State-Selective Vibrational Excitation and Dissociation of H2+ by

Article pubs.acs.org/JPCA

State-Selective Vibrational Excitation and Dissociation of H2+ by Strong Infrared Laser Pulses: Below-Resonant versus Resonant Laser Fields and Electron−Field Following Guennaddi K. Paramonov* and Oliver Kühn Institut für Physik, Universität Rostock, D-18051 Rostock, Germany ABSTRACT: The quantum dynamics of vibrational excitation and dissociation of H2+ by strong and temporally shaped infrared (IR) laser pulses has been studied on the femtosecond (fs) time scale by numerical solution of the time-dependent Schrödinger equation with explicit treatment of nuclear and electron motion beyond the Born− Oppenheimer approximation. Using sin2-shaped laser pulses of 120 fs duration with a peak intensity of I0 > 1014 W/cm2, it has been found that below-resonant vibrational excitation with a laser carrier frequency of ω < ω10/2 (where ω10 is the frequency of the |v = 0⟩ → |v = 1⟩ vibrational transition) is much more efficient than a quasi-resonant vibrational excitation at ω ≈ ω10. In particular, at the below-resonant laser carrier frequency ω = 0.3641 × 10−2 au (799.17 cm−1), dissociation probabilities of H2+ (15.3% at the end of the 120 fs laser pulse and 21% at t = 240 fs) are more than 3 orders of magnitude higher than those obtained for the quasi-resonant laser frequency ω = 1.013 × 10−2 au (2223.72 cm−1). Probabilities of state-selective population transfer to vibrational states |v = 1⟩, |v = 2⟩, and |v = 3⟩ from the vibrational ground state |v = 0⟩ of about 85% have been calculated in the optimal below-resonant cases. The underlying mechanism of the efficient below-resonant vibrational excitation is the electron−field following and simultaneous transfer of energy to the nuclear coordinate. refs 11−16. Ionization and dissociation of H2+ and HD+ in strong IR continuous-wave (CW) like laser fields have been investigated in refs 17 and 18. A different approach has been employed in refs 19−22 where use was made of an extended state representation including both bound electronic states (together with the corresponding vibrational states and the respective dissociation continuum) and electronic continuum states (including again the corresponding vibrational states). The latter approach has been used in particular to study the single photon-induced symmetry breaking during H2 dissociation.22 Until recently, the problem of optimal design of laser pulses that enable steering a molecule to a specified target has not been systematically addressed with the explicit treatment of both nuclear and electronic degrees of freedom of molecules beyond the Born−Oppenheimer approximation. The possibility to control the angular distribution of HD+ and H2+ dissociation by an optimal choice of the carrier envelope phase of an intense 10 fs laser pulse at λ = 790 nm has been reported in ref 15. It has also been shown in our previous work18 that vibrational excitation of H2+ and HD+ by CW-type rectangular-shaped infrared laser fields of 120 fs duration with the below-resonant frequency ω = ω10/2 is much more efficient than that with the quasi-resonant frequency ω ≈ ω10.

1. INTRODUCTION The theoretical foundations of the well-known laser control schemes for manipulation of molecular dynamics have been developed within the Born−Oppenheimer approximation.1−10 In principle, investigation of the laser-driven dynamics of molecules beyond the Born−Oppenheimer approximation, i.e., with an explicit treatment of both nuclear and electronic degrees of freedom,11−18 could extend substantially the range of applications of control schemes and bring new ideas to the field of optimal laser control. Indeed, the problem which will be addressed in the present work, for example, cannot be treated within the Born−Oppenheimer approximation at all. Explicit treatment of electronic and nuclear motion within the time-dependent non-Born−Oppenheimer Schrö dinger equation makes it possible to avoid any assumption concerning the relative magnitudes of intramolecular Coulombic forces and electron−field interactions. Due to the complexity and the computational requirements of the numerical simulations involved, current attention has been focused for the most part to small diatomic molecules, H2, D2, HD, and simple molecular ions, H2+, D2+, HD+. The relative simplicity of small molecular systems has made it possible to treat their laserdriven dynamics on a reasonable time scale. Usually, molecular rotations are not taken into account, while the nuclei are free to move along the polarization direction of the laser electric field, and electrons move in three dimensions with conservation of cylindrical symmetry. In particular, the three-dimensional (3-D) quantum dynamics of H2+ and HD+ in strong laser fields (I0 > 1013 W/cm2) in the ultraviolet and near-infrared domains has been studied with the time-dependent Schrödinger equation in © XXXX American Chemical Society

Special Issue: Jörn Manz Festschrift Received: June 20, 2012 Revised: August 23, 2012

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dx.doi.org/10.1021/jp3060679 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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The present work is addressed to 3-D quantum dynamical simulations of efficient vibrational excitation and dissociation of H2+ by strong temporally shaped IR laser pulses with belowresonant carrier frequencies. By definition, below-resonant or under-resonant carrier frequencies are those which are substantially (e.g., 2 times) smaller than the frequency ω10 corresponding to the |v = 0⟩ → |v = 1⟩ vibrational transition. Note that, if the Born−Oppenheimer approximation would be employed, H2+ could not be excited by the IR laser fields at all, due to its dipole moment being zero. It has been shown previously17 that if the electronic motion is treated explicitly, together with the nuclear motion, the dipole moment of H2+ is induced by the applied IR laser field, and vibrational excitation of H2+ becomes possible, but remains weak at the laser carrier frequency ω ≈ ω10. It is the aim of the first part of our present work to find the optimal laser carrier frequency and carrier envelope phase providing the maximum dissociation probability of H2+ at a fixed field amplitude and using sin2-shaped laser pulses of 120 fs duration. We will also address the problem of controlling the angular distribution of the dissociation fragments of H2+ and calculate the dissociation probability at z > 0 and at z < 0 (z is the electron coordinate along the molecular axis) versus the carrier envelope phase of the applied laser pulse. In the second part, we study the problem of efficient state-selective population transfer from the ground vibrational state of H2+ to vibrational states |v = 1⟩, |v = 2⟩, and |v = 3⟩ by IR laser pulses with optimal amplitude, carrier frequency, and carrier envelope phase. The paper is organized as follows. The 3-D model of H2+ and techniques used to solve numerically the time-dependent nonBorn−Oppenheimer Schrödinger equation are described in section 2. The laser-driven quantum dynamics of H2+ and its free evolution after the end of the 120 fs laser pulses with quasiresonant and below-resonant carrier frequencies are presented in section 3.1, where also the optimal laser frequency is determined and carrier envelope phase control of the z>0 dissociation fragments, Pz>0 D and PD , is demonstrated. Section 3.2 presents the results obtained for the optimal state-selective vibrational excitation of H2+ to |v = 1⟩, |v = 2⟩, and |v = 3⟩ from the vibrational ground state |v = 0⟩. The results are summarized and discussed in the concluding section 4.

Figure 1. The 3-D model of H2+ excited by a laser field, which is linearly polarized along the z axis. The internuclear distance is R, and the distances between the electron and each of the two protons are r1 and r2; see eqs 4 and 5.

vibrational motion can be excited only indirectly, due to the electron motion induced by the IR laser field.17,18 The time-dependent Schrödinger equation describing the dynamics of H2+ in the classical linearly polarized laser field (see Figure 1) has the following form: iℏ

1 ∂Ψ ⎞ ∂ ℏ2 ∂ 2Ψ ℏ2 ⎛ ∂ 2Ψ Ψ=− − ⎟ ⎜ 2 + 2 2mn ∂R 2me ⎝ ∂ρ ρ ∂ρ ⎠ ∂t −

d (z) ∂A(t ) ℏ2 ∂ 2Ψ Ψ + VC(R , ρ , z)Ψ + z 2 2me ∂z c ∂t (2)

In the above equation, mn = Mp/2 is the nuclear reduced mass and me = 2MeMp/(Me + 2Mp) is the electron reduced mass; the Coulomb potential reads VC(R , ρ , z) = e 2(1/R − 1/r1 − 1/r2)

where the electron−proton distances (see Figure 1) are r1(R , ρ , z) = [ρ2 + (z + R /2)2 ]1/2

(4)

and r2(R , ρ , z) = [ρ2 + (z − R /2)2 ]1/2

2. MODEL, EQUATIONS, AND TECHNIQUES The 3-D three-body model with full Coulombic interactions representing the molecular ion H2+ which is excited by the laser field is shown in Figure 1. The applied laser field is assumed to be linearly polarized along the z axis, the nuclear motion is restricted to the polarization direction of the laser electric field, whereas the electron (e) moves in three dimensions with conservation of cylindrical symmetry. Accordingly, only two electron coordinates, z and ρ, measured with respect to the center of mass of the two protons (p) should be treated explicitly together with the internuclear distance R. The component of the dipole moment of H2+ along the z axis reads23 dz(z) = −ez[1 + Me /(2M p + Me)]

(3)

(5)

The z-component of the dipole moment is defined by eq 1, and the vector potential A(t) is chosen in the following form: c A(t ) = , 0 sin 2(πt /t p) cos(ωt + φ) (6) ω where , 0 is the amplitude (for the sake of comparison, it is chosen to be , 0 = 2.98 × 102 MV/cm as in our previous work,17 where 400 fs CW-like quasi-resonant IR laser fields were used), tp = 120 fs is the pulse duration at the base, ω is the laser carrier frequency, and φ is the carrier envelope phase. Accordingly, the electric field , (t) = −(1/c)[∂A(t)/∂t] is given by ,(t ) = , 0[sin 2(πt /t p) sin(ωt + φ)

(1)

−

where e is the electron charge and Mp and Me are the proton and electron masses, respectively. The homonuclear system H2+ does not have a permanent dipole moment; therefore, its

π sin(2πt /t p) cos(ωt + φ)], ωt p

0 ≤ t ≤ tp

(7)

In the above equation, the first term corresponds to a sin2shaped laser pulse with a sine-oscillating carrier field which is B



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Figure 2. Resonant versus below-resonant vibrational excitation of H2+ by strong infrared laser pulses and field-following electronic oscillations. In the upper panels, PD(t) stand for the dissociation yields and PI(t) stand for the overall ionization yields (see eq 9). In the middle panels, ⟨R(t)⟩, ⟨z(t)⟩, and ⟨ρ(t)⟩ stand for the expectation values of the internuclear distance and the electron coordinates, respectively. Lower panels: the sin2shaped IR laser pulses; the pulse amplitude is , 0 = 2.98 × 102 MV/cm in all three cases. Left panels: quasi-resonant vibrational excitation at ω = 1.013 × 10−2 au (2223.72 cm−1). Middle panels: below-resonant vibrational excitation at ω = 0.5065 × 10−2 au (1111.86 cm−1). Right panels: belowresonant vibrational excitation at ω = 0.3752 × 10−2 au (823.6 cm−1). The carrier envelope phase is φ = 0 in all three cases.

switched on at t = 0, while the second, the so-called “switching” term, appears due to the finite pulse duration tp. Note that the “switching” term in 7 cannot be neglected24 if the number of optical cycles per pulse duration is less than about 15, which is the case in the present study. The above definition of the electric field via the vector potential25,26 is also suitable because it assures that the laser electric field , (t) has a vanishing directcurrent (dc) component

∫0

tp

, (t ) d t = 0

The dissociation probability PD will be calculated with the time- and space-integrated outgoing flux for the nuclear coordinate R; the ionization probabilities follow from the respective fluxes for the positive and negative direction of the z axis (denoted as Pz>0 and Pz>0 I I , respectively) as well as for the outer end of the ρ axis, denoted as PρI . Thus, the overall ionization probability PI is given by PI = PIz > 0 + PIz < 0 + PIρ

(9)

Similarly, the overall dissociation yield PD is composed of two z0 D and PD . Note that comparing these probabilities gives an indication whether one has dominantly the products H+ + H or H + H+. The 3-D wave packet was damped with imaginary smooth optical potentials adapted from ref 28 at R > 63a0, at z < −170a0 and z > 170a0, and at ρ > 80a0. Initially, at t = 0, the H2+ is assumed to be in its ground vibrational and ground electronic state. The wave functions and eigenenergies of the ground and three lowest excited vibrational states have been obtained by the numerical propagation of the

(8)

and satisfies Maxwell’s equations in the propagation region.27 The numerical technique used to solve the 3-D equation of motion (eq 2) has been described previously.17,18 The following grids for the R, z, and ρ degrees of freedom have been used (a0 stands for the Bohr radius): Rmin = 0.07a0, Rmax = 81.92a0 (256 grid points); zmin = −197.35a0, zmax = 197.35a0, Rmax = 197.35a0 (260 grid points); ρmin = 0.012a0, ρmax = 98.63a0 (120 grid points). C



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Figure 3. Snapshots of the reduced density P(R, z, t) at representative times for the laser field of Figure 5 at φ = 0.85π. Under these conditions, the z>0 2 −2 au (799.17 cm−1), tp = 120 fs). ratio Pz>0 D /PD takes its maximum value of 5.047. (Parameters: , 0 = 2.98 × 10 MV/cm, ω = 0.36413 × 10

The case of Nc = 8 corresponds to ω = 1.013 × 10−2 au (2223.72 cm−1), which is close to the resonance with the |v = 0⟩ → |v = 1⟩ vibrational transition (ω ≈ ω10). The comparison of the quasi-resonant case (Nc = 8, ω ≈ ω10) and two belowresonant cases (Nc = 4, ω ≈ ω10/2 and Nc = 3, ω ≈ ω10/2.7) is presented in Figure 2. The laser-driven dynamics is illustrated in the upper panels of Figure 2 with the time-dependent overall dissociation probabilities PD(t) and the overall ionization probabilities PI(t) (see eq 9). The time-dependent expectation values of the internuclear distance R and the electron coordinates z and ρ (⟨R(t)⟩, ⟨z(t)⟩, and ⟨ρ(t)⟩, respectively) are presented in the middle panels. The laser fields are shown in the lower panels. In all three cases, the field amplitude is , 0 = 2.98 × 102 MV/cm, which corresponds to a peak intensity I0 = c, 02/8π of 1.18 × 1014 W/cm2. It is seen from the upper panels of Figure 2 that the ionization starts prior to the dissociation, most notably in the

equation of motion (eq 2) in imaginary time with the laser field being switched off: , (t) = 0. The resulting eigenenergy of the vibrational ground state, for example, Ev=0 = −0.59716 au, is in good agreement with the highly accurate value reported in ref 29.

3. RESULTS 3.1. Dissociation of H2+ by IR Laser Pulses: BelowResonant versus Resonant Vibrational Excitation. First, the laser carrier frequency ω has been defined in a stepwise manner, by the number of optical cycles Nc per pulse duration tp as follows: ω(Nc) = 2πNc/t p

(10)

Subsequently, a fine-tuning of ω has been performed in order to find the optimal laser frequency. D



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case of resonant excitation. The ionization probabilities PI(t) reach their maximum values in the second half of the respective laser pulses and remain constant after the end of the pulses. The ionization of H2+ is much less efficient as compared to the dissociation. The maximum ionization probability is PI = 0.001% in the quasi-resonant case of ω ≈ ω10 and strongly increases with the decrease of the laser carrier frequency: PI = 0.029% at ω ≈ ω10/2 and PI = 0.22% at ω ≈ ω10/2.7. In contrast, the dissociation probabilities PD(t), being much higher as compared to the ionization probabilities, continue to increase after the end of the laser pulses and reach their saturation domains at approximately t = 240 fs. It is clearly seen from the upper panels of Figure 2 that the below-resonant vibrational excitation is especially efficient for the dissociation of H2+. At t = 240 fs, for example, the dissociation probability in the quasi-resonant case of ω ≈ ω10 is PD = 0.014% only, while in the two below-resonant cases the dissociation probabilities are more than 2 and 3 orders of magnitude higher: PD = 1.67% at ω ≈ ω10/2 and PD = 17.6% at ω ≈ ω10/2.7. Note, for comparison, that the overall dissociation probability of about 0.28% only was calculated in ref 17 where a 400 fs CW-like quasi-resonant IR laser field with the same amplitude was used. The underlying mechanism of the highly efficient belowresonant vibrational excitation, which results finally in the dissociation of H2+, is the electron−field following illustrated in the middle panels of Figure 2 with the expectation values ⟨z(t)⟩ of the electron coordinate z. Since the electron is coupled to the laser electric field, it follows the oscillations of the field. For this reason, the electron wave packet is transferred from one proton to the other twice per each optical cycle. Such periodic shifts of the electron density to one of the two protons disturb the equilibrium configuration of the entire system and result in the Coulombic repulsion of the protons. In the case of the quasi-resonant excitation (ω ≈ ω10, the middle-left panel in Figure 2), the field-following electronic oscillations are relatively fast, and the relatively heavy protons do not have enough time to substantially change their positions (see curve “⟨R(t)⟩”). It is the concept of the below-resonant vibrational excitation at ω ≤ ω10/2 to substantially increase the time intervals corresponding to the nonequilibrium configurations of the entire system and thus to allow the Coulombic repulsion of the protons to act more efficiently on a longer time scale, as compared to the quasi-resonant vibrational excitation at ω ≈ ω10. It is clearly seen from the middle panels of Figure 2 that the amplitudes of the nuclear vibrations at ω ≤ ω10/2 are very large, as compared to the quasi-resonant case of ω ≈ ω10, and the entire system remains vibrationally excited even after the end of the below-resonant laser pulse. The nuclear vibrations activated by the electronic motion influence the electronic motion as well, and the field-following oscillations of the electron density are substantially modified by the nuclear motion in the below-resonant cases of ω ≤ ω10/2. Also note that the expectation values ⟨ρ(t)⟩ of the electron coordinate ρ oscillate in-phase with the nuclear oscillations (curves “⟨R(t)⟩”) in the below-resonant cases, while it remains almost constant in the quasi-resonant case of ω ≈ ω10. The coupled dynamics of electronic and nuclear degrees of freedom is illustrated in more detail in Figure 3, where we show snapshots of the reduced density for the electronic z coordinate and the nuclear coordinate R. Clearly, due to the excitation of electronic motion along the z coordinate, wave packet motion and accompanying vibrational excitation up to dissociation is seen along the nuclear coordinate.

Changing the laser carrier frequency in a stepwise fashion (see eq 10), it was found that the maximum dissociation probability of H2+ at the end of the laser pulse (t = 120 fs) and at t = 240 fs (when dissociation reaches the saturation domain) is achieved in the vicinity of Nc = 3 [ω = 0.3752 × 10−2 au (823.6 cm−1)] where a fine-tuning of ω has been performed in order to find the optimal laser frequency. The results are shown in Figure 4, where the dashed curve presents the dissociation

Figure 4. Dissociation yield of H2+ versus the laser carrier frequency ω (the upper horizontal scale) of the IR laser pulse. The lower horizontal scale gives the number of optical cycles, Nc, per pulse duration. The pulse duration is tp = 120 fs, the laser-field amplitude is , 0 = 2.98 × 102 MV/cm, and the carrier envelope phase is φ = 0.

yield at the end of the laser pulse (t = 120 fs), and the solid curve at t = 240 fs. The lower horizontal scale in Figure 4 gives the number of optical cycles Nc per pulse duration, while the upper horizontal scale gives the corresponding frequency in wavenumbers. Due to a very small dissociation yield at Nc = 8 through 5, the respective data are given in Figure 4 being multiplied by a factor of 100. It is seen from Figure 4 that the optimal frequencies for dissociation of H2+ belong to the domain of ω < ω10/2 (Nc < 4). Even in the mostly belowresonant case of a one-cycle pulse, Nc = 1, ω = 0.12665 × 10−2 au (277.96 cm−1), the dissociation probability is more than 2 orders of magnitude higher than in the quasi-resonant case of Nc = 8 (ω ≈ ω10). The maximum dissociation yield is achieved at ω = 0.36413 × 10−2 au (799.17 cm−1) and the corresponding dissociation probabilities are PD = 15.3% at the end of the pulse and PD = 21% at t = 240 fs. It is also interesting to get some information on the dissociation yield beyond the frequency domain of Nc = 1 (ω = 0.127ω10) through Nc = 8 (ω = 1.015ω10) presented in Figure 4. We performed several numerical simulations at ω > ω10 and found that, for ω up to ω ≈ 5ω10, the behavior of H2+ is similar to that at ω ≈ ω10. For example, for Nc = 16 − 40 (ω = 2.03ω10−5.075ω10), the dissociation probability lies in the domain of PD = 0.0013−0.0028% at the end of the pulse and PD = 0.0018−0.0039% at t = 240 fs. These values are smaller than those obtained in the quasi-resonant case of Nc = 8 (ω = 1.015ω10). In the opposite domain of very small frequencies, ω ≪ ω10, the case of a half-cycle pulse (Nc = 0.5, ω = 0.063ω10) is of special interest. Half-cycle pulses and trains of half-cycle pulses have proven to be useful for quantum control of molecular dynamics (see, e.g., the recent work30 and references therein.) Our simulations performed for a single 120 fs halfE



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cycle pulse gave a dissociation probability of PD = 12% at the end of the pulse and PD = 19% at t = 240 fs. These values are much larger than those obtained in the case of the one-cycle pulse and only by about 20% smaller than those obtained at the optimal frequency of Figure 4. A more detailed study of halfcycle pulses will be given elsewhere. Due to the small number of optical cycles of the IR light per pulse duration corresponding to the maximum dissociation yield (Nc ≈ 3, see Figure 4), a strong dependence of the dissociation probability on the carrier envelope phase φ can be expected.24 In Figure 5, the overall dissociation yield PD and

the different partial dissociation channels is seen in the wave packet dynamics in Figure 3 for the case of φ = 0.85π (see especially t = 60 and 70 fs). Note finally that a very important feature of Figure 5a is the existence of two almost “nondissociative” domains of the carrier envelope phase, φ ≈ 0.5π and φ ≈ 1.5π, where the overall dissociation probability is very small (almost 1 order of magnitude smaller as compared to the maximum values). In order to understand the physical reasons for the existence of “nondissociative” domains, we remind first that the nuclear motion in homonuclear H2+ and the subsequent dissociation is induced by the laser driven motion of the electron. Therefore, we present in Figure 6 the time-dependent expectation values

Figure 6. The time-dependent expectation values ⟨R(t)⟩ and ⟨z(t)⟩ of H2+ excited by the 120 fs laser pulse at , 0 = 2.98 × 102 MV/cm, ω = 0.36413 × 10−2 au (799.17 cm−1), and φ = 0.1π (solid lines) and φ = 0.5π (dashed lines).

of the internuclear distance, ⟨R(t)⟩, and the z coordinate of electron, ⟨z(t)⟩, calculated at φ = 0.1π (the maximum dissociation yield, solid lines) and at φ = 0.5π (the minimum dissociation yield, dashed lines). It is clearly seen from Figure 6 that electron displacements from the equilibrium position ⟨z⟩ = 0 at φ = 0.5π are substantially smaller than those at φ = 0.1π. Therefore, the equilibrium configuration of the entire system is much less disturbed at φ = 0.5π than at φ = 0.1π. Due to this reason, the Coulombic repulsion of the protons occurring during the nonequilibrium configuration at φ = 0.5π is much less efficient than that at φ = 0.1π, resulting in a small change of the internuclear distance ⟨R(t)⟩ and accordingly to a small dissociation yield. It will be shown in the next section that the existence of aforementioned “nondissociative” domains of the carrier envelope phase is very suitable for vibrationally state-selective population transfer in H2+ controlled by the optimal belowresonant IR laser pulses. 3.2. Below-Resonant State-Selective Vibrational Excitation of H2+. In the following, it will be demonstrated that below-resonant vibrational excitation can also provide an efficient state-selective population transfer in the homonuclear molecular ion H2+. The wave functions and eigenenergies of the initial ground state and three lowest vibrational target states of H2+ have been obtained by the numerical propagation of the equation of motion in the imaginary time, with the laser field being switched off. The following eigenenergies have been calculated: Ev=0 = −0.59716 au, Ev=1 = −0.58718 au, Ev=2 = −0.57936 au,

Figure 5. The overall dissociation yield PD (upper panel) and partial z>0 dissociation yields Pz>0 D and PD (lower panel) at the end of the 120 fs IR laser pulse versus the carrier envelope phase φ. The laser-field amplitude is , 0 = 2.98 × 102 MV/cm, and the optimal laser carrier frequency is ω = 0.36413 × 10−2 au (799.17 cm−1). z>0 partial dissociation yields Pz>0 D and PD achieved at the end of the 120 fs IR-laser pulse are presented versus the carrier envelope phase φ of the pulse. The laser carrier frequency, ω = 0.36413 × 10−2 au (799.17 cm−1), corresponds to the optimal value found at φ = 0 (see Figure 4). It can be concluded from Figure 5a that the maximum of the overall dissociation yield PD is increased from 15.3% achieved at φ = 0 to 16.7% achieved at φ = 0.1π and at φ = 1.1π. Since the field profile for the carrier envelope phase φ is (except for the opposite sign) the same as that for φ + π, the overall dissociation yield is the same in both cases. Results presented in Figure 5b show the possibility to control the distribution of the dissociation fragments of H2+, i.e., H+ + H vs H + H+. At the carrier envelope phase φ = 0.85π, for example, H atoms move with 12.85% probability in the positive direction of the z axis and with only 2.5% probability in the negative direction of the z axis (the corresponding branching ratio is more than 5). At φ = 1.85π, the situation changes to the opposite. The preference of

F



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Figure 7. Resonant versus below-resonant excitation of the |v = 1⟩ state in H2+. (a) Time-dependent populations of the vibrational levels v = 0 and v = 1 at ω = ω10. (b) The resonant IR laser field with , 0 = 2.982 × 102 MV/cm, ω = 0.998 × 10−2 au (2190.4 cm−1), φ = 0, and tp = 120 fs. (c) Timedependent populations of the vibrational levels v = 0, 1, and 2 at ω = ω10/2. (d) The below-resonant IR laser field with , 0 = 2.982 × 102 MV/cm, ω = 0.499 × 10−2 au (1095.2 cm−1), φ = 0, and tp = 120 fs.

and Ev=3 = −0.57373 au. The time-dependent populations of the vibrational states have been calculated by projection of the time-dependent 3-D wave packet obtained by the propagation of the equation of motion (eq 2) in real time on the eigenfunctions of the respective vibrational states. First, we recall the optimization technique developed previously1,6 within the Born−Oppenheimer approximation for heteronuclear molecules excited close to a multiphoton resonance from an initial to a target vibrational state. It usually starts with a reasonable guess for the laser carrier frequency ω. In the case of the v-photon transition |0⟩ → |v⟩, for example, a reasonable initial choice of the frequency is

ω=

Ev − E0 ℏv

populations of the vibrational states, and the lower panels (b and d) present the corresponding resonant and below-resonant IR laser fields. It is seen from Figure 7a that, in the resonant case of ω = ω10, the population goes back to the initial state |v = 0⟩ by the end of the pulse almost entirely. In contrast, in the belowresonant case of ω = ω10/2, more than 25% population transfer to the target state |v = 1⟩ is achieved. Subsequent optimization of the laser-field amplitude and the carrier envelope phase (plus a second application of the threeparameter optimization procedure) made it possible to find the optimal parameters of the below-resonant IR laser pulse, which provides the population transfer to the target |v = 1⟩ state with more than 85% probability. The optimal case is presented in Figure 8, where the upper panel (a) shows the population dynamics and the lower panel (b) presents the optimal belowresonant IR laser field. Note that the optimal below-resonant laser carrier frequency ω = 0.45995 × 10−2 au (1009.5 cm−1) is smaller than the initial below-resonant guess of the laser carrier frequency, ω = ω10/2, suggested by 12. Also note that, due to a small number of optical cycles per pulse duration, the carrier envelope phase φ plays a very important role for the efficient state-selective population transfer to be accomplished. As could be expected, the optimal carrier envelope phase φ = 0.42π falls into the domain of a very small overall dissociation yield (see Figure 5a). The three-parameter optimization procedure described above for the state-selective population transfer |v = 0⟩ → |v = 1⟩ has also been applied to find the optimal parameters of the below-resonant IR laser pulses, providing an efficient stateselective excitation of vibrational states |v = 2⟩ and |v = 3⟩. In both cases, the initial below-resonant guess of the laser carrier

(11)

In the next steps, the laser-field amplitude , 0 and the carrier envelope phase φ are optimized. The three-step optimization procedure can be repeated several times until convergence. It should be noted, however, that the flexibility of this optimization (e.g., step size) in the present case is hampered by the computational demands for solving the 3-D non-Born− Oppenheimer Schrödinger equation. Our numerical simulations performed for the homonuclear molecular ion H2+ showed that the initial guess for the laser carrier frequency given by eq 11 is not a good choice at all. Much better results are obtained with the below-resonant initial guess for the laser carrier frequency:

ω=

Ev − E0 2ℏv

(12)

The resonant (ω = ω10) and below-resonant (ω = ω10/2) excitation of the vibrational state |v = 1⟩ is shown in Figure 7. The upper panels (a and c) present the time-dependent G



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frequency, ω = (Ev − E0)/2ℏv, suggested by eq 12, proved to be successful. The state-selective population transfer from the initial vibrational state |v = 0⟩ to the target vibrational states |v = 2⟩ and |v = 3⟩ controlled by the optimal below-resonant IR laser pulse is presented in Figure 9. The upper panel (a) presents the time-dependent populations of the vibrational levels v = 0, 1, 2, and 3 which acquire substantial populations during the pulse. The lower panel (b) shows the optimal laser pulse, which provides the population transfer to the target state |v = 2⟩ with more than 86% probability. Note that, similar to the previous case of the target vibrational state |v = 1⟩, the optimal belowresonant laser carrier frequency found for the target state |v = 2⟩, ω = 0.42103 × 10−2 au (924.05 cm−1), is smaller than the initial below-resonant guess for the laser carrier frequency, ω = 0.445 × 10−2 au (976.66 cm−1), suggested by eq 12. Due to the small number of optical cycles involved, the carrier envelope phase φ also plays a very important role for the efficient stateselective population transfer, and the optimal carrier envelope phase φ = 0.58π also belongs to the domain of a very small overall dissociation yield (see Figure 5a). Note that the optimal laser-pulse amplitude, , 0 = 3.029 × 102 MV/cm, found for the target state |v = 2⟩ is larger than that required for the target state |v = 1⟩, while the optimal laser carrier frequency found for |v = 2⟩ is smaller than that found for |v = 1⟩. Analogous observations are made for the state-selective population transfer to the target vibrational state |v = 3⟩ in Figure 9c and d. Similarly to the previous cases of the target vibrational states |v = 1⟩ and |v = 2⟩, the optimal belowresonant laser carrier frequency found for the target state |v = 3⟩, ω = 0.38693 × 10−2 au (849.21 cm−1), is smaller than the initial below-resonant guess for the laser carrier frequency, ω = 0.3905 × 10−2 au (857.05 cm−1), suggested by eq 12. The

Figure 8. Optimal below-resonant state-selective excitation of |v = 1⟩ in H2+. Upper panel: time-dependent populations of the vibrational levels v = 0, 1, and 2. Lower panel: the optimal below-resonant IR laser field with , 0 = 3.004 × 102 MV/cm, ω = 0.45995 × 10−2 au (1009.5 cm−1), φ = 0.42π, and tp = 120 fs.

Figure 9. Optimal below-resonant state-selective excitation of |v = 2⟩ (a and b) and |v = 3⟩ (c and d) in H2+. Upper panels (a and c): time-dependent populations of the vibrational levels v = 0, 1, 2, and 3. The optimal below-resonant IR laser fields (tp = 120 fs) are shown in the lower panels: (b) |v = 2⟩, , 0 = 3.029 × 102 MV/cm, ω = 0.42103 × 10−2 au (924.05 cm−1), φ = 0.58π; (d) |v = 3⟩, , 0 = 3.044 × 102 MV/cm, ω = 0.38693 × 10−2 au (849.21 cm−1), φ = 0.51π. H



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magnitude larger than those obtained at the quasi-resonant laser frequency ω = 1.013 × 10−2 au (2223.72 cm−1). The physical mechanism responsible for the high efficiency of the below-resonant vibrational excitation is the electron− field following, which disturbs the equilibrium configuration of the entire system two times per each optical cycle and allows the Coulombic repulsion of the two protons to act on a longer time scale than in the case of a quasi-resonant vibrational excitation. The below-resonant vibrational excitation of H2+ studied in this work is practically non-ionizing: the ionization probability is always much less than 1% and at least 2 orders of magnitude smaller than the dissociation probability at ω < ω10/2. Due to the small number of optical cycles involved in the below-resonant vibrational excitation, the carrier envelope phase plays a very important role both for the overall dissociation yield PD and for the partial dissociation yields Pz>0 D (the H atom moves in the positive direction of the z axis in (z,R) plane) and Pz>0 D (the H atom moves in the negative direction of the z axis in (z,R) plane). It is shown, in particular, that there exist such values of the carrier envelope phase where the probability of the H atom to move in the positive/negative direction of the z axis in (z,R) plane is more than 5 times larger than its probability to move in the opposite direction. This feature makes it possible to selectively control the angular distribution of the dissociation fragments of H2+. It is also shown that there exist two almost “nondissociative” domains of the carrier envelope phase where the overall dissociation yield PD is very small. This feature proved to be very important for vibrationally state-selective population transfer in H2+ controlled by the optimal below-resonant IR laser pulses. Here, the probability of state-selective population transfer to vibrational states |v = 1⟩, |v = 2⟩, and |v = 3⟩ from the vibrational ground state |v = 0⟩ exceeds 85% if the process is controlled by optimally chosen below-resonant IR laser pulses.

optimal value of the carrier envelope phase found for the target state |v = 3⟩, φ = 0.51π, also belongs to the domain of a very small overall dissociation yield (see Figure 5a). The optimal laser-pulse amplitude, , 0 = 3.044 × 102 MV/cm, required for the target state |v = 3⟩ is larger that that required for the target state |v = 2⟩. The optimal laser carrier frequency found for |v = 3⟩ is smaller than that found for |v = 2⟩. A general feature of the vibrationally state-selective population transfer controlled by the optimal below-resonant IR laser pulses presented in Figures 7−9 is that a higher vibrational target state requires a larger amplitude of the laser pulse and a smaller laser carrier frequency. Note that this feature and the population dynamics of the vibrational states presented in Figures 7−9 for the homonuclear molecular ion H2+ are very similar to those obtained previously within the Born−Oppenheimer approximation for many heteronuclear molecules being excited close to a multiphoton resonance from an initial to a target vibrational state; see, for example, refs 31−40. Also note that, in heteronuclear molecules, overtone transitions (ω > ω10) can sometimes be used for an efficient state-selective population transfer.34,37 In contrast, this is not the case for the homonuclear molecular ion H2+. Our investigations have shown that any transitions with ω > ω10, in particular overtone transitions, are not suitable for any noticeable state-selective population transfer. We conclude that the distinctive features of the belowresonant, vibrationally state-selective population transfer in the homonuclear H2+ are as follows: (i) the optimal laser carrier frequency is far off any multiphoton resonance between the initial and target vibrational states; (ii) due to the small number of optical cycles involved, the carrier envelope phase plays a very important role; (iii) the existence of “nondissociative” domains of the carrier envelope phase, similar to those presented in Figure 5a, are critical for the efficient state-selective population transfer. Note finally that the probability of vibrationally stateselective population transfer as high as 86% obtained in the present work for H2+ suggests that an efficient state-selective vibrational excitation of other homonuclear molecules is feasible by making use of optimally chosen below-resonant IR laser pulses. As a caveat, we emphasize that the three-step optimization procedure used here to find optimal below-resonant laser fields is suitable but definitely not unique. Other methods, described, for example, in refs 1, 3, and 41, might provide different laser fields and higher yields. However, it should be noted that the present non-Born−Oppenheimer simulation is rather timeconsuming and methods requiring an iterative solutions such as optimal control theory would be extremely demanding.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been financially supported by the Deutsche Forschungsgemeinschaft through the Sfb 652. The numerical simulations have been performed in part at RQCHP (Canada) under the sponsorship of Prof. A. D. Bandrauk, which is gratefully acknowledged.

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4. CONCLUSION The results presented in this work clearly demonstrate that below-resonant vibrational excitation and dissociation of the homonuclear molecular ion H2+ by shaped IR laser pulses with a laser carrier frequency ω < ω10/2 is much more efficient than a quasi-resonant excitation at ω ≈ ω10. It is shown, in particular, that, at the below-resonant laser carrier frequency ω = 0.3641 × 10−2 au (799.17 cm−1), the dissociation probabilities of H2+ (15.3% at the end of the 120 fs laser pulse and 21% at t = 240 fs) are more than 3 orders of

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State-Selective Vibrational Excitation and Dissociation of H2+ by

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