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High-order harmonic generation (HHG) is a nonlinear nonperturbative process in ultrashort intense laser-matter interaction. It is the main source of coherent attosecond (1 as = 10^{−18} s) laser pulses to investigate ultrafast electron dynamics. HHG has become an important table-top source covering a spectral range from infrared to extreme ultraviolet (XUV). One way to extend the cutoff energy of HHG is to increase the intensity of the laser pulses. A consequence of HHG in such intense short laser fields is the characteristic nonadiabatic red and blue shifts of the spectrum, which are reviewed in the present work. An example of this nonperturbative light-matter interaction is presented for the one-electron nonsymmetric molecular ion HeH^{2+}, as molecular systems allow for the study of the laser-molecule orientation dependence of such new effects including a four-step model of MHOHG (Molecular High-order Harmonic Generation).

Atomic high-order harmonic generation (HHG) was discovered in 1980s [^{−15} s) laser fields to coherent high-energy attosecond (1 as = 10^{−18} s) pulse trains or an isolated attosecond laser pulse [^{14} W/cm^{2}. The general feature of atomic HHG which also appears in molecular high-order harmonic generation (MHOHG) is a fast decay of the lower order harmonics, then a long plateau, and a short cutoff [_{k}_{p}_{e}_{k}_{p}

In MHOHG, the ionized electron can recombine with neighboring ions in the same molecule leading to a much larger cutoff energy depending on the internuclear distance

From Equation (1), one has to increase the ionization potential _{p}

Earlier work has studied the blue shift of HHG. It is the result of mainly two mechanisms. One is the propagation effect of a fundamental laser pulse in an ionized medium [

Free electrons created during the laser pulse induce a change of the refractive index [_{1} induced by the shift of the fundamental field and the other shift _{2} caused by the generated harmonics propagating in the medium. _{1} is proportional _{2} is propational to ^{3}.

Rapid change of a laser pulse shape will lead to a chirp of HHG. The electronic dipole response to the external laser field is nonadiabatic. This effect is intuitive in the above three-step model. As shown in

Illustration of the positive (solid line) and negative (dotted line) chirp of High-order harmonic generation (HHG) by the rapid change of the electric field

From the discussion in the above section, the depletion of the ground state on the RLE will prevent the HHG emission on the FLE. It is therefore hard to observe a noticeable red shift of HHG directly in such a case. However, enhanced excitation (EE) in resonant systems will change the above three-step model to a four-step model. We take the resonant nonsymmetric system HeH^{2+} as an example to explore the nonadiabatic red shift in MHOHG [

We study numerically the following time-dependent Schrödinger equation (TDSE):
_{0} is the field-free Hamiltionian [

where the pulse shape

The molecular ion HeH^{2+} has a large ionization potential (_{p}^{2+} are obtained by numerically solving the TDSE. For the computational details of this one-electron system, we refer to reference [_{q}

The MHOHG spectrum of HeH^{2+} in a laser field at wavelength 400 nm and intensity ^{15} W/cm^{2} with a duration of 15 optical cycles is shown in

We take the resonant EE of long lifetime excited states into account explicitly and generalize the three-step model to a four-step model as presented in

Resonant MHOHG described by the above four-step model have novel features [_{p}

To understand the red shift in detail, we explore the role of EE and enhanced ionization (EI) [

(^{2+} in linearly polarized laser fields; (^{15} W/cm^{2}. The wavelength is 400 nm. The pulse has a cos^{2}(

Four-step model for MHOHG in the resonant molecular ion system HeH^{2+}. Step (1): the laser field pumps the electron from the ground _{1} and _{2} is included [^{+}_{1} and ^{+}_{2} refer to the dressed energies with the field along +^{-}_{1} and ^{-}_{2} refer to the dressed energies with the field along −

Population of the first excited state

The red shift in MHOHG is a nonadiabatic process as it depends on the pulse shape

(

From the above discussion, the resonance of

An obvious feature in

(

We have reviewed in the presented work the nonadiabatic spectral shift of atomic HHG and MHOHG in intense laser fields. The blue shift in HHG has been well studied in theories and confirmed by experiments. In the present work, red shifts of harmonic generation are described and predicted theoretically by a four-step model in resonant systems, thus confirming that the principle of nonadiabatic red shifts is general at high intensities. Since resonant HHG in laser plasma interaction systems [

We thank RQCHP and Compute Canada for access to massively parallel computer clusters and useful discussion with T. Ozaki at INRS on laser-plasma interactions.

The authors declare no conflict of interest.

^{+}-h

_{2}

^{+}system

^{2+}in two-color laser fields

^{2+}and its decay mechanism