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arXiv:2309.01021 · Physical review. B./Physical review. B

Higher-harmonic generation in the driven Mott-Hubbard model

Friedemann Queißer, Ralf Schützhold

Published May 3, 2024• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.205110

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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.
To study excitonic effects on high-harmonic generation (HHG) in Mott insulators, we investigate pumped nonequilibrium dynamics in the one-dimensional extended Hubbard model. By employing time-dependent calculations based on the exact diagonalization and infinite time-evolving block decimation methods, we find the strong enhancement of the HHG intensity around the exciton energy. The subcycle analysis in the sub-Mott-gap regime shows that the intensity region of the time-resolved spectrum around the exciton energy splits into two levels and oscillates following the driving electric field. This excitonic dynamics is qualitatively different from the dynamics of free doublon and holon but favorably contributes to HHG in the Mott insulator.
Using Floquet dynamical mean-field theory, we study the high-harmonic generation in the time-periodic steady states of wide-gap Mott insulators under ac driving. In the strong-field regime, the harmonic intensity exhibits multiple plateaus, whose cutoff energies ε_{cut}=U+mE_{0} scale with the Coulomb interaction U and the maximum field strength E_{0}. In this regime, the created doublons and holons are localized because of the strong field and the mth plateau originates from the recombination of mth nearest-neighbor doublon-holon pairs. In the weak-field regime, there is only a single plateau in the intensity, which originates from the recombination of itinerant doublons and holons. Here, ε_{cut}=Δ_{gap}+αE_{0}, with Δ_{gap} the band gap and α>1. We demonstrate that the Mott insulator shows a stronger high-harmonic intensity than a semiconductor model with the same dispersion as the Mott insulator, even if the semiconductor bands are broadened by impurity scattering to mimic the incoherent scattering in the Mott insulator.
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.
We demonstrate the existence of long-lived prethermalized states in the Mott insulating Hubbard model driven by periodic electric fields. These states, which also exist in the resonantly driven case with a large density of photo-induced doublons and holons, are characterized by a nonzero current and an effective temperature of the doublons and holons which depends sensitively on the driving condition. Focusing on the specific case of resonantly driven models whose effective time-independent Hamiltonian in the high-frequency driving limit corresponds to noninteracting fermions, we show that the time evolution of the double occupation can be reproduced by the effective Hamiltonian, and that the prethermalization plateaus at finite driving frequency are controlled by the next-to-leading–order correction in the high-frequency expansion of the effective Hamiltonian. We propose a numerical procedure to determine an effective Hubbard interaction that mimics the correlation effects induced by these higher-order terms.
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.
We study the Fermi-Hubbard model in the strongly correlated Mott phase under the influence of a harmonically oscillating hopping rate $J(t)={J}_{0}+\mathrm{\ensuremath{\Delta}}Jcos(\ensuremath{\omega}t)$. Apart from the well-known fundamental resonance, where the frequency $\ensuremath{\omega}$ of this oscillation equals (or a little exceeds) the Mott gap, we also find higher-order resonances where multiple particle-hole pairs are created when $\ensuremath{\omega}$ is near an integer multiple of the gap. These findings should be relevant for experimental realizations such as ultracold fermionic atoms in optical lattices or pump-probe experiments using laser pulses incident on correlated electrons in solid-state materials.