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Probing the interatomic potential of solids with strong-field nonlinear phononics


Nonlinear optical techniques at visible frequencies have long been applied to condensed matter spectroscopy1. However, because many important excitations of solids are found at low energies, much can be gained from the extension of nonlinear optics to mid-infrared and terahertz frequencies2,3. For example, the nonlinear excitation of lattice vibrations has enabled the dynamic control of material functions4,5,6,7,8. So far it has only been possible to exploit second-order phonon nonlinearities9 at terahertz field strengths near one million volts per centimetre. Here we achieve an order-of-magnitude increase in field strength and explore higher-order phonon nonlinearities. We excite up to five harmonics of the A1 (transverse optical) phonon mode in the ferroelectric material lithium niobate. By using ultrashort mid-infrared laser pulses to drive the atoms far from their equilibrium positions, and measuring the large-amplitude atomic trajectories, we can sample the interatomic potential of lithium niobate, providing a benchmark for ab initio calculations for the material. Tomography of the energy surface by high-order nonlinear phononics could benefit many aspects of materials research, including the study of classical and quantum phase transitions.

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Figure 1: Experimental set-up and time-resolved optical response.
Figure 2: Spectra of time-resolved optical responses and harmonic field dependences.
Figure 3: Calculated A1-mode potential energy.
Figure 4: FDTD simulations for the phonon-polariton propagation.
Figure 5: Reconstructed A1-mode potential energy.


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We thank R. Merlin and M. Altarelli for discussions. The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 319286 (QMAC). We acknowledge support from the Deutsche Forschungsgemeinschaft via the excellence cluster ‘The Hamburg Centre for Ultrafast Imaging—Structure, Dynamics and Control of Matter at the Atomic Scale’ and the Priority Program SFB925 ‘Light induced Dynamics and Control of Correlated Quantum Systems’.

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Authors and Affiliations



A.C., together with A.v.H. and R.M., conceived this project. R.M., A.v.H. and M. Först built the experimental set-up. A.v.H. and R.M. conducted the experiment and analysed the data. M. Fechner performed the DFT calculations. A.v.H. conducted the FDTD simulation. All authors interpreted the data and contributed to the manuscript.

Corresponding authors

Correspondence to A. von Hoegen or A. Cavalleri.

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The authors declare no competing financial interests.

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Reviewer Information Nature thanks M. Kira and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Figure 1 Experimental set-up.

Pulses (30 fs) from a Ti:sapphire amplifier are used to pump two optical parametric amplifiers (OPA), which are seeded by the same white-light continuum (WLC). CEP-stable, 3 μJ, 150 fs pulses at 17 μm wavelength are obtained by difference frequency generation (DFG) of the two signal beams from the OPAs. The mid-infrared light is focused to a spot size of approximately 65 μm using a telescope and overlapped with the 800 nm probe beam (40 nJ, 35 μm spot size).

Extended Data Figure 2 Sideband generation from phonon harmonics.

The black solid line is the incident spectrum of the 800 nm probe pulses with a bandwidth of about 30 THz. The grey solid lines are the sidebands generated from the phonon harmonics measured at different positions behind the LiNbO3 crystal. Owing to momentum conservation, each sideband propagates in a slightly different direction compared with the unperturbed 800 nm beam. The red line is a guide to the eye of the resulting spectral broadening.

Extended Data Figure 3 Probe spectra and sampling efficiencies.

a, Spectrum of the 800 nm probe pulse before (red) and after (grey) propagation through the unpumped LiNbO3 crystal in units of THz. b, Red curve: sampling efficiency of the 800 nm light calculated with the spectrum shown in a. The grey curve is the penetration depth in the mid-infrared region obtained from FTIR spectroscopy. c, Spectrum of the generated SH light (blue curve) and normalized transmission of the bandpass filter placed in front of the detector (dashed curve), also shown in units of THz. d, Sampling efficiency of the SH light with the spectrum shown in c. The sampling efficiency is almost constant in the 15–45 THz region of the first three phonon harmonics.

Extended Data Figure 4 Phonon-polariton dispersion.

The phonon-polariton dispersion of the two dominant lattice modes in LiNbO3 (black curve) and two light lines ν = vgq for 800 nm (red) and 400 nm (blue) wavelengths are shown. The dots mark the points of intersection with the dispersion relation, which correspond to the observed fundamental frequencies of the driven mode (left and right panels).

Extended Data Figure 5 Assignment of phonon harmonics.

The amplitude spectrum of the time-resolved PR measurement is shown. Blue symbols denote a blueshift of 15 THz (triangles) and 19 THz (circles). Multiple symbols represent shifts by multiples of the corresponding frequencies. Red symbols denote redshifts.

Extended Data Figure 6 Phonon frequency renormalization.

The black circles denote the peak-field-dependent fundamental phonon frequencies extracted from Fourier transformations of the time-resolved signals. Values at the same frequency have been binned (red circles) to account for the limited frequency resolution of the FFT analysis. The error bars denote 1σ (67% confidence interval). The grey line is a fit to the data with the function .

Extended Data Figure 7 Terahertz reflectivity spectrum.

The grey solid line is the measured terahertz reflectivity spectrum of LiNbO3 with light polarized along the c axis. The red line is a fit considering four Lorentzian oscillators. The dashed blue line is a fit considering only the two dominant phonon modes at 7.5 THz and 19 THz. The green line is the FDTD simulated reflectivity when only these two oscillators are considered (see Methods).

Extended Data Table 1 Parameters for the A1 Lorentzian oscillator

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von Hoegen, A., Mankowsky, R., Fechner, M. et al. Probing the interatomic potential of solids with strong-field nonlinear phononics. Nature 555, 79–82 (2018).

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