Sensing coherent phonon dynamics in solids with delayed even harmonics

Abstract

High harmonics have emerged as a powerful ultrafast probe of phonon dynamics and electron-phonon interactions in solids, with most studies focusing on odd harmonics. Here, in a pump-probe setup with variable delay, we theoretically investigate how even harmonics reveal coherent phonon dynamics. If pump and probe pulses overlap temporally, the spatial interference effect resulting from a non-coaxial pump-probe setup suppresses harmonic yields. At longer delays, odd-harmonic yields oscillate in phase at the optical phonon frequency, whereas even harmonics exhibit order-dependent phase-shifted oscillations. We identify a responsive range of even harmonic orders, in which the delay of yield oscillations is highly sensitive to subtle features of phonon dynamics and electron-electron interactions. Our findings highlight the potential of even harmonics to elucidate microscopic effects in systems with dynamically broken inversion symmetry.

Figures (5)

• Figure 1: (a): Sketch of the non‑coaxial pump–probe configuration. The angle $\theta$ between the propagation directions of the pump and probe pulses is indicated. (b): Diagram of the model solid consisting of parallel one‑dimensional diatomic chains. The two different atoms are labeled A and B, with lattice constant $L$ and inter‑atomic distance $D$. (c and d): Integrated harmonic yields as a function of pump–probe delay for selected odd and even harmonics obtained from the TCD calculations. A negative delay corresponds to the pump pulse arriving before the probe pulse. Regions labeled “I” and “II” denote temporal overlap and temporal separation between the pump and probe pulses, respectively. Note that the scale for curves in I and II are very different. For better comparison, the curves are vertically offset and normalized such that their extrema in region II are $+1$ and $-1$, respectively.
• Figure 2: The integrated 12th harmonic yield from MCD (gray curve), its incoherent superposition (red deshed curve) and TCD (black curve), normalized to the integrated yield at asymptotic negative delay ($\tau=-30T_0$) for better comparison. The coordinates of area I is on the left side, while those of area II is on the right side.
• Figure 3: Fitted phases of (a) the fundamental $\Phi_n^1$ and (b) second-harmonic $\Phi_n^2$ components of the phonon frequency in MCD yield oscillations of even harmonics according to Eq. \ref{['eq:fitting-model']}. We show results for full calculations (black circles), as well for the quasi-static (QS, red squares) and the no-feedback (NF, blue crosses) approximations, respectively. Lines are to guide the eye. For all harmonics shown, the fit satisfies $R^2 > 0.99$.
• Figure 4: (a) Fitted $\Phi_n^1$ of the fundamental phonon-frequency components in the QS for even harmonics, obtained with dynamic and frozen KS potentials. (b) Integrated even-harmonic yield for the fixed lattice (atoms at equilibrium positions) with dynamic and frozen KS potentials.
• Figure 5: Amplitude $C^1_n$ for even harmonics (a) and phase $\Phi_n^1$ for odd harmonics obtained with Eq. \ref{['eq:fitting-model']}.