High-harmonic generation

Atoms respond highly nonlinearly to strong laser fields, emitting high harmonics (i.e. coherent light whose frequency is a multiple of the frequency of the external field) up to the 300-th order [1]. These emission spectra present a long plateau, with harmonics of roughly the same intensities, and a sharp cut-off proportional to the intensity of the external field. These features are best reproduced by the so-called ``three-step-model'' [2], in which an electron leaves the atom through tunneling, propagates in the continuum and is driven back by the laser field towards the atomic core, recombining with the ground state.

My work on the subject addresses four main issues:

Physical mechanisms involved in high-harmonic generation:

A first step towards the control of HHG is a better understanding of this phenomenon, as well as of the limitations of the three-step model. With that purpose, we performed a comparison between this model and other existing approaches, such as the fully numerical solution of the time-dependent Schrödinger equation (TDSE) or a strongly driven two-level atom. These studies are:

Computation of the main contributions to HHG within a field cycle:

The three-step model has clear predictions concerning these contributions: they should occur at the times the electron returns to the atomic core. We checked the three-step model against the TDSE using a windowed Fourier (Gabor) transform, and obtained strikingly similar temporal profiles, for monochromatic and bichromatic driving fields.

Influence of excited bound states:

The three-step model assumes that the internal structure of the atom does not contribute to HHG, which is an oversimplification. We use a time-dependent projection method to investigate for which spectral region this assumption holds, using the Fourier and Gabor transforms and model-atoms with several bound states. We conclude that the three-step model describes high-harmonics well, but fails for the low-plateau harmonics. Thus, in order to control high harmonics, one must influence the propagation of the electron in the continuum.

Control of high-harmonic generation:

Using additional fields and/or confining potentials, one can in principle distort the motion of the electron in the continuum, altering its kinetic energy upon return. As a consequence, there are alterations in the resulting harmonic spectra, such as the enhancement or suppression of particular groups of harmonics and the increase of the cutoff energy. We investigated several schemes for controlling HHG, namely:

Bichromatic fields:

This is a scheme which is easily realized experimentally and includes several working parameters, namely the intensity and frequency ratio between both driving waves and their relative phase. With the objective of understanding how to control harmonics in this case, we investigated a cutoff law and the range of validity of the three-step model. We show that the relevance of a group of harmonics is determined by:

The field strength at the time the electron left the atom: Since the "first step" is a tunneling process, a strong field at the emission time means strong harmonics.
The excursion time of the electron in the continuum: shorter excursion times mean little wave-packet spreading for the electron, and, consequently, strong harmonics.
The quantum interference between several possible paths for the returning electron:if this interference is constructive, harmonics resulting from the corresponding trajectories can be enhanced.

We were also able to:

Extract all return times predicted by the three-step model from the TDSE solution using the Gabor transform, showing its validity for bichromatic driving fields.
Show the absence of a simple cutoff law in the bichromatic case.
Enhance/suppress groups of harmonics in particular cases (e.g., a w-2w field).
Explain effects recently observed in experiments on HHG by a strong field and its second harmonic, for which the relative phase between the two driving waves was controlled [3]. These effects are a periodic modulation for the harmonic intensities as functions of the relative phase, which occurs near the cutoff, and an offset phase shift in this modulation for neighboring harmonics. I.e., if, for the n-th harmonic, the modulation presents a maximum for a particular phase, for the n+1-th harmonic this maximum is slightly shifted. In particular, we show that the latter effect is inherent to a particular set of electron trajectories, which can be isolated according to their phase dependence using bichromatic fields already at the single-atom-response level. For monochromatic fields, this is only possible using propagation effects [4].

Additional high-frequency field:

Using a bichromatic field, one can enhance or suppress groups of harmonics very efficiently. However, it is not always possible to effectively extend the cutoff energy. For instance, for a field composed of a wave of frequency w and its second harmonic, there is a double plateau, with two distinct cutoffs. The upper cutoff harmonics, whose energies are up to 30% higher than the monochromatic cutoff, are weak, and correspond to a set of electron trajectories for which tunneling is not pronounced. The lower cutoff harmonics are strong, and correspond to electron trajectories for which appreciable tunneling ionization takes place. Thus, one needs an additional pathway for the electron to reach the continuum in the upper-cutoff case, making both sets of harmonics equally strong. This is provided by an additional driving wave, whose amplitude is much smaller and whose frequency is much higher than those of the bichromatic field. This scheme is most efficient when all three frequencies are commensurate.

Additional confining potential:

We show that placing an atom in a parabolic potential is a very efficient way to increase the cutoff energy in more than 50%. This can be done by choosing the parameters of the confining potential appropriately, with respect to the frequency of the driving field and the excursion amplitude of the electron. An advantage of this scheme over those using static electric and magnetic fields is that there is no increase in irreversible ionization in our case, and therefore no loss in the harmonic intensities. A possible experimental realization of this scheme are solid-state devices, such as an impurity in a quantum dot.

Resonant enhancements of high-harmonic generation:

A very intriguing feature of above-threshold ionization or high-order harmonic generation concerns the dependence of these phenomena on the intensity of the driving field. The photo-electron or high-order harmonic peaks, as functions of the driving-field intensity, present resonance-like enhancements, such that a variation of a few percent in the external-field strength may originate enhancements of up to one order of magnitude in the spectra. These enhancements have been observed experimentally by several groups [5], and occur in the low-plateau frequency region, where, in principle, both the atomic binding potential and the external laser field may influence the spectra.

To the present date, there are two apparently conflicting physical pictures explaining these features, namely resonances involving ponderomotively upshifted Rydberg states or channel-closing effects. Our results, obtained from the one-dimensional TDSE for various model-atoms,show that both pictures are complementary aspects of a more complete physical description. In both cases, the enhancements are due to resonant effects, the resonance and the channel-closing condition being equivalent in this context. The resonances occur, when an appropriate highly excited state or, in the absence of such a state, the continuum threshold are ponderomotively upshifted so that they become multiphoton resonant with the ground state.

High-harmonic generation and periodic level crossings

A physical picture which has been widely used until the late nineties to describe HHG is a strongly driven two-level atom. This picture has been abandoned, for it has proven inadequate to describe high harmonic generation in atoms, and therefore the physical mechanism responsible for HHG in this case was not completely understood. Since nowadays there are many other systems which may survive the required intensity range, and for which this picture may be applicable, as for instance quantum wells, it is of interest to understand HHG by a two-level atom more thoroughly. This is the main objective of this project.

We have shown that HHG in a two-level atom can be described by a three-step process analogous to that occurring for atoms in strong fields. The main difference is that, in the two-level atom case, the three-steps do not involve the ground state and the continuum, but the adiabatic states which come from the diagonalization of the two-level Hamiltonian. In this case, periodic level crossings play a very important role. Hints that a one-to-one correspondence between both physical pictures might exist have been provided in the literature [6]. We go however beyond such studies, giving evidence that a three-step mechanism exists in the two-level atom case and analysing its features in detail.

We also provide examples of how these level crossings, and therefore the harmonic yield, can be controlled using additional fields. Finally, using scaling laws, we establish sharp criteria for the invariance of the physical quantities involved, such that our results can be extended to a broader parameter range, as for instance those characteristic of solid-state systems.

The above studies are mainly performed numerically, and use spectral and time-frequency analysis.

My contributions to the the subject
My collaborators (in alphabetical order): W. Becker, M. Dörr, M.L. Du, R. Kopold, D.B. Milosevic, G.G. Paulus, J.M. Rost, I. Rotter, W. Sandner.

References:

[1] For a review, consult P. Salières, A. L'Huillier, Ph. Antoine and M. Lewenstein, Adv. At, Mot. Phys. 41, 83 (1999)

[2] M. Yu. Kuchiev, JETP Lett. 45 (7), 404 (1987); K. C. Kulander, K. J.Schafer , and J. L. Krause, in: Proceedings of ``Super Intense Laser-Atom Physics IV", B. Piraux et al.(Plenum, New York, 1993); P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993); M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L'Huillier and P. B. Corkum, Phys. Rev. A 49, 2117 (1994); W. Becker, S. Long and J. K. McIver, Phys. Rev. A 41, 4112 (1990) and 50, 1540 (1994)

[3] U. Andiel, G. D. Tsakiris, E. Cormier, and K. Witte, Europhys. Lett. 47, 42 (1999).

[4] M. Bellini, C. Lynga , A. Tozzi, M. B. Gaarde, T. W. Hänsch, A. L'Huillier, and C.-G. Wahlström, Phys. Rev. Lett. 81, 297 (1998); M. B. Gaarde, F. Salin, E. Constant, Ph. Balcou, K. J. Schafer, K. C. Kulander, and A. L'Huillier, Phys. Rev. A 59, 1367 (1999).

[5] M. P. Hertlein, P. H. Bucksbaum, and H. G. Muller, J. Phys. B 30, L197 (1997); P. Hansch, M. A. Walker, and L. D. Van Woerkom, Phys. Rev. A 55, R2535 (1995); M. J. Nandor, M. A. Walker, L. D. Van Woerkom, and H. G. Muller, Phys. Rev. A 60, R1771 (1999). E. S. Toma, Ph. Antoine, A. de Bohan, and H. G. Muller, J. Phys. B 32, 5843 (1999); G. G. Paulus, F. Grasbon, H. Walther, R. Kopold, and W. Becker, Phys. Rev. A 64, 021401(R) (2001)

[6] F. I. Gauthey, B. M. Garraway, and P. L. Knight, Phys. Rev. A 56, 3093 (1997); M. Yu Ivanov, P. Hawrylak, P. Haljan, T. Fortier and P. B. Corkum, unpublished.