in ≃159 fs, i.e., it is longer than the 94 fs that correspond to the 30 fs pulse at these chirp values. The longer time broadening of the 17 fs pulse hence allows the upper state vibration to make several oscillations (T_vib ≃ 60 fs) during the intrapulse interaction, and can thus explain the observed enhancement undulations. The overall highest value of the enhancement factors obtained can be explained by the broader spectral wavelength of the 17 fs laser pulse (≃ 865 cm⁻¹), which in this case covers all the resonant energies of the upper states, and therefore more resonant transitions become available. The maximum enhancement obtained with positively chirped pulses for the population in the upper state is in this case ≃1.4, at $\tilde{\nu }$ = 18,800 and with φʺ = 168.5 fs² (Figure 5f). For the coherence in the upper state the maximum enhancement is ≃3.2 at $\tilde{\nu }$ = 18,800 and for φʺ = 168.5 fs² (Figure 5h).

5. Influence of the Laser Peak Intensity

It is also worth noting that the Rabi period in the present simulations, as mentioned above, is ≃180 fs, which means that during the interaction times that we have considered (≲90 fs) the system is in the exciting half of the Rabi cycle. We have performed simulations by doubling the amplitude of the interacting pulse so that the Rabi period becomes of ≃90 fs. We would expect a response of the system to the fact that most of the interaction is produced during a whole Rabi cycle, but the results that we obtain are qualitatively the same as those described for a Rabi period of ≃180 fs. The only difference that we observe is quantitative. We obtain a higher enhancement factor of ≃3.2 for the population in ρ₁₁ + ρ₂₂. The maximum enhancement of |ρ₁₂| results in ≃1.6, and for ρ₃₃+ ρ₄₄ is ≃2.1. Of relevance, however, is the maximum enhancement factor in |ρ₃₄|, which becomes in this case ≃38.5. This large value is observed close to the resonance of the first vibrational upper state level, with $\tilde{\nu }$ ≃ 19,365 cm⁻¹ and for two almost symmetric chirp values, namely φʺ ≃ −210 fs² and φʺ ≃ 220 fs² (see Figure 6).

6. Influence of the Electronic Dephasing

In all simulations above, the electronic dephasing (T_2,elec ≡ ${\Gamma }_{ij}^{-1}$) was kept constant at 100 fs. The role of the electronic dephasing, however, has been already experimentally observed and theoretically discussed in several coherent control experiments of small and large molecular systems. In general, the magnitude of electronic dephasing of a given electronic transition determines how fast the induced dipole coherently oscillates after interacting with the laser electric field. In this regard, molecular systems with very fast electronic dephasing rates will be potentially less affected by, e.g., long tailored excitations than by shorter laser pulses, since the electronic coherence already dephases faster in the former case. We have considered the influence of electronic dephasing by studying the specific values T_2,elec = 5 fs, 25 fs, 50 fs, 100 fs and 200 fs with a TL laser pulse of 30 fs centered at 19,700 cm⁻¹. The case of T_2,elec = 100 fs corresponds to the one considered in Figure 5 (left panels) and it is shown with thicker blue lines, both in Figure 5 and Figure 7, for a clearer comparison.

The results in Figure 7 are basically expected. They show how a faster electronic dephasing destroys the enhancement/suppression effect of the chirped pulses both in the populations of the ground and excited states (Figure 7a,c), so that for T_2,elec = 5 fs no enhancement/suppression is obtained. This is clearly due to the lack of population transfer at this fast dephasing time. The vibrational coherences of the ground and upper states show a decrease in the enhancement in all cases as the dephasing times are reduced, and they also show a clear symmetric reduction for increasing positive and negative values of chirp as compared with the TL interaction (Figure 7b,d).

7. Discussion and Conclusions

The numerical calculations reported in this work produced two major results. The first one is the clear enhancement of population and vibrational coherence for near-resonant excitation obtained by using chirped laser pulses (Figure 5a,b,e,f). The enhancement of the population and the vibrational coherences in the upper and ground states differ, though, in the sign of the chirp and the direction of the detuning. While population transfer due to pump-dump processes from the ground and back to the ground states is maximized for negative chirp, when the laser is tuned near the lowest upper vibrational state (level $|3\rangle$), enhancement of the population in the excited states is obtained both for positive and negative chirps, with a central laser wavelength slightly tuned above the upper vibrational states. The simulations show that the enhancement of the ground vibrational states is obtained by negatively chirped pulses slightly tuned above and below level $|3\rangle$, while the enhancement of the excited state vibrations is observed in a tiny region with both positive and negative small chirps and the laser slightly tuned below level $|3\rangle$.

The second major result is the absence of population enhancement in the upper states using chirped pulses, when their spectra are resonant with the transition (Figure 5e,f). Also important is the lack of any substantial change in the population enhancement with resonant chirped pulses when molecular electronic dephasing is increased to 200 fs (Figure 7c). This clearly indicates how robust this result is for a wide range of molecular systems. Perhaps even more importantly, our finding hints at why coherent control experiments of photochemical reactions in electronic excited states, which often use electronic resonant tailored excitations, have been challenging in many cases [67,68,69].

In conclusion, we have performed a numerical study on coherent control via chirped pulses in a four-level system by considering specific parameters typical of oxazine systems. We have studied in detail the influence of the detuning of the laser input pulse with respect to the electronic transitions. We have evaluated in some particular cases the effect of the duration of the laser pulse, its peak intensity, and also the influence of the electronic dephasing. We conclude that the time ordering of frequencies inside a pulse makes a fundamental difference in the population transfer and vibrational coherence generation, which can be computed with our four-level model. Intra-pulse adiabatic mechanisms explain most of the enhancement effects that we have observed. Although the results that we have presented are obtained for specific parameter values, the numerical model that we have studied is sufficiently simple to allow for an extension of our findings to other systems. An advance in the understanding of these light-matter interaction processes could open up new pathways and ideas in the field of nonlinear quantum control spectroscopy [70].