## Abstract

The response of a hydrogen atom submitted to a multiharmonic laser field is studied. The relative phases of the harmonic components are critical both for atomic ionization and high-order harmonic generation. In the phase-locked case under adequate interference conditions, the inhibition of ionization is followed by an increase of the intensities of the generated harmonics.

©2006 Optical Society of America

## 1. Introduction

With the introduction of chirped pulse amplification (CPA) lasers in many laboratories across the world, one can state that high-order harmonic generation has become a standard technique to obtain ultraviolet and even soft X-ray coherent radiation.

Moreover, the generated radiation appears in most of the cases as a long plateau with many harmonic components [1, 2, 3, 4]. Having such comb of equally spaced frequencies, the next question to arise is whether or not these harmonics are phase-locked. That is important because, as first noted by Farkas and Toth [5] a decade ago, attosecond pulses can be generated using a set of phase-locked harmonics. In fact, there has been an explosion of research on attosecond pulses in recent years, and in most of the cases those attosecond pulses are made with harmonics of an intense infrared laser pulse [6, 7].

The existence of attosecond pulses is intimately related to the phase locking of the frequencies which compose the field. As it is well-known, in the case of random phases no attosecond pulses are generated. The measurement of single attosecond pulses or attosecond pulse trains has been a very difficult task until recently, since such pulses are arguably the shortest phenomena which can be generated nowadays in a laboratory. Several non-linear techniques have been proposed so far, most of them based on the ionization dynamical effects induced by the multi-harmonic field [8, 9, 10, 11, 12, 13]. One of these techniques is related to a novel type of ultrafast coherent population trapping which appears when the phases of the harmonics are locked and their frequencies are adequately tuned to a resonance of the irradiated system [14].

The coupling of CPA technique with optical parametric amplification has given raise to optical parametric chirped pulse amplification (OPCPA) [15, 16, 17], which can provide us very intense ultrashort pulses over a very broad range of frequencies, making possible the generation of high harmonics of almost any frequency in the visible and near-infrared domains. This will permit the realization of spectral studies involving high harmonics, which so far have been limited by the range of tunability of CPA lasers, mainly based on titanium:sapphire crystals.

In this paper we explore the effects of multiharmonic phase-locked fields on the behavior of a simple atomic system by means of numerical simulations, gaining insight in the processes which take place during the interaction. The effects of the changes in frequencies and phases are discussed over a range of parameters. The paper is organized as follows: Section 2 presents the numerical model for the ionization of hydrogen under a multi-frequency field. Section 3 shows the results obtained with different relevant parameters for the ionization yield. Section 4 analyzes the spectrum of the radiation generated by the field - atom coupling. Finally, our conclusions are summarized in 5.

## 2. Model

Our task consists on solving the time-dependent three-dimensional Schrödinger equation (TDSE) in the dipole approximation for a hydrogen atom

where the Hamiltonian is

We assume the laser field *E*(*t*) is linearly polarized along *z* axis. The laser field is a sum of ten odd harmonics of a fundamental frequency *ω _{L}*

In all the simulations, the pulse envelope has a flat shape during ten or twenty cycles of the fundamental frequency *ω _{L}*. For simplicity, we take the partial harmonic amplitudes

*E*to be equal, i.e.,

_{q}*E*=

_{q}*E*

_{1}=

*E*/√10 for all values of

_{L}*q*. The average amplitude of the total field is ${E}_{L}=\sqrt{\Sigma {E}_{q}^{2}}=0.04$ atomic units. This corresponds to individual irradiances for each harmonic component close to 5.5×10

^{12}W/cm

^{2}.

Regarding the phases φ* _{q}* of the different harmonics, we want to find the behavior of the atom when all the harmonics are phase-locked and when the relative phases are random. In the first case, choosing for instance φ

*=0 for all values of*

_{q}*q*, the laser field may be regarded as a train of attosecond pulses, as it can be observed in Fig. 1 (full blue line). On the other hand, when the relative phases are random the high peaks disappear and the total field is much more irregular, as seen in Fig. 1 (dashed red line).

We have solved the TDSE by expanding the wave function in the spherical harmonics basis and solving the set of ordinary differential equations by means of a Crank-Nicolson algorithm [18, 19, 20]. The numerical parameters (space step, angular and spatial grids) have been chosen in order to ensure us to obtain the energies of the bound eigenstates of the stationary wavefunction up to a radial number *n _{r}*=15 with a difference better than 10

^{-4}a.u. with respect to their theoretical values. Regarding the dipole moments of the transitions between the lower bound states, the difference with the theoretical values is around the 2 percent.

## 3. Ionization spectrum

We have performed simulations of the behavior of the atom submitted to a phase-locked pulse as described above, varying the frequency of the fundamental harmonic from 0.01 a.u. to 0.1 a.u., keeping the average amplitude of the field constant, *E _{L}*=0.04 a.u. This is an interesting range of parameters which can show a great variety of results because, in the case of a monochromatic field, both tunnelling and multiphoton ionization can occur. Moreover, there are lots of possible resonances between the different field harmonics and the transitions among bound and continuum states of the atom.

The results of these series of simulations can be shown in Fig. 2, where we have depicted the population which remains in the ground state after twenty cycles for the different values of frequencies. We show also the population remaining in the bound states up to *n _{r}*=10, which can be considered the total bound population after that time.

At first glance, we can notice that the ionization does not follow a clear dependence with the frequency. Instead, there are many peaks corresponding to high remaining bound population and dips of high ionization along the whole frequency interval. As we have mentioned before, this fact is not completely surprising due to the multiple resonances with transitions between the bound states. Such resonances, well-known in the case of a monochromatic field, can explain the dips of high ionization. For instance, for a fundamental frequency *ω _{L}*=0.089 a.u. there is a distinct dip which can be explained as the effect of the 5-photon resonance between the bound states 1s and 3p (let us remind that 5

*ω*is one of the components of the total field).

_{L}The low-ionization peaks are more difficult to explain if we do not take into account the interactions of the different harmonic components which can give rise to interference phenomena, coupling states of the same parity via continuum states to form some kind of dark states which trap part of the population and, as a final result, yielding a ionization much lower than expected [14].

To gain insight into these resonances and interferences, in Table 1 we show the most relevant frequencies for which these couplings appear in hydrogen.

Resonances are well-known phenomena which happen when the photon energy approaches the energy difference between two states whose angular momenta verify Δ*l*=±1. This occurs, for instance, between the ground state 1s and states *n _{p}*, due to the sharp growth of the transition probability between those states. Once in the excited state, the population is much more easily ionized to the continuum states. In Fig. 2, we can identify some of the resonances summarized in Table 1, for instance at frequencies ω

*=0.042, 0.052, 0.089 a.u.*

_{L}The ionization for a resonant frequency will happen faster when all harmonics are phase-locked than when their phases are random, as stated in references [8, 10, 13]. This is due to the coupling of the transition probabilities associated to the different ionization paths, which enhance the ionization yield when this coupling is constructive, what happens when the different harmonics are phase-locked.

Interferences are strong couplings of two states of the same angular momentum via a common upper level, usually in the continuum, which is resonant with both of them. They occur when the two coupled states are separated by an energy equivalent to an even number of photons, *n*, which guarantees the existence of such common resonant upper level reachable from the lower one with *m* photons, and from the upper one with *m*-*n*. Both *m* and *m*-*n* are odd numbers and both transitions are allowed in the dipole approximation.

As it was shown in ref. [14], concurrent resonant interference phenomena generate population trapping in a state which is a linear combination of several pairs of lower states in the case of a multiharmonic laser field, but only if the phases of the different frequency components are locked. In that paper, the ten-photon interference between states 1*s* and 2*s* which appears at frequency ω* _{L}*=0.0375 a.u. was studied. In Fig. 2, we can also clearly see the population trapping at frequencies 0.031 a.u., 0.047 a.u., 0.063 a.u., or 0.094 a.u., corresponding to even-photon transitions between those same states, as shown in Table 1.

## 4. Harmonic spectra

In recent times, there has been a lot of interest in the radiation emitted by atomic systems when irradiated by multifrequency fields. In the case of bichromatic fields, different effects of the relative phase of both frequencies have been reported [21, 22, 23]. The main interest in the phases (carrier-envelope phase and relative phases) deals with the generation of attosecond pulses or trains of pulses [24, 25, 26, 27]. Here we have checked whether the phases of the different harmonics are only relevant for the ionization dynamics [8, 14] or they have also an effect on the secondary emitted radiation. Carrier-envelope does not play a role here since we simulate relatively long pulses with a flat top envelope.

In Fig. 3 we show the radiation spectrum generated by the hydrogen atom under even-interference conditions (a) and off-resonance conditions (b). As it is expected, the most intense part of the harmonic spectrum corresponds to the harmonics 1 to 19, which are those of the incident field. Regarding the new generated frequencies, we can observe odd harmonics from 21st to roughly 41st order with the parameters used in the simulations. In the case of interference conditions middle harmonics from 20th to 30th order are noticeably more intense (one order of magnitude) when the incident pulse is phase-locked than when the phases are random. For odd-resonance conditions and non-resonant conditions, the intensities of the secondary harmonics do not depend on the relative phases.

It is clear from Fig. 3 and from the animation in Fig. 4, in which the evolution of the generated harmonic intensities is shown as the first frequency is increased from ω* _{L}*=0.03 a.u. (

*h̄ω*=0.82 eV, λ

_{L}*=1520 nm) to ω*

_{L}*=0.04 a.u. (*

_{L}*h̄ω*=1.04 eV, λ

_{L}*=1140 nm), that even-interference conditions (*

_{L}*ω*=0.0313 a.u. and

_{L}*ω*=0.0375 a.u. in this interval) trigger the rising of secondary harmonics, which fade again not immediately but for frequencies a bit above interference. This rising is accompanied by a change in the intensity profile of primary harmonics (from 1st to 19th), which show a characteristic resonance shape instead of a flat top one as the incident field has.

_{L}As the increase of the middle harmonic intensities in the phase-locked case is strongly dependent on the frequency, it is clear that it is a purely coherent effect related to the inhibition of ionization explained previously. It cannot have a classical explanation as the well-known three-step model based upon classical trajectories and rescattering of the ionized electrons, i.e., this effect does not depend on the difference between the temporal shape of the phase-locked and random phase pulses, which obviously would generate different classical trajectories of the ionized wavepackets, but independently of the main frequency of the pulse. Here, the interference condition as well as the locking of the phases are necessary for the increase of the harmonic intensities. This way, bound-bound resonant transitions become a key factor in the amplification of the harmonics, as it was proposed in a different context in [28].

## 5. Conclusions

We have performed simulations on the ionization of a hydrogen atom with a multifrequency laser field which can be obtained in high harmonic generation. The simulations scan a wide range of fundamental frequencies, and they compare the phase-locked case with many cases of arbitrary relative phases.

For certain frequencies, clearly identified, the phase-locked case results in a dramatic reduction of the ionization. This fact could be used for future detection of the relative phases of harmonics and, therefore, for future detectors with attosecond resolution. The physical effect underlying this reduction of the ionization is atomic interference, closely related to coherent population trapping. Here, instead of a single three-level system, the effect of several coupled three-level systems adds coherently in the phase-locked case.

We have also shown that the locking of the phases of the different harmonics of the incident field not only affects the ionization yield, but it has also a great effect on the secondary harmonics emitted by the atom: the intensities of middle harmonics are noticeably enhanced under interference conditions when the incident field is phase-locked. This enhancement has a broad-band nature and may be used to increase the efficiency of the generation of coherent light in the ultraviolet region.

Regarding the experimental feasibility of the effects studied in this paper, we have to notice that the generation of a multiharmonic field like the one used in our simulations is not straight-forward. The harmonic spectrum generated in gases driven by strong laser fields contains a plateau of middle- and high-order harmonics with roughly the same intensity whose phases can be locked. However, this does not naturally occur for low-order harmonics from 1st to 19th. The recent advances in ultrashort pulse shaping and coherent control make us optimistic about the possibility of manipulating individual harmonics so as to prepare a tailor-made pulse similar to the one we use in our simulations in the near future. Moreover, an adequate election of the atomic or molecular system and of the frequencies of the harmonic comb can permit us to find conditions for which a more realistic field, made with only plateau harmonics of roughly the same amplitude and locked phases, generates results similar to the effects reported here.

## Acknowledgments

This work has been partially supported by the Spanish Ministerio de Ciencia y Tecnologia (FEDER funds, grant FIS2005-01351) and by the Junta de Castilla y León (grant SA026 A05).

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