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arXiv:1807.02525 · Physical Review A

Floquet-theoretical formulation and analysis of high-order harmonic generation in solids

Tatsuhiko N. Ikeda, Koki Chinzei, Hirokazu Tsunetsugu

Published Dec 26, 2018• 33 citations
arXivHTMLPDF

Abstract

By using the Floquet eigenstates, we derive a formula to calculate the high-order harmonic components of the electric current (HHC) in the setup where a monochromatic laser field is turned on at some time. On the basis of this formulation, we study the HHC spectrum of electrons on a one-dimensional chain with a staggered potential to study the effect of multiple sites in the unit cell such as the systems with charge density wave order. With the help of the solution for the Floquet eigenstates, we analytically show that two plateaus of different origins emerge in the HHC spectrum. The widths of these plateaus are both proportional to the field amplitude, but inversely proportional to the laser frequency and its square, respectively. We also show numerically that multistep plateaus appear when both the field amplitude and the staggered potential are strong.

DOI
10.1103/physreva.98.063426

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By using the Floquet eigenstates, we derive a formula to calculate the high-order harmonic components of the electric current (HHC) in the setup where a monochromatic laser field is turned on at some time. On the basis of this formulation, we study the HHC spectrum of electrons on a one-dimensional chain with a staggered potential to study the effect of multiple sites in the unit cell such as the systems with charge density wave order. With the help of the solution for the Floquet eigenstates, we analytically show that two plateaus of different origins emerge in the HHC spectrum. The widths of these plateaus are both proportional to the field amplitude, but inversely proportional to the laser frequency and its square, respectively. We also show numerically that multistep plateaus appear when both the field amplitude and the staggered potential are strong.
The spectra produced by high-order harmonic generation (HHG) typically exhibit well-defined peaks at odd integers times the laser frequency. However, in recent investigations of HHG from correlated materials, spectra exhibit signals at noninteger harmonics which do not conform to the well-known symmetry-based selection rules for HHG spectra. Here, we use the Fermi-Hubbard model to study HHG from a linear chain of atoms. This model allows us to study both the correlated and uncorrelated phases through a specification of the amount of onsite electron-electron repulsion. The presence of signal at noninteger harmonics can be interpreted as originating from the population of multiple Floquet states. We show how this coupling to different Floquet states depends on the characteristics of the driving pulse and the strength of the electron-electron interaction in the system.
This paper presents an analytical study of high-order harmonic generation (HHG) by quantum systems possessing permanent dipole moments (PDMs), such as atoms/molecules placed in a static electric field or polar molecules. HHG by conventional, non-PDM systems typically demonstrated a plateau in the envelope of harmonics followed by a steep fall in harmonic intensities when the harmonic number exceeds a certain threshold. This paper shows that the HHG by PDM systems results in a significant extension of the plateau to higher frequencies and a slower decline of intensities at frequencies higher than the end of the plateau . Moreover, there occurs a substantial growth of the total, summed intensity of all components of the scattering spectrum . Other physically interesting distinctive features of the HHG by PDM systems are: the presence of both odd and even harmonics of the laser frequency in the scattering spectrum; a triplet structure of both odd and even harmonics; and the appearance of a component at the doubled Rabi frequency (i.e. at a frequency much lower than the laser frequency). The results are obtained by using the adiabatic basis for the analysis of a multi-quantum resonance in a two-level system possessing PDMs.
We study the high harmonic generation (HHG) in Mott insulators using Floquet dynamical mean-field theory (DMFT). We show that the main origin of the HHG in Mott insulators is the doublon-holon recombination, and that the character of the HHG spectrum differs depending on the field strength. In the weaker-field regime, the HHG spectrum shows a single plateau as in the HHG from gases, and its cut-off energy εcutε_{\rm cut}εcut​ scales linearly with the field strength E0E_0E0​ as εcut=Δgap+αE0ε_{\rm cut}=Δ_{\rm gap} + αE_0εcut​=Δgap​+αE0​, where ΔgapΔ_{\rm gap}Δgap​ is the Mott gap. On the other hand, in the stronger-field regime, multiple plateaus emerge and the mmm-th cut-off scales as εcut,m=U+mE0ε_{\rm cut,m}=U + m E_0εcut,m​=U+mE0​. We show that this difference originates from the different dynamics of the doublons and holons in the weak- and strong-field regimes. We also comment on the similarities and differences between HHG from Mott insulators and from semiconductors. This proceedings paper complements our recent work, Phys. Rev. Lett. 121, 057405 (2018), with additional results and analyses.
Combining Floquet theory with the hierarchy of correlations (a method based on the formal expansion into inverse powers of the coordination number), we study the nonequilibrium dynamics of the Mott insulator state in the Fermi-Hubbard model under the influence of a harmonically oscillating electric field representing the pump laser. After deriving the associated Floquet exponents, we consider multiphoton resonances leading to the generation of higher harmonics. For weak electric field strengths, the strongest signal is obtained for the third harmonic when the driving frequency equals one-third of the Mott gap (because even harmonics are absent due to inversion symmetry). For stronger electric field strengths and lower driving frequencies, the higher-harmonic spectrum can also exhibit plateaulike structures extending over several harmonics (i.e., Floquet channels) below the resonance peak, which is consistent with previous findings.
Recent developments on intense laser sources is opening a new field of optical sciences. An intense coherent light beam strongly interacting with the matter causes a coherent motion of a particle, forming a strongly dressed excited particle. A photon emission from this dressed excited particle is a strong nonlinear process causing high-harmonic generation (HHG), where the perturbation analysis is broken down. In this work, we study a coherent photon emission from a strongly dressed excited atom in terms of complex spectral analysis in the extended Floquet-Hilbert-space. We have obtained the eigenstates of the total Hamiltonian with use of Feshbach-Brilloiun-Wigner projection method. In this extended space, the eigenstates of the total Hamiltonian consisting of the radiation field and the atom system have complex eigenvalues whose imaginary part represents the decay rate. Time evolution of the system is represented by the complex eigenvector expansion so that the correlation dynamics between the photon and the atom is fully taken into account. The HHG is interpreted as the irreversible spontaneous photon emission due to the resonance singularity in terms of the multiple Floquet states that are generated by periodic external field. We have found that the interference between the emitted photons over the different Floquet states causes spatial pulse emission correlated with the decay process of the excited atom.
We present a theory of harmonic generation by a strongly driven atom within the framework of non-relativistic QED. The atom is assumed to possess a finite number N of levels and ionization is neglected. This atom is dressed by the driving field using a unitary time-dependent transformation. Within the quasistatic approximation this transformation is shown to lead from the bare to the Floquet atom. The interaction of the Floquet atom with the vacuum modes is derived and it is shown to give rise both to the high-order harmonic and to the hyper-Raman spectrum. In the case N=2 the spectrum of the light scattered into the vacuum modes can be evaluated analytically and it is shown to consist of two parts. Neglecting interference between these two parts, the first of them is the high-order harmonic spectrum displaying peaks at the odd harmonics of the driving field. The second part is the hyper-Raman spectrum which has finite bandwidth and which displays peaks at the even harmonics of the laser frequency, centred at the Stark-shifted natural atomic frequency. These two spectra partially overlap and they are discussed in various ranges of intensity of the driving field, both in the minimal coupling and in the multipolar scheme. The results support recent suggestions that the presence of the plateau is related to interference between the two spectra. The dependence of the high-order harmonic spectrum from the driving field intensity is found to be related to that of the hyper-Raman spectrum in such a way that when the former increases the latter decreases and vice versa. It is argued that this might help to explain the non-visibility of the hyper-Raman lines when the high-order harmonics are visible.