Ultrafast Control of the Internuclear Distance with Parabolic Chirped

Nov 15, 2011 - Recently, control over the bond length of a diatomic molecule with the use of parabolic chirped pulses was predicted on the basis of nu...
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Ultrafast Control of the Internuclear Distance with Parabolic Chirped Pulses Bo Y. Chang* and Seokmin Shin School of Chemistry (BK21), Seoul National University, Seoul 151-747, Republic of Korea

Jesus Santamaria and Ignacio R. Sola* Departamento de Quimica Fisica, Universidad Complutense, 28040 Madrid, Spain ABSTRACT: Recently, control over the bond length of a diatomic molecule with the use of parabolic chirped pulses was predicted on the basis of numerical calculations [Chang; et al. Phys. Rev. A 2010, 82, 063414]. To achieve the required bond elongation, a laser scheme was proposed that implies population inversion and vibrational trapping in a dissociative state. In this work we identify two regimes where the scheme works, called the strong and the weak adiabatic regimes. We define appropriate parameters to identify the thresholds where the different regimes operate. The strong adiabatic regime is characterized by a quasi-static process that requires longer pulses. The molecule is stabilized at a bond distance and at a time directly controlled by the pulse in a time-symmetrical way. In this work we analyze the degree of control over the period and elongation of the bond as a function of the pulse bandwidth. The weak adiabatic regime implies dynamic deformation of the bond, which allows for larger bond stretch and the use of shorter pulses. The dynamics is anharmonic and not time-symmetrical and the final state is a wave packet in the ground potential. We show how the vibrational energy of the wave packet can be controlled by changing the pulse duration.

I. INTRODUCTION Laser control of chemical processes leading to molecular fragmentation or rearrangement of nuclear positions,1,2 is being regularly achieved in the laboratory nowadays, usually with the aid of femtosecond lasers,3 pulse shaping techniques,4,5 and learning algorithms.6,7 Although the yield of these processes is difficult to estimate and typically not measured, in most control experiments only a fraction of the molecules in the sample undergoes the chemical reaction. The relative positions of the nuclei as, e.g., the interatomic distances, are not precisely defined. A stronger control over the external degrees of freedom of all the molecules or atoms in the sample is achieved in ultracold experiments.8 On the other hand, strong fields are needed to orient or align the molecules in the sample,9 or control in more complicated ways the external degrees of freedom.10 The aim of this work is to explore laser mechanisms that allow a similar control over the internuclear distances of all the molecules in the sample, in a way that could give rise to control of ultrafast structural dynamics11,12 as measured in, e.g., a clear diffraction pattern. In this work we will only be concerned with internuclear distances of diatomic molecules, that is, the control of the bond length of molecules. In a recent work we proposed to control the bond distance with time-symmetrical chirped pulses.13,14 The scheme was termed the laser adiabatic manipulation of the bond (LAMB) r 2011 American Chemical Society

scheme. We showed that under certain conditions, the frequency of the pulse solely determines the dynamics of the bond length. With pulses of sufficient intensity that change slow enough, it is possible to create a coherent superposition of electronic wave functions whose associated potential, called light-induced potential or LIP,1517 creates the force field under which the nuclei, or actually the reduced mass quasi-particle that represents the relative distance, respond. In adiabatic conditions it is possible to have all the nuclei at the minimum of the LIP, avoiding vibrational dispersion and allowing the best possible localization of all the quasiparticles in the sample.18 In this regard it is meaningful to use a classical definition such as the bond length for this fully quantum superposition state, where the bond length is calculated as the expectation value of the interatomic distance, Æræ. Because the LIP is totally controlled by the laser parameters, so is the bond length. In particular, to induce large bond deformations, the quantum superposition must involve a bound, e.g., the ground electronic state Ξg, and an excited dissociative state Ξe.1922 The LIP correlating with Special Issue: Femto10: The Madrid Conference on Femtochemistry Received: August 9, 2011 Revised: October 28, 2011 Published: November 15, 2011 2691

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The Journal of Physical Chemistry A Vg at shorter internuclear distances, Vag, exhibits bond softening, whereas that correlating with Ve, Vae, exhibits vibrational trapping.2326 The localization of the minimum of Vae depends basically on the crossing of Ve with Vg + pω, which can be controlled by the pulse carrier frequency ω(t). Additionally, it is necessary to invert the population from Vg to Ve, which can be done with sequences of two pulses,18,2733 or with a single chirped pulse.34 The scheme implies two steps: population inversion to a dissociative potential, and vibrational trapping in this potential. However, both steps can be controlled by a single pulse.13,35 In this case the ability to largely elongate the bond depends on the bandwidth of the pulse Δω. Ultra broad-band pulses are needed. On the other hand, the time dependent frequency ω(t) uniquely defines the trajectory of Ær(t)æ. In this work we build on the results of previous work to analyze the extent to which quantum control of the bond is possible in time and space.13,14,35 The control over the spatial or structural properties of the bond imply the analysis of maximal elongations, whereas the control over the time dimension is here the timing and duration of the bond stretch. Both aspects are independent in what we define as the strong adiabatic regime and we show how the bond elongation depends on the pulse bandwidth. The period of bond stretching is only limited from below: there is a minimum time duration below which the strong adiabatic regime cannot be reached. The role of adiabaticity and the required conditions for its emergence are the central aspects of this work. We also analyze a different regime, defined as the weak adiabatic regime where dynamic bond elongation is possible, and further bond stretch can be achieved, albeit losing control over the bond trajectory and particularly the time-symmetry of the process. The final outcome of the LAMB process in the weak adiabatic regime is a wave packet in the initial electronic state. We show how to control its vibrational energy by simply adjusting the pulse duration. The weak adiabatic regime is also limited from below, as too-short pulses lead to no bond stretch at all. In this work we define appropriate measures to identify the different LAMB regimes and show the thresholds at which they appear. The paper is organized as follows. In section II we introduce the model system used in all the numerical simulations. In section III we show the degree of control on the duration-displacement coordinates that characterize the bond deformation, in the strong adiabatic regime. Section IV analyzes the process when the dynamics is not fully adiabatic, showing how to control the vibrational energy of the wave packet that is finally prepared. In section V we define appropriate measures of different types of adiabatic conditions, namely the survival probability and the socalled geometrical overlap, to study the thresholds of the strong and weak adiabatic regimes. Finally, section VI is the summary and conclusion.

II. NUMERICAL MODEL In all calculations in this study we work on a simple test model of two electronic states coupled by a field, presented already in refs 13 and 14. We here review the model underlying the most important approximations and we briefly mention the numerical procedure. Assuming molecular orientation9,10 and the dipole approximation for the coupling of the radiation and the molecule in the rotating wave approximation (RWA), the Hamiltonian

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of the system is36 p2 d2 H¼  I þ 2m dr 2

Vg μge EðtÞ=2

μge EðtÞ=2 Ve  pωðtÞ

! ð1Þ

where m is the reduced mass of the molecule, taken as the mass of the Na 2 dimer, and I is the unit 2  2 matrix. The time-dependent Schr€odinger equation for the nuclear wave functions is solved numerically in a grid of 1024 points using the second-order split-operator method. 37 The kinetic energy is evaluated locally in the corresponding momentum grid by fast-Fourier transform techniques.38 Molecular orientation is important as the effects of varying the coupling along the orientation angle θ imply the appearance of laser-induced conical intersections39,40 where the coupling and the control is inefficient. On the other hand, the RWA is only used for a matter of convenience, because it allows the calculation of the LIPs, which provide insightful analysis of the results. However, the results without RWA mostly resemble those with the RWA.14 The LIPs are the diagonal elements obtained after diagonalizing the potential energy matrix (including the coupling with the field) of eq 1. Thus they are adiabatic potentials in the presence of the laser. The control of the bond length in this work implies that the vibrational wave function in this new adiabatic representation ψae is always defined in a single LIP, the one with larger population on Ve (except at initial and final time). The population on the other LIP Vag is usually negligible. In this case the bond length is calculated as the expectation value of the internuclear distance in the adiabatic representation, Ær(t)æ  Æψae(t)|r|ψae(t)æ. The same results could be obtained using ψg(t) or ψe(t) in the calculation. Moreover, in the strong adiabatic regime, as shown in section III, ψae(t) is always the zeroth energy vibrational eigenstate of Vae. Thus, to compute Ær(t)æ, it suffices to know Vae(ω(t)) and calculate its properties, e.g., the equilibrium distance ra0 or its lowest vibrational eigenstate. However, if the dynamics does not proceed on a single LIP, then Ær(t)æg  Æψg(t)|r|ψg(t)æ and Ær(t)æe  Æψe(t)|r|ψe(t)æ follow different trajectories, driven by the different force fields of the potentials. In the latter case the bond length of the molecule is not well-defined until a measurement shows which electronic state is populated in the superposition. As in previous work13,14 the potentials are modeled as Vg = De(1  exp(βΔr))2, with r0 = 7 bohr (a0), De = 0.01 hartree, and β = 0.35 bohr1, and Ve = D1 exp(α(Δr + r1)) + D12, with D1 = 1.2 hartree, α = 1.2 bohr1, r1 = 6.2 bohr, and D12 = 2De. The transient dipole is taken as constant and unit, μge = 1 e a0. The precise shape of the potentials is not important, what is required is one bound potential (with a single minimum) for the initial state, and one dissociative state, where the bond is stretched. We have shown that the laser resources required to elongate the bond depend essentially on the pulse bandwidth Δω, which must be larger than (or of the order of) the dissociation energy De, pΔω g De, thus the need of applying the LAMB scheme to molecules with very low De.41 For the seek of generalization in this work we use a simple model and use scaled units of energy and time. In particular, we scale the energies with respect to the dissociation energy De, as well as the pulse bandwidth Δω with respect to De/p, and we scale all times with respect to the characteristic vibrational period of the ground state: T = (2π2m/Deβ2)1/2. For the set of parameters of 2692

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our model T ∼ 460 fs. Finally, we always give the results of bond elongations, Δr = r  r0, rather than internuclear bond distances. The key control parameter is the pulse frequency. We use a parabolic chirp   4Δω τ 2 ð2Þ ωðtÞ ¼ ω0 þ 2 t  τ 2 As we shall see over the next sections, τ and ω0 determine the duration of the stretch and the maximal elongation of the bond. In principle, chirp functions of this form, which involve cubic variations of the optical phase, can be easily achieved with the use of pulse shape technology.4,5 Other forms of time-symmetrical functions for the chirp could be used as well.13 The choice of the shape of the pulse is important but not decisive. We use rectangular pulses with fast switch on/switch off functions 8 2 > for 0 e t e 0:2T < sin ð5πt=2TÞ 1 for 0:2T e t e τ  0:2T EðtÞ ¼ E 0  > : sin2 ð5πðt  τÞ=2TÞ for τ  0:2T e t e τ

ð3Þ where τ is the pulse duration. The plateau amplitude E 0 must be chosen to allow full population transfer and at the same time minimize ionization and multiphoton processes. Obviously, in our model there cannot be multiphoton processes out of the two electronic states so there is no upper bound for the amplitude. We fix E 0 at 0.002 hartree/(e a0) unless otherwise stated (implying roughly an intensity of 0.1 TW/cm2) as a reasonable limit to avoid ionization and multiphoton absorption in realistic systems. However, because ω(t) varies over a large range, it may become difficult to avoid resonant two-photon transitions that would clearly reduce the yield of the LAMB process. This is in fact the strongest limitation in finding realistic systems where LAMB can be implemented. The duration of the pulse switch on/off should be as small as possible (but not sudden) to avoid wasting pulse bandwidth. In this work we set it as 0.2T. The chosen simple pulse shape clarifies the analysis making the LAMB process solely dependent on ω(t).

III. ROLE OF BANDWIDTH: CONTROL OF BOND LENGTH IN THE STRONG ADIABATIC REGIME In this section we will show how the bond stretch depends on the pulse bandwidth in strong adiabatic or quasi-static conditions, where the bond length does not depend on the pulse duration. But first we illustrate with an example how the dynamics proceeds in this regime. As indicated in the Introduction (see also refs 13 and 14), significant laser-induced stretch of the molecular bond can be accomplished if (i) the molecule is excited to a dissociative state, Ve, and (ii) the dissociative potential is stabilized by a fieldinduced crossing with a bound potential, such as Vg + pω. To adiabatically transfer the initial wave function to Ve, the initial pulse frequency ωmax must be tuned to the blue of the absorption band Vg f Ve. Then, by sweeping the pulse frequency through all the band, the population is transferred to Ve, but the molecule does not dissociate. In the presence of the strong field the molecular potential is then the LIP Vae, which exhibits vibrational trapping. Its equilibrium distance, ra0, is parametrically controlled

Figure 1. Bond dynamics of the system under a parabolic chirped pulse near full adiabatic conditions, with τ = 4.5T. The top panel shows the electronic populations. Whereas in the diabatic representation there is almost full population inversion, in the adiabatic representation (the dotted lines) all the process proceeds in a single LIP, Vae. The bottom panel shows the bond stretched as a function of time. The insets show the pulse profile and the parabolic frequency chirp.

by two laser parameters: ω(t) sets the crossing point rc(t) [where Ve(rc) = Vg(rc) + pω(t)], and the field amplitude E 0 determines the depth of the well of the LIP. To restore the initial bond configuration, the same process must be time-reversed, such that the frequency changes now from red-to-blue unfolding the same pulse bandwidth Δω. The simplest time-symmetrical chirp function is the parabolic chirp of eq 2. However, time-symmetrical dynamics can only be achieved in the strong adiabatic regime, as discussed in section IV below. As an example of the process, we show in Figure 1 the dynamics of bond elongation [Ær(t)æ  r0] in the strong adiabatic regime using the following pulse parameters: pΔω = 1.08 De, pωmax = 2.38 De, E 0 = 2  103 Hartree/(e a0) and τ = 4.5T. As the pulse frequency sweeps the photodissociation band, population inversion (up to 90%) to Ve is achieved, followed by a bond elongation with a maximum value of 2.9 bohr occurring after 2.25 T (half the pulse duration). When the process is reversed, all the population in restored in Vg and the bond length goes back to its initial value. In strong adiabatic conditions the vibrational wave function is always located at the bottom of the LIP, since in the adiabatic representation, obtained by diagonalizing the potential energy matrix of eq 1, the wave function is the zeroth energy vibrational eigenstate of Vae, ϕa0,e. Because this is a stationary state, there is no dispersion and the bond length is always well-defined, corresponding to Æϕa0,e|r|ϕa0,eæ ≈ rae. (There is always a difference between both magnitudes because the LIP is typically very anharmonic and in the wave function the average position is always larger than the most likely value.) Maximum bond elongations will occur at the minimum frequency, when ω(t) = ω0 at time t = τ/2. Because the two parameters, ω0 and τ, are independent, one can fully control the two-dimensional (timedisplacement) motion of the bond length or internuclear motion, as long as the strong adiabatic conditions are satisfied. Then the bond distance changes from Ær(0)æ ≈ r0 to Ær(τ/2)æ ≈ ra0, and backward, only once. The time duration of the whole event is τ. 2693

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Figure 2. Equilibrium distance of the LIP as a function of the minimum detuning Δm [defined as Δm = Vg(∞) + pω0  Ve(∞)], for different pulse amplitudes, E 0 = 0.001, 0.002, and 0.003 hartree/(e a0) (or e/a02), respectively. The equilibrium geometry of the LIP is displaced to larger values as the detuning decreases, involving larger pulse bandwidths. For Δm = De, Ve and Vg + pω0 cross at very short distance and there is almost no bond stretch. For Δm = 0, Ve and Vg + pω0 only cross at the asymptote. Although vibrational trapping in Vae still exists with strong pulses, it is no longer feasible to adiabatically access this state.

In Figure 2 we show how the bond elongation in the LIP, ra0  r0, depends on the laser frequency for different values of the field amplitude E 0. We define the minimum detuning for a given pulse, Δm = Vg(∞) + pω0  Ve(∞) as the energy difference between the asymptotes of the ground potential dressed by the field at its minimum frequency ω0, and the excited state. When Δm = De, both states cross at short bond lengths, whereas when Δm e 0, the states do not cross at all. A small Δm involves a small ω0 and, because ωmax must be fixed to avoid direct photodissociation, a large bandwidth Δω = ωmax  ω0. The results in Figure 2 show that as the detuning decreases or the bandwidth increases, the bond length increases. Part of this bandwidth (exactly 0.38 De/p = ΔωFC) is used in sweeping the photodissociation band, allowing almost full population inversion without barely changing the wave packet position. The remaining bandwidth, Δω BE , is employed in elongating the bond.13,35 When pΔωBE = De, the potentials only cross at the asymptote, but if E 0 is strong enough, Vae bares vibrational states and vibrational trapping and large bond stretching are in principle possible. It is, however, very difficult to reach the strong adiabatic regime in this case. That is, the larger bond distances cannot be reached under normal dynamical conditions and photodissociation is the most likely event. On the other hand, Figure 2 also shows the effect of the field amplitude. When E 0 decreases, the maximum bond length increases too, because the binding region in Vae becomes smaller and the LIP flatter. However, if E 0 is too small, full population inversion does not occur, nor the dynamics is adiabatic. A fraction of the wave packet would then proceed in Vg and another fraction in Ve, moving in independent molecular potentials. For the chosen system, E 0 = 2  103 Hartree/(e a0) is near the minimum required laser amplitude for adiabatic evolution.

IV. ROLE OF PULSE DURATION: STRONG VERSUS WEAK ADIABATIC CONDITIONS In this section we deal with the time dimension of the LAMB process. Specifically, we study how the pulse duration can be used as a different variable for controlling the bond stretch and the

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Figure 3. Bond dynamics as a function of the pulse duration τ for the same bandwidth Δω. In the strong adiabatic regime τ g 6T, the timescaled bond dynamics are similar: Ær(t/τ)æ follow the same curve. Before fully adiabatic conditions (τ = 4.5T) the dynamics can still be timesymmetrical but the bond stretches more at t/τ = 0.5 (half the pulse duration). For shorter pulses the bond dynamics is anharmonic. The wave packet gains momentum, and it is no longer attached to the bottom of the LIP. The bond can be further stretched, and as the chirp is reversed (t/τ > 0.5), the wave function returns to the initial potential as a vibrationally excited wave packet. This is the so-called weak adiabatic regime. If the pulse is too short (τ = 0.85T), the wave packet has not reached the classical turning point before the chirp reverses and the equilibrium geometry of the LIP goes back to r0. Thus the anharmonic bond dynamics are dumped.

vibrational energy of the wave packet after the laser is turned off, in a different intermediate regime where LAMB operates. Perfect time-symmetrical dynamics can only be achieved if all the energy given by the laser in driving the population to Ve and creating the LIP is given back as the process is reversed. Strong adiabatic conditions (also called spatial adiabatic conditions30,31) require slow modifications of the LIP. Then, as the contribution of the dissociative state Ve in Vae increases, by adiabatic following the wave function displaces to larger bond distances and becomes broader. To fully transfer the wave function from Vg to Ve with a typical strong pulse, one needs much shorter pulses than those needed for the strong adiabatic conditions. However, using shorter pulses, some electronic energy is transformed into vibrational (initially kinetic) energy, creating a nonstationary vibrational wave packet in the LIP. This wave packet exhibits dephasing (dispersion) and evolves following the LIP, which is itself dynamically modulated by ω(t). We refer to this situation as the weak adiabatic regime, because the electronic populations still evolve adiabatically, but the vibrational populations do not. Because shorter pulses can be used, it is easier to induce dynamical (as opposed to quasi-static) deformations of the bond. In Figure 3 we show how the dynamics of the bond deformation occur in strong and weak adiabatic conditions. As already stated, in the first case Ær(t)æ = Æϕa0,e|r|ϕa0,eæ strictly depends on Vae(t). For time durations τ longer than those needed for the onset of adiabaticity, τA, the maximum bond elongation is independent of τ, although τA itself depends on the bond length (it requires more time to quasi-statically deform the bond if the bond deformation is very large). This is the regime where twodimensional control of the internuclear motion is possible. However, as τ is made smaller, the vibrational wave function in Vae has less time to change its width and, correspondingly, arrives with larger kinetic vibrational energy to the LIP. Then, instead of 2694

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the wave packet, without changing the pulse spectrum. For shorter pulse durations, as explained before, the system is not given enough time to reach the outer turning point of the LIP or, for even shorter times, the population is not excited to the LIP (below the onset of the weak adiabatic regime). In the strong adiabatic regime all the energy given by the pulse when ω(t) decreases is given back when ω(t) increases. In the weak adiabatic regime a considerable fraction of the pulse bandwidth Δω(t) is used to vibrationally excite the wave packet in the LIP, which at the end of the process is transformed into vibrational energy in the ground state. To increase the final vibrational energy, one has to use pulses with larger bandwidths. Figure 4. Final averaged vibrational energy in the ground potential as a function of the pulse duration. In the strong adiabatic regime the final state recovers the initial state, so that the averaged vibrational energy is the zeroth energy in the ground potential, shown as a dashed line. In the weak adiabatic regime one can control the vibrational energy by using shorter pulses. For even shorter pulses the wave packet cannot reach the LIP turning points or the wave packet is not even fully excited to the LIP.

being attached to the bottom of the well, ψae(t) oscillates in the LIP approximately at the classical turning points.13 Depending on the amount of vibrational energy conferred to the packet, the maximum bond elongation, rmax  r0, will be greater or smaller. Additionally, rmax will not be reached at time τ/2, because the packet needs additional time to reach the outer wall of the LIP: the bond dynamics is anharmonic. In principle, as τ becomes smaller, the process is less adiabatic and rc will be larger. Larger bond oscillations and more excited wave packets can then be prepared using shorter and shorter pulses. This strategy is, however, limited by two factors. The first one is the following: When τ is small, the intervals of switching on and off the pulse cover an important fraction of the pulse’s duration; because ω(t) sweeps the same bandwidth Δω, it may occur that ω(t) starts to sweep the photodissociation band before E(t) is large enough, severely reducing the efficiency of the population inversion from Vg to Ve. This problem can be overcome by forcing the pulses to switch on (and off) at a faster rate or increasing ωmax and therefore Δω. In Figure 3 we have used pulses with switching on/off durations of 0.05 T for the results obtained with τ = T. The second limiting factor is more fundamental. When τ is shorter than T, the unleashed wave packet in Vae is forced back to Vg before it had time to reach the largest possible bond length. The maximum bond elongation is then limited by the “natural” period of oscillation of the wave packet in the LIP, which is typically quite larger than T, because the potential is flatter. In the weak adiabatic regime, at the end of the process a wave packet ψg(t) is formed in Vg. Despite being highly vibrationally excited, this wave function remains spatially localized at least before the anharmonicity forces dispersion. By simply adjusting the pulse duration τ (and to a smaller extent the amplitude of the field E 0), it is possible to control the vibrational energy of the wave packet. In Figure 4 we show the vibrational energy gained by the wave packet as a function of the pulse duration. For τ > 4.5T the LAMB scheme operates in the strong adiabatic regime and the wave packet returns to the initial state, the slashed line marking the zeroth vibrational energy. As τ becomes smaller, we shift to the weak-adiabatic regime. For τ ∼ 2T a maximum vibrational energy of ∼55% of the dissociation energy can be gained by

V. MEASURES AND THRESHOLDS OF ADIABATICITY IN THE LAMB SCHEME In this section we analyze the thresholds of the different LAMB regimes, the strong and the weak adiabatic regimes, as a function of different laser parameters. In strong adiabatic conditions one can easily predict the maximum bond elongation rc for a given pulse and at what time (τ/2) it will occur. In fact, one can easily estimate the bond trajectory Ær(t)æ during the pulse excitation, allowing for a laserinduced engineering of slow molecular vibrations of “arbitrary” period (yet longer than τA) and amplitude (but if the bond is too long, it is not possible to reach the strong adiabatic regime). In weak adiabatic conditions the dynamics of the wave packet in the LIP are entangled with the dynamical changes of the LIP itself, making more difficult the two-dimensional control of the internuclear distance. It is, however, useful to consider the maximal elongations achieved during the laser action. There are several tests to measure the degree of adiabaticity in the strong sense. Here we will consider the simplest, which computes the survival probability after the pulse is switch off. If the dynamics is perfectly time-symmetrical (as it should in the strong adiabatic regime), then Ps(τ) = |Æψ(0)|ψ(τ)æ|2 ∼ 1. On the other hand, for τ < τA, there is a wide range of pulse parameters where the scheme works in the weak adiabatic regime. In this regime the dynamics must proceed on a single LIP. Therefore, simply computing Pae(t) = |Æψae(t)|ψae(t)æ|2, which is the population in Vae, one can estimate the efficiency of the method. But because the goal is to modify the bond length, we require the method to prepare a localized vibrational wave packet in a single LIP. When this wave packet returns to the initial FranckCondon region, it will not overlap with the initial wave function, because it has gained larger vibrational energy. However, the spatial overlap should be maximal. We define the spatial or geometrical overlap as Z jψð0ÞjjψðtÞj dr Pg ðtÞ ¼ Æjψð0ÞjjjψðtÞjæ ¼ where |ψ| is the absolute value of the wave function. Because the anharmonicity of the bond dynamics makes it difficult to predict at what exact time will the wave packet cross again the Franck Condon region, we collect the maximum bond length and let the wave packet evolve after the pulse is off to calculate the largest value of Pg(t), which we call Pg,m. In weak adiabatic conditions Pg,m must be close to unity; thus we use Pg,m as a measure of the degree of adiabaticity in this “weaker” sense. In Figure 5 we show how the survival probability at the end of the pulse Ps(τ), and the maximal geometrical overlap Pg,m, change as a function of the pulse duration. Also shown is the 2695

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Figure 5. Maximum bond stretch as a function of the duration of the pulse in different regimes. In (a) we show the survival probability at final time Ps (solid line), the maximum geometrical overlap Pg,m (yellow line), and the population in the target LIP (dashed line) at half pulse duration Pae (which is practically constant during all the dynamics). In (b) we show the corresponding maximum bond elongation achieved during the process as a function of pulse duration. For τ g 6 T (Ps g 0.95) the LAMB dynamics proceeds in the strong adiabatic regime (regime I), where the maximum bond elongation is independent of the pulse duration. For τ approximately between 3 and 6 T (Pg,m g 0.95) the LAMB mechanism is operating in the weak adiabatic regime, where larger maximum bond elongations can be achieved, but evolving dynamically (regime II). For τ e 2 T the dynamics is not following the LAMB principles, because the population in the LIP is smaller than 0.95. Additionally, the maximum bond elongation becomes smaller again.

population of the LIP calculated at time t = τ/2 and the maximal elongation of the bond length observed during the dynamics. In this calculation we have used pulses with E 0 = 2  103 Hartree/(e a0) and bandwidth pΔω = 1.08De, with switch on/ off times of 0.2T. Populations larger than 0.95 are readily obtained with τ = 2.5T. In principle, as shown in the previous section, one could reach these values with shorter pulses, if the switching on of the pulse occurs much more rapidly. For pulses shorter than 2.5T the measure of the bond length is not welldefined, because the dynamics proceeds in both Vae and Vag (or Ve and Vg) following different trajectories. The average Ær(t)æ = Pg(t)Ær(t)æg + Pe(t)Ær(t)æe (where Pj(t) are the populations in the molecular potentials, Vj) can be computed, but the standard deviation of this measure (or bond spread) is very large, making the intuitively classical concept of bond length meaningless. Although Ps(τ/2) ≈ 1 for τ = 4.5T and the dynamics is time-symmetrical, the bond length is not really independent of τ until τ g 6T, where the dynamics becomes “strongly” adiabatic. With τ between 3T and 6T the weak adiabatic regime is fully at work. The geometrical overlap is greater than 0.95, which means that the wave packet moving in the LIP is totally localized and its shape is very similar to that of the initial state as it crosses the FranckCondon region. In this regime the maximum bond elongation increases as τ decreases, as expected.

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Figure 6. Adiabatic thresholds as a function of (a) the pulse bandwidth and (b) the pulse amplitude. We show the survival probability Ps (red lines) as a measure of the strong adiabatic regime and the maximum geometrical overlap Pg,m (black lines) as a measure of the weak adiabatic regime. In (a) pΔω = 0.3De (solid line), 0.4 De (dotted line), 0.5 De (dashed line). The maxima of both Ps and Pg,m displace to larger pulse durations as the minimum frequency decreases or the pulse bandwidth increases. In (b) E 0 = 2  103 Hartree/(e a0) (solid line), 3  103 Hartree/(e a0) (dotted line), 4  103 Hartree/(e a0) (dashed line). The maxima of both Ps and Pg,m displace to larger pulse durations as the pulse amplitude decreases.

Figure 6 shows how the adiabatic thresholds change when the pulse bandwidth (Figure 6a) or the pulse amplitude (Figure 6b) varies. As the minimum frequency of the field, ω0, gets smaller, and therefore the bandwidth larger, the onset of strong adiabaticity appears at longer pulses, although not necessarily much longer. Also the interval of validity of the scheme in the weak adiabatic regime is shifted to longer time durations. This is expected as rc becomes larger. On the contrary, as E 0 increases, the binding region of the LIP gets deeper and rc is displaced to smaller bond lengths; thus the onset of strong adiabaticity occurs at shorter pulse durations.

VI. SUMMARY AND DISCUSSION Using a simple model of an aligned diatomic molecule with a bound and a dissociative state coupled by dipole moment, we have tested a novel scheme that allows us to control the internuclear motion by means of parabolic chirped pulses. The method implies population inversion to the dissociative state and vibrational trapping in the excited state, at a bond distance and at a time directly controlled by the pulse. At the end of the pulse all the population is restored in the bound potential and in principle, the dynamics can be totally time-symmetrical. A sequence of pulses could thus create arbitrary bond vibrations where both the amplitude and the period of vibration would be driven by the laser. We have identified two regimes where the scheme operates. One involves strong (full) adiabaticity where maximal control is achieved, requiring large pulse durations and involving smaller 2696

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The Journal of Physical Chemistry A bond stretch. The other only requires adiabatic evolution of the electronic populations. In this weak adiabatic regime a wave packet is formed in the vibrationally trapped LIP, which is then transferred back to the ground potential in the form of a localized vibrationally excited wave packet. For the set of parameters here studied the bond elongation varies between 2 and 3 bohr in the strong adiabatic regime, and the minimum pulse durations required are typically longer than 56T. Using parabolic chirped pulses, one can therefore induce and control “slow” vibrations with periods at least 5 times longer than the vibrational period in the ground potential, and amplitudes that imply around 50% of the dissociation energy De. Larger vibrational amplitudes and shorter time periods can be obtained in the weak adiabatic or dynamic approach, at the expense of loosing control on the internuclear position at each instant of time. In this regime one can achieve larger bond elongation, up to 4 bohr in our model system. Additionally, in this regime a vibrationally highly excited wave packet is prepared in the ground potential at the end of the process. The pulse duration can be used as a single control parameter to modify the averaged vibrational energy of the system. Although extremely large bandwidths are required to induce very large bond elongations, which are inaccessible with current technology if the dissociation energy of the molecule is not small, the effect of bond elongation should be detectable with Δω a fraction of De/p, in the range of ultrashort femtosecond or attosecond pulses. The main problem to implement the LAMB scheme remains in the need of decoupling the LAMB process from any other multiphoton process, which may become resonant at some frequency of the excitation pulse spectrum, thus the need of finding molecules with a dissociative excited state relatively isolated from other states of the same symmetry.

’ AUTHOR INFORMATION Corresponding Author

*Electronic mail: B.Y.C., [email protected]; I.R.S., isola@ quim.ucm.es.

’ ACKNOWLEDGMENT This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (2010-0005643) by the NRF grant (No. 2011-0001211), funded by the Korean government (MEST), and by the Direccion General de Investigacion of Spain under Project No. CTQ2008-06760. ’ REFERENCES (1) Rice, S. A.; Zhao, M. Optical Control of Molecular Dynamics; Wiley: New York, 2000. (2) Brumer, P.; Shapiro, M. Principles of the Quantum Control of Molecular Processes; Wiley: New York, 2003. (3) Zewail, A. H. Angew. Chem. Int. Ed. 2000, 39, 2586. (4) Weiner, A. M.; Heritage, J. P.; Thurston, R. N. Opt. Lett. 1986, 11, 153. (5) Spano, F.; Haner, M.; Warren, W. S. Chem. Phys. Lett. 1987, 135, 97. (6) Judson, R. S.; Rabitz, H. Phys. Rev. Lett. 1992, 68, 1500. (7) Assion, A; Baumert, T.; Bergt, M.; Brixner, T.; Kiefer, B.; Seyfried, V.; Strehle, M.; Gerber, G. Science 1998, 282, 919. (8) Letokhov, V. Laser Control of Atoms and Molecules; Oxford University Press: Oxford, U.K., 2007.

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