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Coherent Control of Population Transfer via Linear Chirp in Liquid Solution: The Role of Motional Narrowing Porscha L. McRobbie and Eitan Geva* Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, United States ABSTRACT: The conditions under which linear chirp can be used to control population transfer between the electronic states of a chromophore dissolved in liquid solution are investigated. To this end, we model the chromophore as a twostate system with shifted electronic potential energy surfaces and a fluctuating electronic transition frequency. The fluctuations are described as an exponentially correlated Gaussian stochastic process, which can be characterized by the average fluctuation amplitude, σ, and correlation time, τc. The time-dependent Schrödinger equation is solved numerically for an ensemble of stochastic histories, at different values of σ and τc, and under a wide range of pulse intensities and linear chirp coefficients. In the limit τc → ∞, we find that control diminishes rapidly as soon as σ exceeds the bandwidth of the pulse. However, we also find that control can be regained by reducing τc. We attribute this trend to motional narrowing, whereby decreasing τc narrows down the effective bandwidth of the solvent-induced fluctuations. The results suggest that the choice of methanol as a solvent in the actual experimental demonstration of chirp control by Cerullo et al. [Chem. Phys. Lett. 1996, 262, 362−368] may have contributed to its success, due to the particularly short τc (∼20 fs) that the rapid librations of this hydrogen bonded liquid give rise to. The results also give rise to the rather surprising prediction that coherent control in liquid solution can be strongly dependent on the choice of solvent and be improved upon by choosing solvents that correspond to lower values of στc.

I. INTRODUCTION The goal of quantum coherent control is the design of laser pulses for controlling molecular dynamics.1−12 Although many of the early studies in this field have been focused on gas phase systems, there is a rapidly growing interest in extending its range of applicability to condensed phase systems.13−32 This interest is fueled by the fact that many important chemical and physical processes that one would like to control take place in the condensed phase, as well as by the development of the closed loop self-adaptive learning approach,10−12,33−40 which made it possible to explore the prospects of controlling complex systems without a detailed knowledge of the full Hamiltonian. However, the theoretical analysis of coherent control in the condensed phase is significantly more demanding than its gas phase counterpart. This is because one has to account for the influence of relaxation processes induced by interactions between the, typically few, photoactive degrees of freedom (DOF) of the chromophore and the bath of remaining photoinactive DOF. Importantly, one of the fastest relaxation processes in the condensed phase is electronic dephasing, which threatens to suppress the very same coherences coherent control relies upon. Thus, understanding the interplay between coherent control and dephasing is essential to determine the conditions under which coherent control is possible in the condensed phase. One class of problems that lends itself to such a study of the interplay between dephasing and control is based on using linear chirp to control population transfer between the © 2015 American Chemical Society

electronic states of a dye molecule in room temperature liquid solution.41−48 In this case, the laser frequency is a linear function of time and the chirp sign determines the temporal ordering of the pulse frequency components. When the chirp is positive, the low frequency components reach the sample before the high frequency components, whereas for a negative chirp the order is reversed. The ability of positively chirped pulses to transfer more population from the ground to the excited state than negatively chirped pulses has been demonstrated experimentally on dye molecules dissolved in liquid solution.42 For such a system, the electronic transition frequencies are known to fluctuate on the experimental time scale (i.e., the time period during which the pulse is on). It is also known that these fluctuations may lead to relaxation of electronic coherences, or electronic dephasing. The main goal of the current study is to elucidate the effect of electronic dephasing on population transfer control via linear chirp, and to shed light on the conditions under which one can expect to use this control scheme in the presence of electronic dephasing. To this end, we choose to model electronic dephasing in terms of two parameters, namely, the average amplitude, σ, and Special Issue: Ronnie Kosloff Festschrift Received: October 5, 2015 Revised: November 22, 2015 Published: November 23, 2015 3015

DOI: 10.1021/acs.jpca.5b09736 J. Phys. Chem. A 2016, 120, 3015−3022

Article

The Journal of Physical Chemistry A

from the vertical transition frequency, which is given by ωveg(t) = ωeg(t) + ω2Qd2/2 ℏ. The above choice to focus on the effect of interactions with the solvent on the 0−0 transition frequency is motivated by the fact that the dye molecules of the type used in experimental demonstrations of chirp control of population transfer are known to be rigid. Thus, one does not expect the ground and excited intramolecular PESs to be affected significantly by the solvent. However, because photoexcitation is accompanied by changes to the charge distribution within the dye molecule, one expects that the gap between the intramolecular PESs to be affected by the interaction with a polar solvent. Following the same phenomenology underlying the commonly used Kubo model for line-broadening,54 we assume that δω eg (t) follows exponentially correlated Gaussian statistics.55−58 This implies that the initial value of δωeg(t), δωeg(0), is sampled from a normal distribution with zero average, ⟨δωeg⟩ = 0, and width σ:

the correlation time, τc, of the electronic transition frequency fluctuations. This in turn allows us to interpolate smoothly between the limits of fast and slow fluctuations of the electronic transition frequency (i.e., the limits of homogeneous and inhomogeneous broadening, respectively).49 In this context, it should be noted that previous attempts to understand the effect of electronic dephasing on linear chirp control were based on describing it in terms of a dephasing rate constant of the form 1/T2, which corresponds to implicitly assuming that we are in the homogeneous limit (where 1/T2 = σ2τc).50−52 Another study by Fainberg et al.53 focused on the effect of a distinctly different relaxation mechanism, namely, vibrational energy relaxation on the excited potential energy surface (PES), which typically occurs on time scales slower than electronic dephasing in the type of systems considered here. Finally, we note that an approach similar to the one described here was taken by Demirplak and Rice in their study of the effect of dephasing on another population transfer control scheme which is based on stimulated Raman adiabatic passage (STIRAP).30 The remainder of this paper is organized as follows. The model system and simulation techniques are outlined in section II. The results are presented and discussed in section III. Summary and outlook are provided in section IV.

P(δωeg ) =

⟨δωeg (t ) δωeg (t ′)⟩ ≡ C(t −t ′) = σ 2 exp( −|t − t ′| /τc)

(1)

(2)

where ℏ2 d2 1 Ĥg = − + ω 2Q 2 2 2 dQ 2 ℏ2 d2 1 Ĥ e = − + ω 2(Q + Q d)2 2 dQ 2 2

Ĥ int(t ) = −W (t )|e⟩⟨g| − W *(t )|g⟩⟨e|

(3)

∞

Here, ω is the vibrational frequency (which for the sake of simplicity is assumed the same for the ground and excited states), Qd is the displacement between the excited state and ground state equilibrium geometries, and ωeg(t) is the 0−0 transition frequency. Importantly, the latter is treated as a stationary stochastic variable whose fluctuations reflect interactions with solvent molecules, which exhibit random thermal motion. More specifically, ωeg (t ) =

ωeg0

+ δωeg (t )

(6)

It should be noted that σ and τc define the characteristic amplitude and time scale of the solvent-induced electronic transition frequency fluctuations. An ensemble of stochastic trajectories traced by the stochastic variable δωeg(t) over the relevant experimental time scale (the pulse length) was generated by the colored noise algorithm in ref 57. The timedependent Schrödinger equation for each stochastic trajectory was solved numerically (see below), yielding a different prediction of the excited state population at the end of the pulse. The actual excited state electronic populations reported below were obtained by averaging over the ensemble of stochastic trajectories. The number of stochastic trajectories sampled over was determined by convergence. Within the rotating-wave approximation, the interaction of such a two-state system with a linearly chirped pulse that has a Gaussian envelope and a central frequency in resonance with the average vertical transition frequency, ⟨ωveg(t)⟩ = ω0eg + ω2Qd2/2ℏ, is described by59

Here, g and e correspond the ground and excited electronic states, respectively, ψg(Q,t) and ψe(Q,t) are the corresponding ground and excited vibrational wave packets (∫ dQ |ψg(Q,t)|2 + ∫ dQ |ψe(Q,t)|2 = 1), and Q is the intramolecular nuclear coordinate. The field-free Hamiltonian is assumed to be of the following form: Ĥ 0 = |g⟩Ĥg⟨g| + |e⟩[Ĥ e + ℏωeg (t )]⟨e|

(5)

Consistent with Gaussian statistics, we also assume that the fluctuations of δωeg(t) are completely characterized by the twotime correlation function, ⟨δωeg(t) δωeg(t′)⟩, which is assumed to be exponential with correlation time τc:

II. MODEL SYSTEM AND SIMULATION TECHNIQUES We consider an electronic two-state system whose state is defined by a wave function of the form ⎛ ψ (Q ,t )⎞ g ⎟ |Ψ(t )⟩ = ψg(Q ,t )|g⟩ + ψe(Q ,t )|e⟩ ≐ ⎜ ⎜ ψ (Q ,t )⎟ ⎝ e ⎠

⎡ (δω )2 ⎤ eg ⎥ exp⎢ − 2 ⎢⎣ 2σ 2 ⎥⎦ 2πσ 1

=−

∑

[SkjW (t )|ek⟩⟨gj| + SjkW *(t )|gj⟩⟨ek|]

j,k=0

(7)

where ⎤ ⎡ t2 W (t ) = μge E0 exp⎢ − 2 − iωegv t − iαt 2 2⎥ ⎦ ⎣ 2τ

(8)

Here, μge is the magnitude of the electronic transition dipole moment vector (a constant within the Condon approximation), E0 is the peak laser field amplitude along the direction of the transition dipole moment vector, τ is the pulse length, α is the linear chirp, {|gj⟩, |ek⟩|j, k = 0, 1, 2, 3, ...} are the vibronic states and Sjk ≡ ⟨gj|ek⟩ are the Franck−Condon (FC) coefficients.

(4)

where ω0eg is the solvent-free value (a constant) and δωeg(t) is the solvent-induced deviation, whose fluctuations reflect the corresponding fluctuations in the instantaneous solvent configuration. We also note that ωeg(t) should be distinguished 3016


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⎛ ⎛ ⎞ ̂ −W̃ *(t ) ⎞⎛ ψg(Q ,t )⎞ d ⎜ ψg̃ (Q ,t )⎟ i ⎜ Hg ⎟⎜ ⎟ =− dt ⎜⎝ ψ̃ (Q ,t )⎟⎠ ℏ ⎜⎝−W̃ (t ) ℏΔ(t ) + Ĥ ⎟⎠⎜⎝ ψ (Q ,t )⎟⎠ e e e

The latter can be obtained in closed form for this model and are given by 1 j!

Sjk =

⎛ mω 2 ⎞ 1 exp⎜ − Q ⎟ ⎝ 4ℏ d ⎠ k!

⎡ j ⎛ j ⎞⎛ ⎞ j − n dn k ⎤⎥ mω ⎢ ⎜ ⎟ ⎜ Q ⎟ (λ ) × ∑ ⎜n⎟ − ⎢⎣ ⎥⎦ ⎝ 2ℏ d⎠ dλ n n=0 ⎝ ⎠ λ=

(15)

= ωeg(t) − ⟨ω ⟩ and ⎡ ⎤ t2 2 W̃ (t ) = μge E0 exp⎣ − 2 − iαt 2⎦. It should be noted that 2τ the populations of the electronic states are invariant to the rotating-frame transformation, such that Here

mω /2 ℏ Q d

(9)

In practice, it is more convenient to express the pulse parameters in terms of the frequency-domain bandwidth, Γ, and spectral chirp, α′. The results reported below were obtained for a fixed value of Γ, which leads to the following relationship between α, and α′:59

α = α′

Γ4 1 + (α′Γ 2)2

Pg/e(t ) =

(10)

⎛ i ⎞ Û [(n+1)Δt ,nΔt ] ≈ exp⎜ − H̃ (nΔt )Δt ⎟ ⎝ ℏ ⎠

(18)

Û [(n+1)Δt,nΔt] can then be calculated by either a grid-based method46,60 or direct diagonalization. Both methods were used in the dephasing-free case, and were confirmed to give the same results as a way of verifying numerical convergence. However, because electronic dephasing is easier to implement and interpret within the energy representation, direct diagonalization was the method of choice when dephasing was included. Diagonalization of H̃ was performed via the Fortran90 LAPACK routine. For the value of Q d = ℏ/mω used in this study, converged results were obtained for 12−22 vibronic states per electronic state.

1 (12)

where I0 ∝ E02τ0. The equation of motion of the field-driven system is given by ⎛ ⎛ ⎞ ̂ −W *(t ) ⎞⎛ ψg(Q ,t )⎞ d ⎜ ψg(Q ,t )⎟ i ⎜ Hg ⎟⎜ ⎟ =− ⎜ ℏ −W (t ) ℏωeg (t ) + Ĥ e ⎟⎜⎝ ψ (Q ,t )⎟⎠ dt ⎜⎝ ψ (Q ,t )⎟⎠ ⎝ ⎠ e e

III. RESULTS AND DISCUSSION Following ref 52, we will present our results in terms of contour maps of the excited state population at the end of the pulse, Pe(tf), as a function of α′ and Ẽ 0. In what follows, we will refer to those plots as “population maps”. Dimensionless variables are used throughout. More specifically, time, length, energy, and frequency are scaled as follows: t ̅ = ωt, Q̅ = mω/ℏ Q , ϵ̅ = ϵ/ℏω, Ω̅ = Ω/ω (in what follows, the overbars are dropped for the sake of simplicity). The results reported below were obtained for a laser bandwidth of Γ = 1.27 and equilibrium geometry shift between the ground and excited state of Qd = 1.0. The α′ range was set to (−2.0, +2.0) and the Ẽ 0 range was set to (0, 10), with 33 grid points along each axis. It should be noted that the pulse duration increases with chirp according to eq 11. For the α′ range considered here, the maximum pulse duration is τ = 3.4τ0. The time step was chosen to be Δt = min(τ0,τc)/200, to ensure that it is sufficiently smaller than the fastest dynamical time scale. The population map in the dephasing-free case is shown in Figure 1. It should be noted that this plot is the same as that reported in ref 52 and is shown here for completeness and as a reference for comparison with the results in the presence of electronic dephasing. As expected, the amount of population

(13)

We assume that the initial state, prior to interaction with the pulse, is given by |Ψ(0)⟩ = ϕg0(Q)|g⟩, where ϕg0(Q) is the vibrational ground state wave function on the ground state PES. Thus, the initial ground and excited state populations are Pg(0) = 1 and Pe(0) = 0, respectively. Population transfer between electronic states is therefore pulse-induced. The excited state population at time tf following the interaction with the pulse is given by Pe(tf) = ∫ dQ |ψe(Q,tf)|2. In the next step, we make the transformation to a rotating f rame. The state vector in the rotating frame is defined by |Ψ̃(t )⟩ = exp[i⟨ωegv ⟩t |e⟩⟨e|]|Ψ(t )⟩ = ψg(Q ,t )|g⟩ + exp[i⟨ωegv ⟩t ]ψe(Q ,t )|e⟩ ⎛ ⎞ ⎛ ψ̃ (Q ,t )⎞ ψg(Q ,t ) ⎜ ⎟≡⎜ g ⎟ ≐ ⎜ exp[i⟨ω v ⟩t ]ψ (Q ,t )⎟ ⎜ ψ̃ (Q ,t )⎟ eg ⎠ ⎝ ⎠ ⎝ e e

(17)

Here, the overall time, t, is divided into N intervals of length Δt = t/N. The time interval is chosen small enough so that H̃ is quasi-constant within each time interval, i.e., Δt ≪ nin(τ0, τc), such that

(11)

1 + (α′Γ 2)2

∏ Û [(n+1)Δt ,nΔt ] n=0

Thus, for a given Γ, the narrowest pulse in the time domain is the transform limited one, which corresponds to zero chirp, α′ = 0, and whose length is given by τ0 = 1/Γ. The integrated intensity of such a pulse is proportional to E02τ. Thus, assuming the integrated intensity is fixed, this requires that the peak intensity becomes chirp-dependent:

I = I0

(16)

N−1

Û (t ) ≈

1 + (α′Γ 2)2 Γ

̃ (Q ,t )|2 ≡ ∫ dQ |ψg/e(Q ,t )|2 ∫ dQ |ψg/e

The rotating-frame time-dependent Schrödinger equation is solved by writing the time evolution operator from time 0 to time t in the following form:

It should also be noted that, for a fixed Γ, the pulse length is chirp-dependent:59 τ=

Δ(t)

v e g

(14)

The time-dependent Schrödinger equation in the rotating frame is then given by 3017


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for the relatively small value of Qd employed here. More specifically, the FC coefficients associated with transition frequencies below the vertical transition frequency are larger than those associated with transition frequencies above the vertical transition frequency. In the case of negatively chirped pulses, the high frequency components precede the low frequency ones, such that the FC coefficients increase with time, thereby leading to more population transfer back to the ground PES via stimulated emission. In contrast, in the case of positively chirped pulses, the low frequency components precede the high frequency ones, such that the FC coefficients decrease with time, thereby suppressing stimulated emission and maximizing population transfer. Interestingly, one expects that increasing Qd would give rise to a more symmetrical distribution of the FC coefficients and thereby diminished effectiveness of chirp control of population transfer. In this respect, it should be noted that chirp control of population transfer was demonstrated on rigid dye molecules where one expects Qd to be rather small. Also noteworthy is the behavior at α′ = 0, which corresponds to transform limited pulses. In this case, the pulse duration is sufficiently shorter than the vibrational period so as to be viewed as impulsive.52,62,63 The oscillatory pattern in the population transfer efficiency as a function of Ẽ 0 can therefore be viewed as arising from Rabi cycling, with maxima corresponding to π, 3π, ... pulses and minima to 2π, 4π, ... pulses. We next consider the case where the system is subject to electronic dephasing, in the limit when τc → ∞. In this case, the solvent-induced deviation of the electronic transition frequency relative to its solvent-free value, δωeg, remains constant throughout the duration of the pulse. The population maps for different values of σ are shown in Figure 2. As can be seen,

Figure 1. Population map in the dephasing-free case. The map is presented as a contour plot of the excited state population at the end of the pulse, Pe(tf), as a function of the linear chirp coefficient, α′, and field amplitude, Ẽ 0. Red and blue correspond to 100% and 0% population transfer from the ground to the excited electronic states, respectively.

transfer from the ground to the excited state is minimal at small values of the driving field amplitude, Ẽ 0 ≲ 1.0. Furthermore, the amount of population transfer is also seen to be independent of chirp in this region. This is consistent with early experimental studies that found population transfer to be chirp invariant when weak driving fields are used.42,61 This behavior changes rather dramatically at more intense laser fields, Ẽ 0 ≳ 1.0. Positive chirps are then seen to give rise to ∼100% population transfer, whereas negative chirps give rise to population transfer efficiencies that vary between ∼0% and ∼100%, depending sensitively on the specific values of field amplitude and chirp. Those trends can be explained on the basis of the asymmetrical structure of the distribution of the FC coefficients

Figure 2. Population maps for different values of σ (the RMS amplitude of the solvent induced electronic transition energy fluctuation) in the τc → ∞ limit (infinite solvent correlation time). The maps are presented as contour plots of the excited state population at the end of the pulse, Pe(tf), as a function of the linear chirp coefficient, α′ (Y axis), and field amplitude, Ẽ 0 (X axis). Red and blue correspond to 100% and 0% population transfer from the ground to the excited electronic states, respectively. 3018


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Figure 3. Population maps for different values of τc (the solvent correlation time) at σ = 0.20. (σ is the RMS amplitude of the solvent induced electronic transition energy fluctuation.) The maps are presented as contour plots of the excited state population at the end of the pulse, Pe(tf), as a function of the linear chirp coefficient, α′ (Y axis), and field amplitude, Ẽ 0 (X axis). Red and blue correspond to 100% and 0% population transfer from the ground to the excited electronic state, respectively.

components of the Bloch vector that are perpendicular to the Z axis. This in turn diminishes the amplitude of the Bloch vector, thereby reducing the efficiency of population transfer.29,32 In the next step, we consider the case where the system is subject to electronic dephasing with a finite τc. We do so for a fixed value of σ = 2.0, for which the ability to control population transfer via linear chirp is negligible when τc → ∞ (Figure 2). Figure 3 shows what happens to the population maps for σ = 2.0 when τc is decreased. The results clearly show that the population transfer efficiency increases with decreasing τc. Furthermore, the asymmetry between positive and negative chirps is observed to re-emerge when τc−1 approaches values comparable to σ. In other words, one’s ability to control population transfer by linear chirps can be recovered by decreasing τc. This interesting trend can be explained by invoking the concept of motional narrowing. To this end, it should be noted that decreasing τc corresponds to making δωeg(t) fluctuate more rapidly. As the fluctuations become more rapid, the effective electronic transition frequency experienced by the system is increasingly more the average of the instantaneous ones rather than the distribution of instantaneous ones. The net effect is therefore to narrow down the effective bandwidth of the solvent-induced fluctuations, thereby slowing down electronic dephasing and increasing the population transfer efficiency. It should be noted that the underlying physics is closely related to the motional narrowing mechanism responsible for the narrowing down of an inhomogeneously broadened absorption line shape upon making the fluctuations more rapid (e.g., by increasing the temperature), to obtain a homogeneously broadened absorption line shape whose width decreases with decreasing τc.54,64,65 To put the above results in context, we compare them to the experimental study by Cerullo et al.42 who demonstrated population transfer via linear chirped pulses in the oxazine dye

the ability to use linear chirp to control population transfer diminishes rapidly as soon as σ becomes larger than the bandwidth of the pulse. This can be traced back to the fact that increasing σ corresponds to averaging over a wider distribution of δωeg, which in turn means moving further away from resonance with the vertical transition frequency, where the FC coefficients are the largest. Thus, increasing σ effectively weakens the ability of the laser field to couple the ground and excited electronic states, thereby diminishing one’s ability to use the field to induce population transfer. Another perspective on the effect of electronic dephasing on one’s ability to control population transfer by chirped pulses can be obtained by viewing the transitions as occurring within an ensemble of two-level systems that correspond to the different vibronic transitions. Each of those two-level systems can be described by a Bloch vector.29,32 The initial state corresponds to having the Bloch vectors of the two-level systems associated with transitions from the ground vibrational state on the ground electronic PES to vibrational states on the excited electronic PES aligned along the negative Z axis. The corresponding 2 × 2 density matrices are then diagonal in the vibronic representation, which implies that the coherences between vibronic states (represented by the off-diagonal matrix elements) are zero. Thus, being coherence-free, the initial state is not impacted by electronic dephasing. However, the effect of the interaction with the laser pulse is to rotate the Bloch vector around the rotating Y axis and toward the positive Z axis.29,32 For example, a true π pulse would correspond to rotating the Bloch vector, without changing its amplitude, so as to align it exactly with the positive Z axis by the end of the pulse, thereby leading to 100% population transfer. Importantly, rotating the Bloch vector in the XZ plane corresponds to the creation of electronic coherences (nonzero off-diagonal density matrix elements). This exposes the state to electronic dephasing, while the Bloch vector is rotating, which acts to diminish the 3019


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Finally, although the two-state model used in this study is rather simple, our main conclusions rely on the interplay between the pulse length and the amplitude and time scale of the frequency fluctuations, rather than on fine details of the PESs. Thus, assuming the same vibrational frequency in the ground and the excited electronic states should not have a major impact on them. Furthermore, the methodology is general and can be extended in a straightforward manner to account for higher dimensional and anharmonic intramolecular PESs as well as explicit estimations of σ and τc from MD simulations. Work on such extensions is underway and will be reported in future publications.

LD690 dissolved in methanol. Those authors reported a chirp dependence of population transfer, with positive chirp leading to significantly higher population transfer efficiency in comparison to negative chirp. Those authors also argued that the system can be modeled in terms of a single vibrational mode of frequency 170 cm−1, which is consistent with the model employed here. Setting ω = 170 cm−1, the parameters used to generate the results in Figure 3 correspond to τ0 = 25 fs, σ = 340 cm−1, and στ0 = 0.26. Bardeen and Shank have made a systematic comparison of the effect of different alcohol solvents on the absorption spectrum of LD690.66 MD simulations suggest that rapid librational-type motions dominate electronic dephasing in those systems.67 For LD690 in methanol, those authors reported σ ≈ 200 cm−1 and τc = 20 fs, which implies that στc = 0.120. This is comparable to the case shown in Figure 3e, corresponding to στc = 0.158. It should be noted that the case in Figure 3e corresponds to a significant population transfer that favors positive chirp over negative chirp. Interestingly, Cerullo et al. have observed excited states population of 0.55 and 0.3 for positive and negative chirps, respectively.42 This compares well with the corresponding values in Figure 3e. On the basis of this analysis, it appears likely that chirp control in the LD690/methanol system was made possible by the choice of solvent with a relatively short τc, which can be traced back to the rapid librations in methanol. This also leads one to expect that performing the same experiment in an aprotic solvent that lacks rapid librational motions would have lowered the efficiency of population transfer and asymmetry with respect to chirp. This is consistent with the experimental observation of significantly weaker chirp dependence of a similar dye molecule (LDS750) in liquid acetonitrile.42

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AUTHOR INFORMATION

Corresponding Author

*E. Geva. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project was supported by the National Science Foundation through grant CHE-1464477.

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REFERENCES

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IV. SUMMARY AND OUTLOOK Understanding the interplay between coherent control and electronic dephasing is essential to determine the conditions under which coherent control is possible in a condensed phase environment. In this paper, we explored this interplay in the context of using linear chirp to control population transfer between the ground and excited electronic states of a dye molecule in liquid solution. Our findings indicate that the ability to use chirp for controlling population transfer in liquid solution relies on a delicate balance between the amplitude, σ, and time scale, τc, of the solvent fluctuations. More specifically, we showed that for coherent control to work, σ and τc must satisfy στc ⩽ 1. This condition appears to be satisfied in the LD690/methanol system of Cerullo eta al.,42 where the rapid librations lead to relatively short τc. It should be noted that the model studied here only accounts for pure electronic dephasing and therefore cannot account for relaxation processes that involve energy exchange with the solvent. This should be contrasted with the commonly encountered interpretation of chirp control in terms of the ability (in the case of negative chirp), or lack thereof (in the case of positive chirp), to follow the system’s dynamical Stokes shift as it undergoes vibrational energy relaxation on the excited electronic PES.42,47,53 Our choice to focus on pure electronic dephasing as opposed to vibrational energy relaxation is motivated by the fact that the former occurs on the time scale of the pulse (∼10−20 fs), whereas the former typically occurs on significantly longer time scales (>0.100 ps). Thus, one does not expect vibrational energy relaxation to play a dominant role during the pulse. 3020


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