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

Tatsuhiko N. Ikeda, Koki Chinzei, and Hirokazu Tsunetsugu

Phys. Rev. A 98, 063426 – Published 26 December 2018

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.

Received 4 July 2018

DOI:https://doi.org/10.1103/PhysRevA.98.063426

Physics Subject Headings (PhySH)

High-order harmonic generation Quantum description of light-matter interaction

Crystalline systems

Bloch-Floquet theorem Tight-binding model

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Tatsuhiko N. Ikeda, Koki Chinzei, and Hirokazu Tsunetsugu

Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

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Issue

Vol. 98, Iss. 6 — December 2018

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Images

Figure 1
(a) Floquet eigenvalues for $F = 0$ and $Ω = 1.0$ . The solid lines correspond to the energy bands $ε_{\pm} (k)$ [Eq. (4)], and the dashed lines to the other Floquet bands. The shaded area represents the representative interval $[- Ω / 2, Ω / 2)$ , in which the point of band crossing is encircled. For $F = 0.5$ , (b) the representative Floquet eigenvalues and (c) the weight difference $| w_{2} {(k) |}^{2} - {| w_{1} (k) |}^{2}$ are plotted against $k$ . (d)–(f) Plots are similar plots to those in (a)–(c), where $Ω$ is changed to 0.25. In all panels, we have set $Q = 0.1$ .
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Figure 2
The HHC spectrum in the single-band limit $Q = 0$ [Eq. (35)]. Each data set corresponds to the field strength $F = 0.1$ (circle), 1.0 (square), 5.0 (triangle), and 8.0 (diamond).
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Figure 3
(a) Typical behavior of $| B_{M}^{α} (k) |$ [Eq. (53)] calculated for $F = 0.5$ , $Ω = 0.15$ , $α = +$ , and $k = 7 π / 20$ . The plateau region $M ≲ 2 z_{k}$ is shown by arrows. (b) Schematic illustration for the overlap between $B_{M}^{α} (k)$ and $B_{N + M}^{α} (k)$ . For $N ≳ 4 z_{k}$ , the overlap rapidly decays. (c) The second-order correction $I_{H}^{α, 2} (k, N)$ to the HHC spectrum (55) for $k = 7 π / 20$ (square) and for the average over $k = (π / 2) (m / 10)$ with $m = 0, 1, \dots, 9$ . The arrow indicates $4 z_{k}$ for $k = 7 π / 20$ .
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Figure 4
The HHC spectrum calculated for the four parameter sets $(F, Ω)$ as indicated in the figure. The parameter $Q$ is set to 0.01. The arrow indicates the end of the plateau that is identified from the plot.
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Figure 5
Numerically calculated HHC spectrum in the nonperturbative regimes including the three cases (a) small $Q$ but large $F$ , (b) small $F$ but large $Q$ , and (c) large $F$ and $Q$ . In all panels, $Ω = 0.1$ . (a) The three data sets correspond to different field strengths $F = 5.0$ (circle), 1.0 (square), and 0.1 (triangle) with $Q = 0.01$ . (b) The staggered potential strength is varied: $Q = 0.20$ (circle), 0.10 (square), and 0.01 (triangle) with $F = 1.0$ . (c) The parameter set is $(F, Q) = (3.0, 0.30)$ (solid circle), (1.0,0.30) (open circle), (3.0,0.01) (solid square), and (1.0,0.01) (open square). Dashed lines indicate the multistep plateaus.
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Figure 6
(a) Absolute value of $Δ I_{H} (N)$ [Eq. (A11)] calculated for $F = 0.5$ and $Ω = 1.0$ plotted against $Q$ in the log-log scale. Data are shown for the harmonics $N = 1$ (solid circle), 3 (solid square), 5 (solid triangle), 7 (open circle), 9 (open square), and 11 (open triangle). The solid lines are the polynomial fit (A12) of the fourth order. (b) The determined fitting parameters normalized as $a_{N} / a_{1}$ (solid) and $b_{N} / b_{1}$ (open) plotted against the harmonic order $N$ , where $a_{1} = 6.0 \times 10^{- 2}$ and $b_{1} = - 1.0$ . The error bars show the fitting errors.
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