Effects of Zero-Point Motion in the High Harmonic Generation Spectrum of Solids

TL;DR

The paper addresses why solid-state high-harmonic generation (HHG) spectra appear clean without invoking unrealistically short dephasing times. It introduces a microscopic 1D diatomic-chain model that incorporates optical-phonon zero-point motion as per-site jitter and compares the resulting spectra to those obtained with phenomenological dephasing models, showing that optical jitter intrinsically suppresses long-range coherence and cleans the HHG spectrum. The key findings are that optical zero-point motion, not acoustic phonons or static local strain, drives the spectral sharpening; distance-dependent dephasing or a finite $T_2$ cannot fully replicate this effect, though they can mimic some features, and CEP behavior is sensitive to the presence of long-range coherence. The significance lies in providing a microscopic mechanism for decoherence in solid HHG, offering a practical modeling approach via site-distance dephasing, and proposing CEP-resolved measurements as a probe of coherence lengths and atomic fluctuations in crystalline materials.

Abstract

The interpretation of high-harmonic generation (HHG) in solids typically relies on phenomenological dephasing times far shorter than what is expected from microscopic scattering processes. Here we show that zero-point fluctuations associated with optical phonons naturally suppress long-range electronic coherences and generate clean harmonic spectra without introducing ad-hoc decoherence parameters. Using a 1D semiconductor composed of two distinct sites per unit cell and realistic phonon amplitudes, we demonstrate that random per-site optical-phonon jitter reproduces the spectral sharpening typically attributed to ultrafast $T_2$ dephasing. In contrast, acoustic phonons and local strain, whose distortions are correlated over nanometer scales, produce negligible spectral cleaning. We further show that such long-range site coherence leads to carrier-envelope-phase-dependent effects in the HHG spectrum driven by long pulses, but these effects collapse once optical-phonon-induced decoherence is included. Our results (i) identify optical zero-point motion as a key mechanism governing coherence in solid-state HHG, (ii) demonstrate that it can be qualitatively modeled in periodic solids through site-distance-dependent dephasing, and (iii) suggest that CEP-resolved measurements can probe electronic coherence lengths and atomic fluctuations in crystalline materials.

Figures (6)

• Figure 1: Comparison of the high-harmonic generation spectra between a periodic 1D chain of atoms and a finite chain of $N=1199$ atoms (see text for tight binding parameters). In both cases, atoms have fixed, well-defined positions. (a) Logarithmic scale. (b) Linear scale.
• Figure 2: Comparison of the HHG spectra between the periodic chain (gray) and two finite chains of $N=1199$ atoms with optical-phonon-like site jitter at temperatures of $T=4$ K (blue) and $T=300$ K (orange). Coherent averaging was performed over 20 random simulations, which was enough to attain convergence due to self-averaging effects of the long chain. The inset in (b) illustrates the chain with the width of the Gaussian function representing the per-atom root-mean-squared displacements, alternating between the heavier (A) and lighter (B) atoms, and the center of the Gaussians corresponding to the equilibrium static positions. (a) Logarithmic scale. (b) Linear scale.
• Figure 3: (a-c) Time-frequency plots of the current for a 1D chain of $N=1199$ atoms. a) Chain with no atomic jitter and no phenomenological dephasing terms, displaying multiple trajectories per cycle. (b) Chain with atomic jitter consistent with optical phonon amplitudes. (c,d) Chain with no atomic jitter and a distance-dependent dephasing term of $\gamma(x; x_{cut}=2a_0/3,\beta=4\times 10^{-5}~\text{a.u.})$. (d) HHG spectrum for the case of panel a (gray curve), b (blue curve) and c (orange curve), in logarithmic scale.
• Figure 4: HHG spectra of a chain of $N=1199$ atoms as a function of different dephasing parameters: (a) root-mean-square of atomic displacement for a system with optical-phonon per-site jitter, (b) $\gamma(x; x_{cut},\beta)$ for fixed $\beta=2.4\times 10^{-3}$ a.u. and varying $x_{cut}$, (c) $\gamma(x; x_{cut},\beta)$ for fixed $x_{cut}=2a_0/3$ and varying $\Delta_0$, which corresponds to the distance for which the effective dephasing times is one quarter of the laser cycle, (d) dephasing time $T_2$. The white dashed line in panels (a,c) corresponds to the blue and orange HHG spectra in Fig. \ref{['fig:gabor']}e, respectively.
• Figure 5: Effect of acoustic-phonon-like correlated jitter. HHG spectrum of a chain of $N=1199$ atoms with a correlated bond jitter with mean square atomic fluctuation $\langle q^2\rangle=0.15$ Å (three times larger than that of optical phonons) and different correlation lengths $\ell_{corr}$. For all $\ell_{corr}$, spectrum is essentially the same as the frozen chain in Fig. \ref{['fig:hhg_periodic']}, with negligible noise reduction. (a) Logarithmic scale. (b) Linear scale.