“Stirred, Not Shaken”: Vibrational Coherence Can Speed Up

Article pubs.acs.org/JPCA

“Stirred, Not Shaken”: Vibrational Coherence Can Speed Up Electronic Absorption Bo Y. Chang,† Seokmin Shin,† and Ignacio R. Sola*,‡ †

School of Chemistry (BK21+), Seoul National University, Seoul 151-747, Republic of Korea ‡ Departamento de Química Física, Universidad Complutense, 28040 Madrid, Spain ABSTRACT: We have recently proposed a laser control scheme for ultrafast absorption in multilevel systems by parallel transfer (J. Phys. Chem. Lett. 2015, 6, 1724). In this work we develop an analytical model that better takes into account the main features of electronic absorption in molecules. We show that the initial vibrational coherence in the ground electronic state can be used to greatly enhance the rate and yield of absorption when ultrashort pulses are used, provided that the phases of the coherences are taken into account. On the contrary, the initial coherence plays no role in the opposite limit, when a single long pulse drives the optical transition. The theory is tested by numerical simulations in the first absorption band of Na2.

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Without modulating9,10 (e.g., chirping) the pulse, one way to avoid this effect is by creating an initial superposition state of vibrational levels, such that the absorption to the excited electronic state is much faster; that is, it proceeds via parallel transfer.6 Hence, rather than disturbing the system with a strong pulse (the shaking), it is necessary to stir the system away from its initial ground state by preparing the required vibrational coherences, before a much less intense pulse becomes effective. A simple model of degenerate levels with equal couplings was used to obtain analytical results, whereas the applications to other systems involved using a novel variational approach termed geometrical optimization.6,11 In this work we generalize the analytical model to include different transition dipole couplings or Franck−Condon factors, thus making it a more realistic approximation of the Hamiltonian of a molecule. We show how the parallel transfer scheme must be extended to increase the yield or the rate of absorption, remarking the importance of the phase of the initial vibrational coherence. We then use the geometrical optimization approach to show results of the optimization of absorption at different carrier frequencies and pulse intensities, and involving different numbers of vibrational levels contributing to the vibrational coherence. We also show how the same approach can be applied in the opposite limit of using long (i.e., CW) laser excitation in molecules. In that case the vibrational coherence plays a very minor role. Because coherent control12 is not possible in absorption induced by a single frequency field, the optimization is mostly reduced to the role of selecting the initial state such that Rabi oscillation13 is maximized for the chosen pulse duration.

INTRODUCTION When observed at low resolution, as when ultrashort pulses are used, the electronic spectrum of molecules looks like a congested band. It is thus not surprising that quantum features seem absent from the spectrum and that simple models or classical calculations often provide good enough guidance to simulate or interpret the relatively unstructured absorption spectrum under these conditions.1 The other side of this observation is that one can easily reach saturation, thus not being able to increase the rate or yield of absorption at larger pulse intensities, regardless of the coherence of the pulsemolecule interaction, i.e., of the shortness of the pulse. In this work we show that one can prevent this from happening by increasing the quantumness of the system. This can be achieved by starting in a superposition state, instead of a single vibrational level, that is, by using vibrational coherences in the ground electronic state. There have been several proposals aimed at controlling molecular dynamics, e.g., bond fragmentation, that have shown improved yields and particularly simpler control mechanisms by applying a short infrared (IR) pulse that prepares the initial state in the ground electronic potential, before an optical or ultraviolet (UV) pulse moves the population to the excited electronic state.2−5 The IR pulse creates vibrational coherences that were visualized as imparting momentum along the necessary coordinates in the ground potential energy surface. But the importance of the vibrational coherence can be detached from the spatial formulation (typically limited to low dimensional systems or few-atom molecules) and be formally established in the energy representation. We have recently proposed a novel scheme that allows us to increase the yield of absorption by using ultrashort pulses in general multilevel systems, based on the general Hamiltonian representation.6 We showed that the initially unpopulated levels that fall within the pulse spectrum contribute to create strong Stark shifts7,8 that decouple the initially populated state. © XXXX American Chemical Society

Received: June 23, 2015 Revised: August 6, 2015

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DOI: 10.1021/acs.jpca.5b05994 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

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ANALYTICAL MODEL: OPTIMIZING ULTRAFAST ELECTRONIC ABSORPTION IN A MOLECULE We will first propose a general model Hamiltonian that can fairly reproduce the main features of electronic absorption by using an ultrashort pulse. We write the molecular Hamiltonian states or kets as |g, n⟩ and |e, n⟩ where g and e refer to the electronic state (the ground and the excited state) and n to the vibrational levels assigned to the electronic states. We supposed there are Ni and Nf accessible vibrational levels in each state. We can therefore expand the wave function of the system as Ng

|Ψ(t )⟩ =

where Ng − 1

MR 2 =

μj 2

j≠k

is the normalization constant, such that ⟨E|E⟩ = ⟨R|R⟩ = 1. Together with the initial state |g, 1⟩, the three form an orthonormal basis, |Ψ(t)⟩ = ak(t)|g, k⟩ + B(t)|E⟩ + C(t)|R⟩ with |ak(t)|2 + |B(t)|2 + |C(t)|2 = 1. The effective Hamiltonian in this basis (including nonadiabatic terms, which exactly cancel) is

Ne

H=−

∑ aj(t )|g, j⟩ + ∑ bn(t )|e, n⟩ j

∑

1 Ne ,(t )(μk |i⟩⟨E| + MR |E⟩⟨R | + cc) 2

(5)

n

with analytic eigenvalues and eigenvectors (aka dressed states

In the limit when the pulse bandwidth is much larger than the energy splitting between adjacent vibrational levels, Δω ≫ ΔE, a reasonable approximation necessary to obtain analytical expressions14,15 is to regard all the levels degenerate. However, we include an essential feature that influences the structure of the electronic spectra, namely, the presence of different Franck−Condon factors, μnj = ⟨e, n|μ|g, j⟩, that weight the likelihood of the different possible transitions. Crucially, the transient dipoles can be both positive and negative. For a resonant transition in the rotating wave approximation,13 that is, when the pulse bandwidth Δω is clearly smaller than the pulse carrier frequency ω, the time-dependent Schrödinger equation (TDSE) in the representation of the eigenstates of the molecular Hamiltonian, is

or adiabatic basis), ε0 = 0, ε±(t) = ±Mav NgNe ∫ ,(t ′) dt ′/2 = 0 ±φ(t), where φ(t) is the laser-induced dynamical phase and

13

t

Mav = ∑j μj 2 /Ng is the root-mean-square of the dipoles. The eigenvectors or dressed states are |Φ0⟩ = cos ϑ|g, k⟩ − sin ϑ|R ⟩

iaj̇ = −,(t ) ∑ μnj bn(t )/2

|Ψ(t )⟩ = (cos 2 ϑ + cos φ sin 2 ϑ)|g, k⟩

n

− sin 2ϑ sin 2(φ /2)|R ⟩ − i sin ϑ sin φ|E⟩ 2 ⎞ μk 2 1 ⎛ MR ⎜⎜ ⎟⎟|g, k⟩ = + φ cos Ng ⎝ Mav 2 Mav 2 ⎠ 2 μk MR i μk − sin 2(φ /2)|R ⟩ − sin φ|E⟩ 2 Ng Mav Ng Mav

Ng

ibṅ = −,(t ) ∑ μnj aj(t )/2 (1)

where ,(t ) is the slowly varying pulse envelope (times its peak amplitude). As we are interested in the overall electronic excitation, the individual vibrational levels of the excited electronic state will not play a distinctive role in the theory. To simplify the mathematical treatment, we will now suppose that every |e, n⟩ level will be excited identically, that is, that all the Franck−Condon factors only depend with the quantum number j and not n, μnj = μj. Then we can define the collective excited state |E ⟩ =

1 Ne

(7)

because the nonadiabatic terms are exactly zero. The probability amplitude of reaching the excited state (in whatever |e, n⟩ vibrational level) is given by Ne

∑ bn(t ) = ⟨E|Ψ(t )⟩ = −

Ne

n

∑ |e, n⟩

and therefore the excitation probability or absorption yield,

with the corresponding B(t) amplitude, such that PE(t ) =

iaj̇ = − Ne ,(t )μj B(t )/2 Ng

(3)

First consider that a single vibrational level |g, k⟩, is initially populated, ak(0) = 1. Then we can define the collective Raman state

|R ⟩ =

1 MR

Ng − 1

∑ j≠k

μk 2 NgMav 2

sin 2 φ(t )

shows Rabi oscillations damped by the number of levels in the ground state, weighted by the rate of the particular Franck− Condon factor of the transition with respect to the average. This is a natural extension of the well-known pulse area theorem, which determines the outcome of the Rabi oscillations under time-dependent fields.13 Sometimes it is more interesting to look at the rate of probability of reaching the excited state or rate of absorption, ṖE. When all the Franck−Condon factors have the same magnitude (μk = sign(μk)μ), μk/Mav = sign(μk)/ Ng , and

iḂ = − Ne ,(t ) ∑ μj aj(t )/2 j

i μk sin φ(t ) ≡ UEk(t ) Ng Mav (8)

(2)

n

(6)

where tan ϑ = μk/MR and sign(x) = x/|x|. In the adiabatic representation the initial wave function is |Ψ(0)⟩ = cos ϑ|Φ0⟩ + sin ϑ[|Φ+⟩ + |Φ−⟩]/ 2 . Therefore, the wave function at time t back in the original representation is given by

Ne

j

1 [sin ϑ|g, k⟩ ± sign(μk )|E⟩ + cos ϑ|R ⟩] 2

|Φ±⟩ =

μj |g, j⟩ (4) B

DOI: 10.1021/acs.jpca.5b05994 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A PĖ (t ) =

1 sin(2φ(t )) φ̇(t ) Ng

equal to one. The result is that the optimal initial amplitudes must obey the eigenvalue equation

(9)

Ne

where φ̇ (t) = Ne μ,(t )/2 . Clearly, the excitation probability goes down as the number of vibrational levels in the ground manifold increases, but the rate of absorption does not. For small φ(t) (corresponding to first-order perturbation theory, that is, small pulse amplitude or short times),16 PĖ (t ) ≈

1 2 μ Ne 2

∫0

t

,(t ) ,(t ′) dt ′ =

1 2 μ Ne :(t ) 2

k

(10)

PĖ (t ) ≈

∑ ak(0) UEk(t ) k N

g μ i sin φ(t ) ∑ k ak (0) Ng k Mav

(11)

The absorption yield is therefore Ng

PE(t ) =

Ng

∑ Pk(0)|UEk(t )|2 + 2 ∑ k

* ′(t ) UEk(t )] ℜ[ρk ′ k (0) U Ek

k′,k≠k′

=

Ng ⎡ Ng μ2 μ μ ⎤ 1 sin 2 φ(t )⎢∑ Pk(0) k 2 + 2 ∑ ρk ′ k (0) k ′ 2k ⎥ ⎢⎣ k Ng Mav Mav ⎥⎦ k′,k≠k′

=

0 sin 2 φ(t ) Ng

(12)

where Pk(0) are the initial vibrational populations and ρk′k(0) = ak′(0) ak(0) are the initial vibrational coherences, which we assume real in the last part of the equation, for which case, ρk′k(0) = Pk ′Pk . The first term in the brackets does not add any enhancement to the probability with respect to the single initial state case. However, the second term adds probability whenever the initial coherence has the right sign, that is, whenever it is aligned with the product of the dipoles. To observe this effect more easily, assume now that all the dipoles have the same absolute value, μk = sign(μk)μk. Then the 0 term that incorporates all the structure and initial coherence simplifies to

∑

sign(μk ′) sign(μk ) ρk ′ k (0)]

k′,k≠k′

Ng

Ng

sign(μk ′) sign(μk ) ρk ′ k (0) =

k′,k≠k′

∑ k′,k≠k′

20 1 φ(t ) φ(̇ t ) = 0Ne :(t ) Ng 2

(15)

Ni j=0

where Fj′j = ∑n=0NfUnj′ * Unj, and χi are the optimized absorption yields. The validity of the theory is numerically tested in the following application.

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NUMERICAL EXAMPLE: ENHANCING THE YIELD OF ABSORPTION IN THE A BAND OF NA2 In this section we apply the previous ideas to the electronic absorption in the first band of Na2, between the 1Σg(3s) or X state and the 1Σu(3p) or A state. The potential energy surfaces and dipole moments were obtained from ab initio calculations.17 Using the Fourier-grid Hamiltonian method,18 we obtained all the bound vibrational eigenstates of the first two electronic states, from which the state-to-state transition dipole moment or Franck−Condon factors were calculated. The potential energy curves as well as a sketch of the scheme is shown in Figure 1. The dynamics is integrated in the energy representation by solving the TDSE with a Runge−Kutta method.19 We first analyze the results at small pulse area, corresponding either to the small pulse amplitude or to the short time limit, where first-order perturbation theory gives reasonable results for the rate of absorption. Although we are studying the dynamics of a diatomic molecule under a realistic pulse, where one cannot assume that the vibrational levels are degenerate, as

(13)

Whenever the initial vibrational coherence ρk′k(0) has the same sign as the product of Franck−Condon factors, it adds to the final absorption probability. The probability can be maximized when all the initial amplitudes are equal and with the sign appropriately chosen. Then

∑

(14)

∑ Fj ′ jaj(0) = χi aj

Ng

0 = μ2 [1 + 2

2

Given the right sign for the vibrational coherences, 0 is always larger than μ12 (for an initial state k = 1), so that the rate of absorption is always enhanced. The theory developed so far is valid in the limit that ΔE = 0. The energy splittings cause an additional dynamical phase due to the vibrational energy differences. As a consequence, 0 becomes time dependent, the dynamics is modulated in time by additional dynamical phases and the optimal initial coefficients are in general complex. In addition, λmax is no longer necessarily Ng. However, as long as we choose Ni initial levels that can be excited under the pulse bandwidth, such that ENi − E1 < Δω, the modulation is slow and qualitatively the same conclusions can be applied by changing Ng and Ne for N, the approximate number of excited vibrational levels that fall under the pulse bandwidth, and allowing different complex phases for the optimal initial amplitudes. In addition, the optimized initial amplitudes can be obtained in the general (nonanalytical) case, solving the eigenvalue equation11

Ng

=−

⎝ Mav

⎞ − δk ′ kλ⎟ak (0) = 0 ⎠

where the λ give different possible values of 0 . The maximum eigenvalue is λmax = Ng. Its corresponding eigenvector represents the parallel transfer condition, for which PE(t) = sin2(φ(t)). However, even any small vibrational coherence, given the right sign, improves the yield and rate of absorption. For instance, in the perturbative regime

where :(t ) is a function of the shape. Now consider that initially a certain nonstationary wave function is prepared, |Ψ(0)⟩ = ∑Nk gak(0)|g, k⟩. Following the same analysis as before (given the linearity of the TDSE) and taking into account that φ(t) does not depend on the specific transition dipole, ⟨E|Ψ(t )⟩ =

⎛μ μ k′ k

∑⎜

Ng − 1 1 = Ng 2

so that 0 = Ng and PE(t) = sin φ(t), following the typical full Rabi oscillations. In general, one can find the initial state that maximizes 0 with the constrain that the sum of populations is 2

C

DOI: 10.1021/acs.jpca.5b05994 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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first excited vibrational level. However, a further enhancement is gained by cleverly playing with the initial vibrational coherence, without needing such exceeding vibrational populations on excited states. We show the results when both |g, 0⟩ and |g, 1⟩ are initially populated. The black and magenta curves give the results starting in |g, 0⟩ and |g, 1⟩ respectively, whereas the blue and red curves give the results when P1(0) = 0.2, in-phase or out-ofphase with respect to the Franck−Condon sign, respectively. Finally, the orange curve gives the absorption yield starting in the optimized initial state, obtained by using the geometrical optimization procedure. To estimate the relative absorption rates, we fit our curves to the previous expression (eq 15) and compare the different proportionality constant α obtained. By changing the initial vibrational level, we enhance a factor of ∼80% the absorption rate. However, a 100% enhancement is achieved with just 20% population in the first excited vibrational level, provided the amplitudes of the initial superposition have opposite phases, |Ψ(0)⟩ = 0.89|g, 0⟩ − 0.45|g, 1⟩. A change in the sign of the superposition implies a significant 66% loss in the rate. Finally, a careful engineering of the initial state, optimizing both the amplitudes and phases, but using just the first two vibrational levels, gives the optimal initial state |Ψopt(0)⟩ = 0.6|g,0⟩ + 0.8e−1.36iπ|g,1⟩, leading almost to a 150% improvement. As the Fermi golden rule16 does not apply for the used pulses, the rates do not translate linearly to the final absorption yields. In addition, because of the dynamical phases, the optimal sign of the superposition is not necessarily aligned with the sign of the respective dipoles. In Figure 2 the electronic absorption probability is shown as it evolves in time for the previous cases. Starting in the ground vibrational level, the final yield is 0.21. One can notice that the final improvement using the modest initial vibrational coherence, 0.32 (blue line), is not as large as was the case with the rate. However, the dynamics starting from the optimized initial state gives a great enhancement of the yield, to 0.60. If the initial state is the optimal superposition state formed with the first three vibrational levels (green dashed line), the yield at this pulse intensity is not greatly improved. This is because μ9,2 is very small, because the Franck−Condon region is located near one node of the |g, 2⟩ vibrational wave function. On the contrary, the analytical model gives results in qualitative agreement, as shown in the dashed orange line. The analytical model implies keeping all the exact transition dipole couplings but making all the vibrational energies equal zero. With the optimal initial state obtained under this approximation, |Ψ′opt(0)⟩ = 0.52|g, 0⟩ − 0.85|g, 1⟩, the absorption yield is 0.43. The results can be improved (and, indeed, can be adjusted at final time) if the number of vibrational levels in the model are taken as a fitting parameter. The results shown here correspond to using all the bound vibrational levels, as in the numerical results. The agreement is much better for the rate at initial times, as expected, and in the inset of Figure 2 the result from |Ψopt(0)⟩ and |Ψ′opt(0)⟩ fully overlap. In simple cases, such as diatomic molecules, the amplitudes and phases of the initial superposition state can be interpreted in the position representation, by looking at the wave function. Typically, the sign is such that the optimal initial state is spatially shifted toward the region of space where the transition dipole is larger. However, this is usually only observed for weak pulse intensities. We now study how the absorption yield can be maximized as a function of the pulse amplitude, when the initial state is

Figure 1. Ground and excited potential energy curves of Na2 used for the simulations. The sketch shows how to optimize the yield of absorption using an ultrashort pulse ϵ(ω) centered at the transition between e.g., v = 7 ← v′ = 0. Initially, a superposition of different vibrational levels spanning roughly the same spectral bandwidth is prepared, Ψ(0), after which the same optical pulse leads to a faster transition with a larger yield. The vibrational levels are not drawn to scale.

long as Δω ≫ ΔE the dynamical phases due to the vibrational energy differences at short times are small, and the rates follow closely the shape, α :(t ), predicted by the theory. This is shown in the inset of Figure 2, where we used a 650 nm, 20 fs (fwhm) Gaussian pulse with a peak amplitude of 0.47 GV/m (roughly corresponding to a moderately intense 30 GW/cm2 pulse).

Figure 2. Yield of absorption in the A band using an ultrashort weak field, starting from different quantum states: v′ = 0 (square, black line); a superposition with 20% population in v′ = 1 and 80% population in v′ = 0 with negative (right triangle, blue line) and positive (diamond, red line) phase; and an optimized superposition involving just v′ = 0 and v′ = 1 (left triangle, orange line). With the dashed-orange line we show the results starting from the optimized superposition state obtained by using the analytical model for the Na2 molecule (keeping the couplings but making all vibrational levels degenerate). Also shown as the green dashed line is the yield starting from an optimized initial superposition of the first three vibrational levels. In the inset we show the absorption rates. In this case we show in dashed magenta line the absorption rate starting in v′ = 1. The parameters of the pulse are given in the text. The time is measured in pulse width units.

The carrier frequency is approximately in resonance with the v = 7 ← v′ = 0 transition, although many other transitions fall under the pulse bandwidth, e.g., v ∈ [3,12] as well as transitions starting from different initial vibrational states, v = 8 ← v′ = 1, v = 9 ← v′ = 2, etc. The Franck−Condon factors for the latter transitions are μ7,0 = −0.7, μ8,1 = 1.05, and μ9,2 = 0.02. The first two are similar, so that the rate can increase around 50% by playing with the different dipoles starting the dynamics in the D

DOI: 10.1021/acs.jpca.5b05994 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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involving more levels to saturate the yield and reach full population inversion. A few numbers display the power of the scheme. With 40 fs pulses and by using the vibrational coherence of the first 10 vibrational levels, full population inversion can be achieved with , 0 = 0.35 MV/m at a peak intensity 8 times smaller than that needed to obtain the 0.11 maximal population transfer in the absence of initial vibrational coherence. Moreover, already with , 0 = 0.24 MV/m (a remarkable 21 times smaller intensity) and just two vibrational levels, one can reach a maximum absorption of ∼0.2, increasing 13 times the absorption yield obtained from the ground vibrational level at the same laser conditions. The enhancement reaches almost a factor of 50 if one uses the vibrational coherences of the first three levels.

prepared as a superposition of different number of vibrational levels, with optimized amplitudes and phases, obtained by the geometrical optimization procedure. In Figure 3 we show the

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APPLYING THE METHOD TO POPULATION ABSORPTION WITH LONG PULSES We study now the effect of using initial vibrational coherences for population transfer in the opposite excitation limit, when the pulse bandwidth is much smaller than the vibrational energy splitting, Δω ≪ ΔE. Under this condition, each stateto-state transition is addressed independently and one can study the absorption process as that of Ni-independent systems, where in each system only the two levels in resonance or in near resonance, |g, k⟩ and |e, k′⟩, participate. Let us call ΔEk = Ee,k′ − Eg,k − ℏω, where ω is the pulse carrier frequency. Due to the different anharmonicity of the electronic states, if the electronic transition is driven by a single laser field (that is, a single carrier frequency) at most one transition will be exactly resonant. Very few analytic models exist for two-level systems nonresonantly driven by pulses. One of the simplest is the Rosen−Zener model where ,(t ) = , 0 sech(t ). However, the numerical solutions for other smooth pulses are very similar.20,21 In the Rosen−Zener model, PE(t) depends on hypergeometric functions but the asymptotic probability has the simple form

Figure 3. Absorption yield as a function of the pulse amplitude when the initial state is an optimized superposition of different vibrational levels, as indicated in the figure. In (a) we use a 20 fs pulse with frequency tuned to the v = 7 ← v′ = 0 transition and in (b) we use a 40 fs pulse tuned to the v = 0 ← v′ = 0 transition.

results when the initial state is |g, 0⟩ (black line), a superposition of the first two vibrational levels (orange line), of the first three levels (green line), the first five (red line), and the first 10 levels (blue line). In (a) we use 20 fs 650 nm pulses, tuned to the v = 7 ← v′ = 0 transition. In (b) we show the results using 40 fs, 687 nm pulses, tuned to the v = 0 ← v′ = 0 transition. Without using the vibrational coherence, a maximum population transfer of 0.34 is achieved with a 1.1 GV/m pulse (peak intensity of 0.16 TW/cm2) for the first case. The results are even poorer in the second case. As the number of levels increases, the yield is enhanced and electronic population inversion can be achieved with weaker fields. With 650 nm pulses, the third level adds little improvements, due to the small Franck−Condon factor implied in its transitions, as already noticed. The improvement is greater in the second case, because the starting dipole, μ0,0 ∼ 0.11, is very small. Hence, starting in superposition states one uses both larger dipoles and the vibrational coherence. On the contrary, the second vibrational levels does not lead to such great enhancement in the yield using the 687 nm pulses. In general, one can reach higher yields at smaller pulse amplitudes with the longer 40 fs pulses. This is because the yield depends on the integrated pulse envelope or pulse area, ∞ ( = ∫ ,(t ) dt ,13 which linearly depends on the pulse −∞ duration. On the contrary, if we change the pulse amplitude by the pulse area in the abscissa of Figure 2, we would observe that the yield rises faster by using the 20 fs pulses. This is because the bandwidth is larger and more vibrational levels of the excited state can participate adding to the rate of absorption. For this reason, one needs to use vibrational coherences

PEk(∞) = sin 2(μk (/2) sech(π ΔEk /2) ∞

with ( = ∫ ,(t ) dt . Now if the initial state is a quantum −∞ superposition, |Ψ(0)⟩ = ∑Nk iak(0)|g,k⟩, because the pathways starting from different vibrational levels never interfere (they reach different levels), the overall probability is the weighted sum of the probabilities for the different pathways Ni

PE(∞) =

∑ Pk(0) sin 2(μk (/2) sech(π ΔEk /2) k Ni

≈

⎛

∑ Pk(0) sin 2(μk (/2)⎜1 − k

⎝

π 2ΔEk2 ⎞ ⎟ 8 ⎠

(16)

where the last equality is valid for small energy detuning. In the limit of small bandwidth, starting from a superposition state the rate does not increase and the final yield is not improved by the initial vibrational coherence. In particular, for equal dipoles, μk = sign(μk)μ, the best yield is obtained starting from a single vibrational level: that with ΔEk smallest. Even if all the transitions are approximately resonant (e.g., the force constant of the ground and excited potentials are similar or more than one monochromatic field is used to drive the electronic transition), ΔEk ≈ 0 and eq 16 gives E

DOI: 10.1021/acs.jpca.5b05994 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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∑ Pk(0) sin 2(μ(/2) = sin 2(μ(/2)

Although the fully optimized preparation of the initial state can be too demanding, in practice, even the coherence implied by starting in a superposition with a small contribution of an excited vibrational level is enough to observe 50% or more yield enhancements. Different Raman spectroscopic procedures (e.g., impulsive stimulated Raman scattering22 or STIRAP23) can be used to generate this coherence. However, it is very important to prepare the system with the right phase, or at least sign, of the superposition. In simple molecules with few degrees of freedom, this sign reflects the alignment of the transition dipole with the field. Depending on the case, the parallel transfer procedure may be more difficult to implement than other quantum control scenarios based on pulse shaping. Loosely speaking, in parallel transfer the complexity lies in the initial wave function |Ψ(0)⟩ whereas in quantum optimal control it is hidden in the pulse structure ,(t ). However, the parallel transfer solution is more constrained and in general requires less physical resources. The optimal control pulse is regulated by factors such as ⟨Ψe(t)|μ|Ψg(t)⟩ that involve the overlap of the ground and excited electronic wave packets, at each time. Without further constraints, it implies in general the use of both the ground and excited state vibrational coherences, as well as the electronic coherence. The parallel transfer solution, however, shows that full population inversion only requires to control the vibrational coherence of the ground state at the initial time. Additional control on the excited state vibrational coherences is only needed for vibrationally state selective transitions.11 Alternatively, parallel transfer can be seen as a particular solution of optimal control that involves a sequence of an IR and a UV pulse. Finally, we have shown that the vibrational coherence is only a useful resource when ultrashort pulses are used. With long fields that resolve any state-to-state transition, engineering the initial state does not lead to any real control when a single pulse (a single driving frequency) is used, although one can still use the freedom of starting from different vibrational levels as a way to handle different Rabi frequencies. The vibrational coherences can only play a role when several interfering pathways are coherently controlled.

k

Therefore, the yield is independent of the choice of the initial state. However, one can exploit the different Franck−Condon factors in a molecule so that for any given pulse, the initial populations are chosen such that PE(∞) is maximal. At low intensities or short times, the gain in the yield or rate of absorption can only be due to using a transition with a larger Franck−Condon factor, as the vibrational coherence does not play any role. In Figure 4 we show the absorption yield as a function of the pulse amplitude for moderate intensities, using a long 320 fs

Figure 4. Absorption yield as a function of the pulse amplitude using a long 320 fs pulse. The results show the first and second Rabi oscillation starting in the v′ = 0 state and in optimized superposition states involving the first 2 vibrational levels (dashed orange line) and the first 3 vibrational levels (green line).

pulse. The pulse duration ensures that all the different transitions are energy resolved. The populations approximately show the pattern of Rabi oscillations. The yield is not greatly enhanced starting in a superposition of vibrational levels. Practically the same results are obtained when the superposition involves v′ = 0 and v′ = 1, as the Franck−Condon factor of the respective transitions are very similar. Using also v′ = 2, with a clearly smaller Franck−Condon factor, one can improve the absorption yields at pulse intensities that, following the Rabi oscillation, imply substantial stimulated emission from v′ = 0 and v′ = 1. In summary, starting in different vibrational levels one provides the opportunity to apply different Rabi frequencies. As such, one is effectively controlling the amplitude or duration of the pulse.

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AUTHOR INFORMATION

Corresponding Author

*I. R. Sola. E-mail: [email protected]. Notes

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The authors declare no competing financial interest.

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CONCLUSIONS We have shown that the vibrational coherence in the ground electronic state can be used as a very powerful resource to increase the rate of absorption and enhance the final yield of population inversion to other electronic states. In principle, if all the vibrational levels are used, one can achieve full population inversion almost regardless of the pulse intensity, once a minimal threshold value is overcome. This is achieved by parallel transfer. Indeed, in using all the possible coherences, the required pulse area can be less than 20 times smaller than the area required starting in a single eigenstate. This can be very practical when one needs to excite the system in very short times to avoid competing processes, but the transition dipole is weak. Then, without properly engineering the initial state, one would need to use very strong laser pulses, which can easily induce many other competing processes such as ionization.

ACKNOWLEDGMENTS This work was supported by the NRF Grant funded by the Korean government (2007-0056343), the International cooperation program (NRF-2013K2A1A2054518), the Basic Science Research program (NRF-2013R1A1A2061898), the EDISON project (2012M3C1A6035358), and the MICINN project CTQ2012-36184. I.R.S. acknowledges support from the Korean Brain Pool Program.

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DOI: 10.1021/acs.jpca.5b05994 J. Phys. Chem. A XXXX, XXX, XXX−XXX