Fig. 5: Nonlinear phononics demonstrated via time-resolved optical spectroscopy, X-ray scattering, and 2D terahertz spectroscopy.

a Transient reflectivity changes of La_0.7Sr_0.3MnO₃ at 800 nm for two different pump wavelengths, 1.5 μm and 14.3 μm. Inset: Fourier transform spectra of the oscillatory component and schematics of the corresponding phonons. b Amplitude of the coherently-driven E_g phonon at different pump wavelengths, indicating a resonant enhancement at the mode frequency in a. Horizontal error bars indicate the bandwidths of the mid-IR pump pulses. The red curve is the calculated linear absorption due to the IR-active E_u phonon. c Terahertz-driven phonon up-conversion in SrTiO₃. When the soft transverse optical (TO) phonon (yellow, TO₁) is resonantly driven by a strong terahertz pulse (red), energy is exchanged with higher-frequency phonon modes through nonlinear phononic couplings, resulting in the TO₂ (turquoise) and TO₃ (purple) mode. A schematic of the two lowest-frequency zone-center TO modes are indicated at the top. d Equilibrium X-ray intensity of SrTiO₃ at 135 K, where selected high-symmetry points are labeled, including R (1/2 1/2 1/2) and M (1/2 1/2 0). The R point hosts the antiferrodistortive fluctuations of the cubic-to-tetragonal phase transition at 110 K. e Time-resolved changes in the X-ray diffuse scattering intensity at the R point after photoexcitation by a mid-IR pulse with a fluence of 60 mJ/cm². f-i 2D terahertz spectroscopy of nonlinear phononics in MnBi₂Te₄. f Schematic diagram of the measurement. g Normalized 2D fast Fourier transform (FFT) of the nonlinear response, \({\widetilde{\theta }}_{{\rm{NL}}}({f}_{{\rm{ex}}},{f}_{\det })\). \({f}_{{\rm{ex}}}\) and \({f}_{\det }\) denote the excitation and detection frequency, respectively. h Normalized 2D FFT of the simulated nonlinear response, \({\widetilde{\theta }}_{{\rm{R,NL}}}({f}_{{\rm{ex}}},{f}_{\det })\), modeled by three different mechanisms: photonic (∝ E²), combined photophononic (∝ Q_IRE), and phononic (\(\propto {Q}_{{\rm{IR}}}^{2}\)), where E is the electric field of the terahertz excitation pulse, Q_IR is the normal coordinate of the IR-active mode E_u, and Q_R is the normal coordinate of the Raman-active mode E_g. The photophononic scenario shows the best agreement, which confirms that the excitation of the Raman-active E_g phonon is mediated by the IR-active E_u phonon via the photophononic mechanism. \({f}_{{E}_{g}}\) and \({f}_{{E}_{u}}\) are 3.14 THz and 1.47 THz, respectively. i, Schematic illustration of the excitation mechanism in a model with two oscillators, corresponding to the E_u and E_g phonons. The arrows indicate the stimulated transitions. Panels a and b adapted from ref. ¹⁶⁷, Springer Nature Ltd. Panel c adapted from ref. ¹⁸⁷, Springer Nature Ltd. Panels d and e adapted from ref. ¹⁷⁴, CC BY 4.0. Panels g-i adapted from ref. ¹⁸⁹, American Physical Society.