Pump and probe ultrafast electron dynamics in LiH: a computational study

A time-dependent multiconfiguration method with a large electronic basis set is used to compute the response of all the electrons of LiH to a few-cycle intense pump field followed by a probe pulse. The ultrashort pump pulse excites a coherent superposition of stationary electronic states and, by changing the pump parameters such as intensity, duration, polarization and phase of carrier frequency, one can steer the motion of the electrons. Particular attention is given to the control provided by the polarization and by the phase. For example, a change in polarization is used to select an electronic wave packet that is rotating in a plane perpendicular to the bond or rotation in a plane containing the bond. The electronic wave packet can be probed by a delayed second pulse. This delayed probe pulse is also included in the Hamiltonian with the result that the frequency dispersed probe spectrum can be computed and displayed as a two-dimensional plot.

3 pulses means that a few cycles of light can be contained in an envelope whose width is subfemtoseconds, thereby giving rise to new physics. The chemical properties of molecules are governed by the valence electrons. To excite bound motions of the valence electrons requires light of lower frequencies, in the range of up to, say, 15 eV. The envelope of the pulse must then necessarily be longer [44], from about one to a few femtoseconds. It is currently not yet possible to experimentally generate few-cycle ultrashort laser pulses in the energy range of interest for electronic spectroscopy, but progress is being made [44,45].
We explore here the interaction of a molecule with such intense few-cycle UV pulses at a realistic quantitative level of computation. It is shown that even at the high powers required to pump a sizeable fraction of molecules it is possible to maintain tight control over the dynamics of the electrons. Other recent studies of ultrafast processes include the work on localization of electrons by the laser pulse [46]- [48], on LiCN [49], on inducing ring currents [50,51], on the importance of correlation effects in charge migration [52]- [54] and on other intense field effects on molecules [55]- [66]. We pay special attention to the control of the electron dynamics that is allowed by the physical parameters of the pump and to the probing of the resulting coherent motion.

Time-dependent multiconfiguration method
The dynamics is unfolding in time using a multideterminant approach [10,11] where the N electron wavefunction (x 1 , . . . , x N ; t) is expressed as a linear combination of Slater determinants with time-dependent coefficients and where the determinants themselves are made from n time-dependent (spin) orbitals ϕ j (x; t). As will be discussed shortly, these orbitals are chosen to adapt to the temporal changes in the electronic structure (x 1 , . . . , x N ; t) = J A J (t) det ϕ j 1 (x 1 ; t) · · · ϕ j N (x N ; t) , J is an N -tuple index that enumerates the N spin orbitals that appear in the determinant and a shorthand notation for the determinant is |J (t) . Because of the antisymmetry, there are only n!/N !(n − N )! determinants. x i specifies the space and spin coordinates of electron i and the n/2 spatial orbitals are expressed as a linear combination of M atomic basis functions. As emphasized below, the orbitals ϕ j (x; t) are allowed to adjust with time. Therefore we need only n/2 = 5 < M spatial orbitals to achieve high accuracy, where M is the size of the atomic basis set. We do not freeze the core electrons so that for LiH, N = 4. The atomic basis set is the triple zeta Gaussian basis set with polarization and diffuse functions, 6-311+ + G(2df,2p) [67][68][69][70], which comprises M = 49 Cartesian atomic orbitals. We compute the electronic dynamics for excited singlet states. For the 4 electrons/5 orbitals system under consideration, there are 210 singlet determinants and all are included in the basis set, meaning that all are included in equation (1). We emphasize that because the 5 molecular orbitals (MO) used to generate the 210 configurations are reoptimized at each time step, the basis of determinants that we use actually corresponds to a much larger number of configurations than if a fixed basis of MOs is used. This is analogous to the multi-configuration self-consistent field (MC-SCF) approach to time-independent electronic structure theory. As discussed below, we verify, by comparing to well-established quantum chemistry procedures, that all valence excited singlet states of LiH are accessed by an intense pump pulse acting on the GS. It takes an intense pulse because, of course, some of the excited states can only be reached by multiphoton transitions.

4
To determine the post-Born-Oppenheimer dynamics, namely for such short times that the atomic nuclei do not yet move, it is sufficient to take the Hamiltonian of the system for frozen nuclear positions. In addition, we assume that the molecular axis is aligned with the z-axis of the laser field, which is reasonable for the strong fields considered here [71]. The Hamiltonian consists of the conventional N electron Hamiltonian, including, as usual, the nuclear Coulomb repulsion terms plus any time-dependent external field that can perturb the electrons. We couple the molecule to the electric field E(t) of the pump pulse in the dipole approximation, resulting in a sum of one electron operators, The field envelope is Gaussian, centered at t 0 with a width σ and a carrier frequency ω 0 . A few cycle pulse means that ω 0 σ > 2π but not much larger. φ is the phase of the carrier wave with respect to the envelope. The vector E is the field strength whose components in the different directions define the polarization of the pulse When a pump is followed by a probe pulse, then it too has the same functional form for the electric field E (t), except that it is delayed with respect to the pump, meaning that its envelope is centered at t 0 + t, where t is the time delay. Also, the width in time of the envelope and the frequency and phase of the carrier wave can be different. As is intuitively reasonable, we find that it is advantageous to have the pump pulse as short as necessary for creating a nonstationary electronic state. That it will then be rather broad in frequency does not appear to be a disadvantage. Also, the pump field strength |E| is chosen high so as to drive as many molecules as possible to states above the threshold for ionization.
To propagate the wavefunction, we use equations of motion derived from a variational principle [10,11]. This Dirac-Frenkel variational principle has been very extensively used [72] to derive the equations of motion for distinguishable particles, e.g. nuclei. A recent example is the work of Meyer and co-workers [73]- [75]. The time evolution of the amplitudes A J (t) of the different determinants that make up the wavefunction is generated by the full Hamiltonian Then the time dependence of the orbitals can be restricted to be due only to the need to change the space spanned by the n/2 spatial functions and this is possible when (n/2) < M, where M is the size of the atomic basis set [10,11]. The equations of motion are solved by representing operators as matrices in the M-dimensional atomic basis set. The initial state for the time propagation is taken to be the GS so that any subsequent time dependence is due to the switching on of the pump field. To monitor the dynamics, the pump pulse is followed by a probe pulse. The time dependence of the spatial orbitals and of the expansion coefficients A J (t) does not necessarily cease when the field is switched off. This is because a very short pulse sets up a coherent superposition of eigenstates and so the electronic charge density will continue to move with time as is discussed below. The orbitals dynamically adjust to this changing density. Even so, one can define stationary states and the time-dependent wavefunction can be expanded as a coherent combination of these states. In other words, the stationary states and eigenstates of the electronic Hamiltonian are not single configurations but are superpositions of determinants.  Figure 1(a) Figure 1( This is also what is found [10,44] when quantum chemistry procedures that allow for configuration interaction compute the excited states of LiH. Table 1 shows the energies and transition dipoles of the excited states of LiH computed using the well-established MOLPRO electronic structure package [76]. The MOLPRO computation is at the MC-SCF (multi) level for four active electrons using the same basis set as for the dynamics. Including the GS, ten states are computed simultaneously: six 1 + , three 1 and one 1 states. The 1 state is above the vertical ionization threshold of LiH (about 0.29 au [77]) and it is used below as the state that the probe pulse pumps to. The preliminary step is to compute a time-dependent electronic wavefunction with the pump pulse as part of the Hamiltonian and with the GS as the initial state at time zero. Then we can extract the excited states energies and amplitudes and validate the procedure by comparing with the results shown in table 1.

Pumping the excited states of LiH
To compute the spectrum, the wavefunction (t), equation (1), is propagated up to a time t when the pump pulse is effectively over and then for an additional time τ during which interval the molecule is unperturbed. The autocorrelation function [78] (t)| (t + τ ) is computed and its Fourier transform with respect to τ is performed. The longer the time interval τ , the better will be the resolution of the Fourier transform, that defines the excitation spectrum. In terms of the, initially unknown, orthonormal eigenstates of the electronic Hamiltonian, the autocorrelation function is diagonal. Upon Fourier transform, it will result in a series of peaks. The width of the peaks is not inherent but determined by the range of the time interval τ . For LiH, the location of the peaks is reported in table 1 and their heights are shown in the excitation spectra that are the subject of figure 1. The autocorrelation function is computed numerically as a bilinear sum The cross correlation of the determinantal wavefunctions is computed from the time evolution of the orbitals ϕ j (x; t), cf equation (1). The autocorrelation function can also be written as an expansion in eigenstates. From such an expansion, equation (6), it is clear that the Fourier  table 1.) It is seen that at this intensity and carrier frequency, ω 0 = 0.133 au, the primary states that are populated are the lowest excited + and states that are both allowed by the one-photon selection rules and are well within the frequency span of the pulse. (b) At higher field strength (|E| = 0.05 au, peak intensity = 8.75 × 10 13 W cm −2 ), the next group of excited states (four of + symmetry and two of ) is accessed by two-photon transitions. Also seen in the spectrum is a metastable state, whose energy is above the ionization potential (IP) of LiH (0.29 au) [77]. This state cannot be accessed from the GS by a one-photon transition because of selection rules, and it is essentially absent in (a). That is why we use the singlet to probe the coherent superposition of the lowest excited + and states created by the pump pulse, see figure 9.
transform of the autocorrelation function is the excitation spectrum. It shows frequency, the locations ω n = E n /h and the intensity |c n | 2 of the different transitions in figure 1, The delta function is broadened because the interval of propagation τ is finite.
Because the laser field is not introduced as a perturbation but is part of the Hamiltonian, electronic states n that are not allowed in the first order of perturbation theory, meaning that their dipole strength from the GS is zero, cf table 1, can still be seen in the spectrum. However, this only occurs when the pump intensity is high and this possibility of multiphoton transitions is emphasized by noticing the differences between the two panels of figure 1. Figure 1(a) shows the excitation spectrum generated by a moderately intense pump pulse, (|E| = 0.025 au 2.2 × 10 13 W cm −2 ). The states accessed are primarily the excited states of LiH that can be pumped by dipole-allowed one-photon transitions from the GS. The pump pulse is short (σ = 40/ √ 2 au ≈ 0.68 fs) and polarized at an angle of π/4 to the molecular axis (along the x-z-direction), so that both + and states can be reached. The carrier frequency (ω 0 = 0.133 au) is chosen to be about midway between the lowest excited + and states so that they are about equally populated. The phase shift φ in equation (3) is set to 0. Higher-lying excited + and states that do have, see table 1, oscillator strength from the GS are only weakly excited because their excitation frequency is barely contained within the frequency span of the pump pulse, also shown in the plot. At the same carrier frequency and pulse width but with a higher laser power (|E| = 0.05 au 8.75 × 10 13 W cm −2 ) (figure 1(b)), the higher states are reached by two photon transitions. At the higher intensity, the spectrum also shows such states as that by dipole selection rules cannot be excited by one photon from the GS. At the carrier frequency of the pulse, ω 0 = 0.133 au, the singlet state whose excitation energy is 0.356 au, cf table 1, is most likely reached by a three-photon transition. Since in this computation the laser is polarized in the x z-plane, the 1 is most likely accessed by one x-polarized followed by two z-polarized photons or by other suitable combinations.
In section 4, we use a delayed probe pulse to excite to the state as a diagnostic of which states are coherently accessed by the pump pulse. Of course, the energy range in question is above the ionization threshold so any such states will autoionize. Equation (6) shows that the height of a peak in the Fourier transform with respect to τ of the autocorrelation function (t) | (t + τ ) of the wavefunction | (t) at a time t after the pulse is the weight, |c n | 2 of the nth eigenstate after the excitation is over. Figure 2 illustrates the so-called state-averaging effect, which is intrinsic for the multi-configuration time-dependent Hartree-Fock (MCTDHF) procedure. As mentioned above, the time-dependent orbitals adjust with time to the state that is to be represented. Initially the orbitals are optimized for the GS. At the higher intensities as shown in figure 1(b), there is significant depletion of the GS to a point where a population inversion is achieved. The orbitals then adjust to the excited state and the GS energy seems to rise. This correlation between the depletion of the GS and the quality of its description is shown in figure 2. This is an expanded scale of the spectrum shown in figure 1 in the vicinity of the GS peak.
Another view of the role of the pulse intensity is obtained by plotting the autocorrelation function of the GS as advocated by Heller [78] for ordinary electronic spectroscopy. In the present notation, this is the function C GS (τ ) = (0)| (τ ) . Since we invariably start the system in the ground electronic state, another form of this autocorrelation function is C GS (τ ) ≡ GS| (τ ) . This definition allows one to think of C GS (τ ) as the 'survival' probability of the GS [78,79]. As shown in figure 3, at the higher intensity that is used to compute the results shown in figure 1(b) and when the laser is polarized along the molecular axis, about 60% of the GS is depleted. Also note in figure 3 that the GS is depleted most effectively when the laser field is at maximal amplitude. This correlation is at the heart of the coherent control of the outcome that we discuss below. At the same intensity shown in figure 3 and if the laser is polarized along the zx-direction, already about 90% of the GS is depleted. Dipole selection rules determine which states can be excited. For polarization along the z-direction, the + GS can be one photon pumped to the lowest excited + and then higher + states can be reached by multiphoton transitions. The allowed transitions for x-polarization are that the GS can be pumped to the lowest state and then multiphoton transitions can reach higher states, of both + and symmetry. For polarization along the x z-direction, it is also possible to connect the GS coherently to the first excited + state and to the higher-lying first excited state, what is not possible for a pure z-polarization. When the z-polarized field is reduced by a factor of 5 the depletion is only about 10% and one reaches the perturbative limit of low depletion, about 1%, when the field is further reduced by another factor of 5. Then, for a z-polarized field only the lowest excited state is populated. As a check we verify that under the same intensity and duration, for an x-polarized field only the lowest excited state is accessed.
An important control parameter is the polarization of the laser pulse. We first discuss the role of laser polarization in the frequency domain, as shown in figure 4. The three panels in figure 4 show that up to and including intermediate intensities, 0.025 au, same as used in figures 1-3 and same carrier frequency, one-photon selection rules act as reliable propensity rules as to which states can be accessed. For a laser polarized perpendicular to the molecular axis, the main transition is to the lowest excited state with much weaker, two-photon transitions via the state to higher excited + states. For a laser polarized along the molecular bond, middle panel, the spectrum is complementary: the main transition is to the lowest excited + state with much weaker, two-photon transitions via the + state. The lower panel is for a polarization along the zx-direction at the same total intensity, meaning that the field in each direction is 1/ √ 2 of its value in the other two panels. The occupancy of the lowest excited state is indeed reduced by 1/2.
To reliably compute the Rydberg and other high-lying states of LiH [9] one needs a much larger atomic basis set and this is prohibitive in the present stage of development of the timeadapting code that is needed to compute the dynamics.
The role of the polarization of the laser pulse in controlling the dynamics of the electrons is made vivid by examination of the charge distribution in the time domain. Before the pump, the distribution in the GS is cylindrically symmetric about the bond. When the origin of coordinates is placed at the center of mass, it is near the Li atom. The lighter H atom is further away from the origin. We expect that in the GS there is an excess of negative charge on the H atom, so that the center of (negative) charge is on the z-axis, off the center of mass and in a direction that we define to be negative. Upon using a pump polarized in the z-direction, we expect to excite a + state where the charge is more evenly distributed on the two atoms. Unless the intensity is high, a significant amplitude remains on the GS. So the time-dependent wavefunction is a coherent superposition of the ground and first excited + states, meaning that the center of the charge distribution (or, equivalently, the electronic dipole moment) remains along the z-axis, . But the center of charge varies with time due to the beating between the ground, S 0 , and first excited, S 1 , + states and this variation is used here as a diagnostic of the electron dynamics Using the same intensity but a longer pulse and/or using a much more intense pulse produces a motion that is more multiply periodic because more excited states are significantly populated.
Here E exc is the excitation frequency of the lowest + state, 0.114 au as reported in table 1, and φ mol is the phase of the matrix element of the transition dipole, S 1 |N −1 N i=1 z i |S 0 , if there is one plus the phase of c * 0 c 1 [18]. The last plus sign indicates that other, higher-lying excited states can contribute to the sum and that they will do so when the pulse intensity is high and multiphoton transition probabilities are not small. It is already implied by figures 1 and 4 and it is shown explicitly in figure 5 that it is possible to operate in a regime where there is one clear beat term in equation (7).  an increase in the duration of the pulse (to a level where the GS is extensively depleted) before higher excited states make the motion of z (t) multiply periodic. The beat frequency implied by equation (7) is high because it is the excitation frequency of S 1 , 0.114 au, as expected for an interference between an excited state and the GS. The essential point is therefore not the dynamics shown in figure 5(a) but the contrast between figures 5(a) and (b). The results shown in figure 5(b) are for a laser polarized in the x z-direction whereby a coherent superposition of the first excited + and states is accessed. Due to the excitation of the state, charge is expected to move off the z-axis and in the x-direction as is clearly seen. The motion is multiply periodic but a high amplitude component with a long period of about 4 fs stands out. This is the beating between the first excited + and states at a frequency of, cf table 1, 0.152 − 0.118 = 0.034 au. The about fourfold faster periodic motion also seen in figure 5(b) is the beating with the GS.
The electronic motion set up when a coherent superposition of the first excited + and states is pumped is essentially a rotation of the electronic density about the H atom. This is best examined in terms of the two components of the dipole. Along the z-axis, the dipole (= center of charge cloud) moves back and forth from about the location of the H atom towards the Li atom. Along the x-axis, the dipole moves up and down perpendicular to the bond (which is at The steering of the electronic dynamics that is made possible by the polarization is visualized in another manner in figure 6. This shows the same data as figure 5 but as a parametric representation of the dipole in the z-x-plane. In such a projection, it is less easy to identify the periods of the motion. But this way of plotting makes it very clear that for a polarization along z the electronic dipole oscillates exclusively along the z-axis without any component along the x-axis. For a pump polarized in the zx-direction, there is clear rotational motion of the dipole about the H atom. Using a different polarization, it is also possible to excite a wave packet that rotates about the bond without moving in the z-direction. This requires a pulse polarized in the xyplane and such a pulse creates a coherent superposition of states. The results are shown in figure 7. The strength of the field |E| is 0.025 au. Two aspects of control are shown. First suppose that we maintain the field shape as in equation (3). Then the center of charge r (t) = (t)| N i=1 r i | (t) /N will oscillate with equal phase along the x-and y-directions (see upper panel of figure 7). The same is true when a phase difference, φ of π is introduced between the x-and the y-components of the pump pulse. Suppose however, we use a field that is circularly polarized. Then one can set the charge rotating in the x-y-plane as shown in the bottom panel of figure 7 where the phase difference φ between the x-and the y-components of the pump pulse is π/2. This section documented that coherent excitation of superpositions of + and states of LiH is possible with high probability and yet at such laser intensities that the motion of the electronic wave packet remains amenable to analysis. In particular, it is possible to pump a high fraction of the GS to a coherent superposition of the two lowest + and excited states. The interference of these two states results in an electronic wave packet that rotates about the H atom with a period of about 4 fs. In the next section, we discuss the probing of this motion and conclude that it is sufficient if one of the two pulses is shorter than the period of 4 fs. The other pulse, which in our simulations will be the probe, can be as long. Few-femtosecond pulses can therefore be used as a pump. An ultrafast pulse is only required as a probe.

Pump-probe spectroscopy
We use the control afforded by polarization and the selection rules that are still usefully valid as propensity rules up to quite high intensities, to probe a coherent superposition of the lowest excited + and states. Time-resolved information is provided by the delay between the pump and probe pulses.
The pump pulse is polarized in the zx-direction with pulse parameters designed to have about equal amplitudes on the + and states, as in figure 1(a). In the first series of computations a probe pulse polarized in the y-direction is applied. Figure 8 shows the location of the center of charge in the y-direction, when there is no probe and when a probe polarized in the y-direction whose envelope peaks at t = 3.5 fs after the pump peak. It is clear that the probe sets up an oscillating dipole in the y-direction where none was present before. (The probe is centered at the time t 0 + t = 5.9 fs but its envelope has a width so its effect is already felt earlier.) Analysis of the Fourier transform of the oscillating dipole shown in figure 8 verifies that the oscillations are due to interference of the 1 state with the lower-lying + and states. To make sure that the pump can effectively access the state the probe carrier is at a higher frequency, ω p = 0.21 au, so that the state (E = 0.35 au, see table 1) can be accessed by a 1 + 1 photon process where the first photon is from the pump, ω 0 = 0.13 au. Using a probe at a shorter wavelength has two further advantages. One is that one can interpret the results using one-photon selection rules. The second advantage is that The pump pulse is polarized in the x z-direction and therefore cannot give rise to a dipole in the y-direction. The very low amplitude oscillations in y that are present in the absence of the probe pulse are due to numerical noise in the integration procedure. The parameters of the pump are |E x | = |E z | = 0.025/ √ 2 au and t 0 = 100 au, ω 0 = 0.13 au. The probe pulse has the same intensity, |E y | = 0.025 au, and a higher frequency ω p = 0.21 au and is centered at t 0 + t = 245 au so that it is delayed by 145 au (about 3.5 fs) from the peak of the pump pulse. Both the pump and the probe pulse have a width σ = 0.68 fs.
we propose to use a short probe pulse and this is easier to implement with a higher carrier frequency.
The first-order effect of a y-polarized probe pulse is to transfer population from the state to the 1 state, a transition with a relatively high transition dipole ( 1 |µ y | 1 = 1.656 au). Another one-photon transition that is induced by the y-pulse is to access from the + state to a second excited state and then to 1 . This second state is about 0.25 au above the GS and the energy difference from it to 1 is about 0.1 au. A probe-induced transition from the first excited + state to the 1 state is not first order allowed.
The pump pulse excites a coherent superposition of the lowest excited + and states. The delay between the pump and probe is varied so that, since essentially not the pump but the probe can excite the 1 state, it serves as the readout for the amplitude of the state. The 1 state is autoionizing so that the excitation of this state is (relatively easily) monitored by the emitted photoelectrons.
To probe the coherent superposition of the lowest excited + and states, we use a probe polarized along xy. The spectra as a function of the delay time t between the centers of the pump and probe pulses are shown in figure 9. The pump pulse is as in figures 1-3 with a field strength of |E| = 0.025 au. The probe is short, σ = 15/ √ 2 au, with a fourfold higher power, |E| = 0.025 au, and polarized in the x y-direction. The carrier frequency of the probe is 0.12 au and, because of the high intensity, this allows strong multiphoton transitions. The x ypolarization of the probe allows reaching the target 1 state from the excited states and from the GS and excited + states with one-, two-and three-photon transitions. The time varying population of the 1 state as a function of the delay time between pump and probe is shown in the top panel of figure 9 because it is this variation that will need to be measured. The middle panel shows how the populations of the lowest excited + and states respond to the presence of the probe pulse. These change because the probe bleaches these states by up-pumping to the 1 state. The probe is intense so it also affects the population of the GS, as shown in the lowest panel of figure 9. Not shown are changes induced by the pulse in the populations of the higher excited + and states.
A simplistic analysis of why the probe is state-selective is to look only at selection rules. Below we do so and this is equivalent to using a 'white' probe without any frequency limitations. In the appendix, we carry out a model analysis of the enabling role of selection rules for a threestate model with the correct temporal envelope and carrier frequency of the probe pulse, but this analysis can only be done numerically. The results shown in figure 9 are therefore more realistic than those discussed in the appendix because all 210 determinants are kept in the computations leading to figure 9, while only three states are used in the appendix. The model used in the appendix is similar in spirit to what is known as the STIRAP excitation scheme [80,81]. The one feature brought in by the model and not evident from selection rules alone is the important role of the carrier frequency of the probe pulse in determining the influence of the time delay between pump and probe on the observed spectrum.
We start the analysis of the role of selection rules with a simple approximation for the wavefunction at a delay t after the pump pulse: This retains only the GS and the two (stationary) excited states of interest. We then consider an intense probe pulse E short enough that it can be regarded as a source of (coherent) white light. We express the polarization of the probe as E 1 = ax + by. The subscript 1 is for use below to designate the one photon selection rule. The two-photon term is E 2 = (ax + by) 2 = a 2 x 2 + 2abx y + b 2 y 2 . If only the y-component of the field takes from the first excited state to the target state and if only x y takes from the first excited to , then the transition probability to the target state is If the field is intense enough that three-photon transitions are not small, then there will be a third term describing transitions from the GS to the state. The yield of the state, | |E | | 2 , monitors the beat term. If three-photon transitions are allowed, then there is a second beat term. In figure 9, which includes all states, there is a superposition of several beat terms and so the result is not as large a modulation as shown in equation (9).  figure 8. The probe pulse is polarized in the x y-plane; its intensity is higher than in figure 8: |E x | = |E z | = 0.05/ √ 2 au; and its width is shorter than in figure 8: σ = 0.26 fs. We regard the figure as a photoelectron spectrum and the oscillation in the population of the 1 state versus delay time as reflecting the coherent superposition of states set up by the pump pulse. Middle panel: the response of the populations of the lowest excited + and states to the probe pulse. Bottom panel: for the GS. The panels show that the second pulse is not a weak probe but shifts populations in a significant way. By using a probe polarized in different directions, we can take advantage of the selection rules and so probe different aspects of the electron dynamics.
〈x〉(t) Figure 10. High amplitude steering of the charge density along the LiH bond by a strong pump pulse, |E| = 0.05 au. Shown are the profiles of two fields along the z-direction versus time. The two fields are identical, barely more than one cycle pulses, but with different phases φ = −π/3 and φ = 2π/3, as defined in equation (3). The choice of phase means that the two fields oscillate in opposite directions along the z-axis as seen in the plot. The field pulls the charge density preferentially in the direction in which its amplitude is larger or smaller, depending on the sign of the transition dipole. Physically, this means that the two opposite fields send the charge density either towards the Li nucleus or away from it.
The dipole matrix elements are available from MOLPRO and the C coefficients in equation (8) can be varied by choice of the polarization and phase of the pump, as shown in figure 11 below. The ratio a/b is determined by the polarization and the frequency spectrum of the probe. Here, the spectrum is taken to be flat. In the appendix, we do a more realistic job.

Phase control
It is not yet possible to experimentally generate a nearly one-cycle pulse in the energy range of interest for electronic spectroscopy, but progress is being made [44,45]. The problem is that it is not possible to squeeze several cycles of near-UV light into an envelope whose width in time is in the sub-femtosecond to few-femtosecond range. With just one cycle or so, the frequency spectrum of the pulse is rather broad and does not have a protruding maximum. For the present formalism, this spectrum is well defined and, in this section, it is shown that pulses for which ω 0 τ is not much over 2π allow us an even finer steering of the electronic density. This control is achieved by a judicious choice of the phase φ of the carrier wave, see equation (3).
To illustrate the control that may be possible, we contrast the results of computations at two values of the phase φ = −π/3 and φ = 2π/3. The temporal shape of the two pulses is shown in figure 10. The laser field oscillates about the z-direction. Initially the center of the charge  Figure 11. Phase control for a weaker pulse so that the control is even more selective. Examination of the spectrum shows that when the pulse is polarized in the x z-plane and for a moderate strength (|E| = 0.025 au), short (t = 0.26 fs) pulse, a phase shift of −π/3 gives a population ratio for / + of 2.7, while 2π/3 gives a ratio of 1.4. density is towards the H nucleus. Depending on the sign of the laser field at the time when its amplitude is maximal, it sends the electronic density even further away from the Li nucleus, which is roughly at z = 0, or, as in figure 3, the field sends the density towards the Li. As is to be expected for a field in the z-direction, x (t) = 0 throughout. Figure 11 shows a different manifestation of the phase control and it shows this for a weaker pulse so that the control is even more selective. Examination of the spectrum shows that when the pulse is polarized in the x z-plane and for a moderate strength (|E| = 0.025 au), short (τ = 0.26 fs) pulse, a phase shift of −π/3 gives a population ratio for / + of 2.7, while 2π/3 gives a ratio of 1.4.
Finally, note that figure 7 also shows phase control, but in this case it is different phases for different field components. By varying the relative phase of the x-and y-components of the field, one can change the nature of the electronic dynamics from a linear to a circular motion.

Concluding remarks
Detailed computational results for the response of the four-electron LiH molecule to a few-cycle intense UV pulse are provided by a time-dependent multiconfiguration Hartree-Fock method using a large basis set that is, moreover, time-adaptive. The pulse intensity, duration, polarization and phase of carrier frequency can all be tuned to steer the motion of the electrons. A physical understanding of the effects of the pump is possible, because even at higher intensities, selection rules provide useful guidance as to which states are accessed and to what extent. Furthermore, unless the pulse is ultraintense, the dynamics of the electronic wave packet has very few dominant characteristic frequencies so that its principal components of motion are readily visualized. For example, it is shown possible to direct the valence electrons to move along the Li-H bond or, by using a different pump polarization, normal to it. When the laser is polarized other than along the bond, the result is a rotation of the charge density. Shifting the phase of the carrier wave drives the electrons towards the Li nucleus or away from it. The resulting coherent dynamics induced by the pump pulse can be usefully probed by a delayed probe pulse. By means of probe pulse parameters such as polarization that are different from the pump pulse, detailed access to the more intimate aspects of the electron dynamics is possible.