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Nonlocal nonlinear phononics


Nonlinear phononics relies on the resonant optical excitation of infrared-active lattice vibrations to induce targeted structural deformations in solids. This form of dynamical crystal structure design has been applied to control the functional properties of many complex solids, including magnetic materials, superconductors and ferroelectrics. However, phononics has so far been restricted to protocols in which structural deformations occur within the optically excited volume, sometimes resulting in unwanted heating. Here, we extend nonlinear phononics to propagating polaritons, spatially separating the functional response from the optical drive. We use mid-infrared optical pulses to resonantly drive a phonon at the surface of ferroelectric LiNbO3. Time-resolved stimulated Raman scattering reveals that the ferroelectric polarization is reduced over the entire 50 µm depth of the sample, far beyond the micrometre depth of the evanescent phonon field. We attribute this effect to the anharmonic coupling between the driven mode and a polariton that propagates into the material. For high excitation amplitudes, we reach a regime in which the ferroelectric polarization is reversed, as revealed by a sign change in the Raman tensor coefficients of all the polar modes.


Generally, optical radiation tuned to frequencies immediately above that of a transverse optical phonon, where the real part of the dielectric function ε1(ω) < 0, is evanescently screened and does not propagate. The interaction of the light field with the material, and the ability to manipulate its properties1,2,3,4,5,6,7,8,9,10, remains inherently local. This is generally true in centrosymmetric materials, in which not only the driven vibration but also Raman-active modes that are displaced by anharmonic mode couplings do not propagate. However, if the material breaks inversion symmetry, the optically excited mode can exert a displacive force also onto symmetry-odd modes, which can propagate as polaritons. This type of phonon–polariton coupling opens up a broad class of phenomena that we term nonlocal nonlinear phononics.

Polariton excitation through phononics builds on extensive previous work in which polaritons were excited using impulsive stimulated Raman scattering from propagating near-infrared pulses11,12. In these studies, steering, focusing and amplification of terahertz radiation was shown to occur in regions of the crystal beyond those traversed by the excitation pulse13,14. However, even in cases in which phase matching was optimized, polariton fields never reached amplitudes that could alter material functional properties. In this Article, we show that nonlocal nonlinear phononics can alter the ferroelectric polarization, and even reach a regime in which the polarization is reversed15.

In the experiments reported here, LiNbO3 single crystals were illuminated with mid-infrared pulses of 20 THz central frequency and 150 fs duration, with fluences that extended up to nearly 200 mJ cm2, that is, peak electric fields up to 30 MV cm−1. Previous work has shown that under these conditions a sizable reduction of the ferroelectric polarization, and even a reversal for excitation intensities above a certain threshold, can be achieved8. However, these effects were thought to occur only immediately beneath the surface of the material, that is, over the micrometre extinction depth of the resonantly driven phonon, which was probed by time-resolved second-harmonic generation. Here, by using a bulk-sensitive Raman scattering probe, we show that both reduction and reversal of the ferroelectric polarization occur over the entire 50 μm thickness of the crystal, far beyond the excitation region.

The physical situation leading to nonlocal manipulation of the ferroelectric polarization is sketched in Fig. 1a and 1b. We only consider the c-axis-polarized 18.3 THz A1 mode (QIR), which is driven resonantly by the pump pulse, and the 7.3 THz A1 ‘soft mode’16 (QP), which is excited indirectly by anharmonic coupling to the driven mode. Note that there also exist two additional A1 modes near 8 and 10 THz. However, their effective charges are respectively a factor of 30 and 8 smaller than those of QP and hence they make negligible contributions to the time-dependent response of the material at terahertz frequencies.

Fig. 1: Nonlocal nonlinear phononics.
figure 1

a, An evanescent excitation pulse drives a mode QIR beneath the surface, which launches a propagating polariton mode QP via anharmonic phonon coupling. b, Dispersion relation for LiNbO3 along the crystallographic c axis. The parameters used to generate the curves were taken from ref. 26. c, Simulations of the mode amplitudes QIR and QP upon excitation with a 200 fs pulse tuned slightly above ωIR at 20 THz for an incident field strength of E = 0.1 MV cm−1. The vertical axis corresponds to the depth in the LiNbO3 sample. In addition to the evanescent excitation of QIR at frequencies within the Reststrahlen band, the frequency components of the pulse just below the transverse optical phonon frequency ωIR also excite a polariton of QIR near 16 THz, which propagates deeper into the material and itself forces QP to oscillate weakly at the same frequency. However, anharmonic phonon coupling is negligible and no frequency components outside the bandwidth of the excitation field are generated. d, The same as c, except with an incident field strength of 20 MV cm−1. Here, the nonlinear force on QP becomes appreciable, resulting in the launching of a low-frequency polarization wave, which propagates throughout the medium. a.u., arbitrary units.

Source data

The nonlinear response of the LiNbO3 during and after excitation is described by the following potential17,18:

$$\begin{array}{lll}V\left( {Q_{\mathrm{IR}},Q_{\mathrm{P}}} \right) &=& \frac{1}{4}\left( { - \omega _{\mathrm{P}}^2Q_{\mathrm{P}}^2 + \frac{{\kappa _{{\mathrm{P}}}}}{2}Q_{\mathrm{P}}^4} \right) + \frac{1}{2}\left( {\omega _{\mathrm{IR}}^2Q_{\mathrm{IR}}^2 + \frac{{\kappa _{\mathrm{IR}}}}{2}Q_{\mathrm{IR}}^4} \right)\\ && + V_{\mathrm{c}}\left( {Q_{\mathrm{IR}},Q_{\mathrm{P}}} \right)\ + V_E\left( {Q_{\mathrm{IR}},Q_{\mathrm{P}}} \right)\end{array}$$

Here, ferroelectricity is modelled by a double-well potential along the soft mode coordinate QP with quartic nonlinear coefficient κP and minima located at \(Q_{\mathrm{P}} = \pm \frac{{P_0}}{{Z_{\mathrm{P}}^ \ast }}\), where P0 is the magnitude of the equilibrium ferroelectric polarization, and \(Z_{\mathrm{P}}^ \ast\) and ωP are, respectively, the effective charge and transverse optical frequency of the soft mode. The potential along QIR is accounted for in the form of a single parabolic well and additional quartic anharmonicity with coefficient κIR, which is important to account for nonlinearities of its excitation process. The anharmonic potential term \(V_{\mathrm c}(Q_{\mathrm{IR}},Q_{\mathrm P})=\alpha Q_{\mathrm P}^{2}Q_{\mathrm{IR}}^{2}+\beta Q_{\mathrm P}Q_{\mathrm{IR}}^{3}+\chi Q_{\mathrm P}^{3}Q_{\mathrm {IR}}\) describes the coupling between QIR and QP, but also accounts for additional symmetry-odd terms in the QIR potential that lead to higher-order phonon harmonics documented in ref. 19. Finally, the term \(V_E\left( {Q_{\mathrm{IR}},Q_{\mathrm{P}}} \right) = E(Z_{\mathrm{P}}^ \ast Q_{\mathrm{P}} + Z_{\mathrm{IR}}^ \ast Q_{\mathrm{IR}})\) accounts for the coupling of the modes to the excitation field E, where \(Z_{\mathrm{IR}}^ \ast\) is the effective charge of QIR. Once the charged modes are set in motion, a time-dependent polarization \(P = {\it{\epsilon }}_0{\it{\epsilon }}_\infty E + Z_{\mathrm{IR}}^ \ast Q_{\mathrm{IR}} + Z_{\mathrm{P}}^ \ast Q_{\mathrm{P}}\) is created, where ε0 is the vacuum permittivity and ε is the high-frequency dielectric constant.

Using the model above, the response of LiNbO3 to an incident mid-infrared pulse of duration 200 fs and central frequency 20 THz is readily simulated. Figure 1c shows the time- and space-dependent mode amplitudes QP and QIR, simulated using finite-difference time-domain methods for excitation with a 100 kV cm−1 field. The frequency components of QIR excited at the transverse optical phonon frequency ωIR and within the reststrahlen band have the largest amplitudes and penetrate the material evanescently to depths on the order of a few micrometres. The frequency components of QIR that are excited immediately below ωIR correspond to a polariton that propagates throughout the material. This polariton, which contains frequencies in the 16 THz range, also off-resonantly excites QP, causing it to weakly oscillate at the same frequency. Most notable is that all features shown in this panel are the result of a linear excitation in which anharmonic phonon coupling via the potential term Vc(QIR,QP) is negligible.

Figure 1d shows the calculated QIR and QP in the case of an excitation with a higher field strength of 20 MV cm−1. The features from the linear-excitation regime are still present; however, the nonlinear force on QP arising from αQIR2QP2 in the Vc(QIR, QP) term is now considerable. This anharmonic coupling term symmetrically displaces both wells of the potential towards QP = 0. The rectified component of this force excites QP oscillations over a range of frequencies, which spans nearly the entire polariton branch. This results in a broadband, low-frequency (f ≈ 3 THz) polarization wave, which propagates throughout the crystal as shown in Fig. 1d. In real ferroelectric crystals, there exists also a contribution to the potential from uncompensated charges, which becomes relevant when QP is excited near the regime of polarization reversal17. However, for the perturbative regime considered so far, we neglect this effect.

The effects discussed above were probed through measuring the response of all the A1 modes using femtosecond stimulated Raman spectroscopy (FSRS)20,21,22. In these FSRS experiments, two incident probe pulses were superimposed on the material. The first pulse, with electric field Ew, was a white-light continuum pulse with a spectral bandwidth spanning tens of terahertz and temporal full-width at half-maximum (FWHM) of 100 fs. The second pulse, with electric field Er, was centred at a wavelength of 400 nm and had a narrow spectral bandwidth less than 1 THz and corresponding time duration of 3 ps. A sketch of the experimental set-up is shown in Fig. 2. The theory for FSRS has been discussed elsewhere23,24 and can be summarized as follows. When the two pulses impinge upon a given material, a force F = R*ErEw is imparted onto a Raman-active mode Q, which then oscillates at its eigenfrequency ωQ. The oscillations at ωQ modulate the refractive index of the material, causing the narrowband pulse field Er to develop Raman sidebands at ωr ± ωQ. These sidebands are emitted in the direction of the white-light field Ew, which after being sent through a spectrometer, interfere on a detector to yield heterodyned peaks that are positive at ωr − ωQ and negative at ωr + ωQ. FSRS is especially fruitful because it allows for the detection of all Raman modes in a single probing event as well as ultrafast changes of the Raman coefficient R*.

Fig. 2: Femtosecond stimulated Raman scattering.
figure 2

A narrowband pulse Er and a white-light continuum field Ew excite a coherent vibration with frequency ωQ with a force proportional to ErEw. The stimulated mode mixes with Er via the Raman tensor element, resulting in the emission of sidebands at frequencies ωr ± ωQ that propagate in the same direction as Ew. When sent to a spectrometer, the sideband fields and Ew interfere, resulting in the amplification of Ew at frequency ωr − ωQ and depletion at ωr + ωQ. Shown is a typical FSRS spectrum of the A1 modes in LiNbO3.

Source data

In our experiments, FSRS spectra were recorded at each pump–probe delay τ between the FSRS pulse pair and the mid-infrared pulse, as depicted in Fig. 3a (see Supplementary Section 4 for details). Figure 3b shows the evolution of the experimental FSRS spectra with the time delay τ after illumination with a mid-infrared pulse of fluence 38 mJ cm−2 (field strength of 14 MV cm−1). Two notable changes to the spectra were observed and are highlighted in Fig. 3c, which displays the pumped spectrum for τ = 0.05 ps together with the equilibrium response. The transient spectra show (1) a reduction in the peak amplitudes of all the equilibrium Raman peaks (highlighted in red) and (2) the appearance of several new peaks (indicated in blue).

Fig. 3: Time-resolved FSRS measurements of the A1 modes in LiNbO3.
figure 3

a, Experimental geometry. Time-resolved changes to the FSRS spectra were recorded by varying the time delay τ between the FSRS pulse pair and the mid-infrared pump pulse. b, FSRS spectra as a function of τ for a mid-infrared excitation fluence of 38 mJ cm−2 (14 MV cm−1 field strength). The two prominent features that result from mid-infrared excitation are a reduction in the amplitudes of the A1 modes (highlighted in red) and the emergence of new peaks (highlighted in blue). c, Equilibrium FSRS spectrum and pumped spectrum from b for τ = 0.05 ps. As discussed in the text, the features in the pumped spectrum can be explained under the assumption that the Raman tensors for the A1 modes change dynamically.

Source data

The most important observation is the reduction in the Raman peak amplitudes of QIR and QP. This effect is modelled by assuming that the equilibrium components of both \(R_{\mathrm{IR}}^ \ast\) and \(R_{\mathrm{P}}^ \ast\) become transiently reduced upon excitation with the mid-infrared pulse (Fig. 4a). Because the peak amplitudes observed in FSRS scale exponentially with the sample thickness23,24, we conclude that the optical excitation must be reducing the ferroelectric polarization, and thus \(R_{\mathrm{IR}}^ \ast\) and \(R_{\mathrm{P}}^ \ast\) (ref. 25), throughout the entire crystal (50 µm thick). Indeed, if these Raman coefficients were only modified within the few-micrometre penetration depth of the pump, the pump-induced changes to the FSRS spectra would not be observable in our experiment (Supplementary Section 2).

Fig. 4: Calculations of pumped FSRS spectra in LiNbO3.
figure 4

a, Equilibrium and pumped FSRS spectra for τ = 0.05 ps. The calculations were produced using a time-domain model for the FSRS process that is detailed in Supplementary Section 8. b, Group velocity of the FSRS probe projected onto the spacetime maps of QIR and QP from Fig. 1d. Over the course of the 50 µm sample depth, the probe velocity (vprobe) is approximately matched with the phase velocity of the soft-mode polariton. Here, the effective velocity of the probe was calculated given the group refractive index of LiNbO3 at 400 nm as well as the angle between the mid-infrared pump and white-light continuum pulse. c, Black, the dynamic Raman tensor used to produce the pumped calculations displayed in a. Red, the transient reduction of the Raman tensor seen in the rest frame of the probe after accounting for the effects of velocity mismatch throughout the 50 µm crystal thickness. The component that oscillates at 16 THz imparts sidebands to the frequencies of the equilibrium modes, giving rise to the blue-highlighted peaks in a.

Source data

The observed appearance of new peaks in Fig. 3c is well explained by the direct modulation of the Raman tensors \(R_{\mathrm{IR}}^ \ast\) and \(R_{\mathrm{P}}^ \ast\) at the 16 THz polariton frequency (Fig. 1c). Second-order Raman scattering is evidenced here by satellite peaks appearing at ωIR ± 16 THz and ωP ± 16 THz.

The calculated spectra shown in Fig. 4a were obtained by assuming that the Raman tensors reduced over a timescale similar to those of the transient polarization reductions observed in ref. 8. Our model assumed that these reductions of the Raman tensor propagated throughout the crystal at the phase velocity of the soft-mode polariton. To account for the propagation mismatch with the probe, the velocity line from Fig. 4b was projected onto the spacetime map for the Raman tensor, and an integral along the velocity line was carried out for each time step. The result is highlighted in red in Fig. 4c. A damped sinusoid with a frequency of 16 THz was then added to the result to account for the phase-matched detection of the 16 THz modulation of the Raman tensor. The curve shown in black in Fig. 4c models the effective R*(t) seen by the probe and was used to produce the spectra shown in Fig. 4a. Complete details regarding the calculations are provided in Supplementary Section 8.

Figure 5a reports the experimentally determined fluence dependence of the peak reductions for a fixed τ = 0.08 ps. As the mid-infrared excitation fluence was increased, the FSRS peaks decreased monotonically in amplitude and then reversed sign at the maximum fluence of 185 mJ cm−2 (field strength of 30 MV cm−1) used in the experiment. Note that the ability to detect a sign change of R* is unique to the FSRS probe (discussed further in Supplementary Section 1). The results of our calculations, shown in Fig. 5b, confirm that if the ferroelectric polarization is reversed, the signs of both Raman tensors \(R_{\mathrm{IR}}^ \ast\) and \(R_{\mathrm{P}}^ \ast\) also reverse. Taken together with the experimental data, this theoretical consideration evidences a transient reversal of the ferroelectric polarization in the bulk crystal.

Fig. 5: Reversal of the ferroelectric polarization in the bulk.
figure 5

a, Fluence dependence of the A1 modes for τ = 0.08 ps. A vertical offset was applied to each spectrum for clarity. The FSRS peaks reverse sign at the highest fluence of 185 mJ cm−2 (30 MV cm−1 field strength). b, First-principles calculations of the normalized Raman tensors for QIR and QP as a function of the ferroelectric polarization P/P0, where P0 is the value of the static polarization at room temperature and equilibrium. c, Amplitudes of the Raman peaks associated with QIR and QP. Horizontal error bars indicate the systematic uncertainty in the fluence measurements. At fluences above approximately 60 mJ cm−2 (17 MV cm−1 field strength), the reductions of the peak amplitudes begin to exhibit a saturation-like behaviour, possibly indicating a renormalization of the frequency of the driven mode due to a QIR4 term in the crystal potential. d,e, The peak amplitudes as a function of τ at the highest excitation fluence of 185 mJ cm−2 for QIR (d) and QP (e). The calculations were performed in the same manner as those of Fig. 4, except that here it is assumed that a transient sign reversal of the Raman tensor propagates throughout the medium, rather than a simple reduction. The results of b were used to model the corresponding Raman tensors for QIR and QP. The peak amplitudes shown in ce were extracted via Lorentzian fits to the FSRS spectra. Vertical error bars indicate the uncertainty of the extracted amplitudes as determined from the fit residuals.

Source data

The amplitudes of the FSRS peaks corresponding to QIR and QP as a function of mid-infrared pump fluence are shown in Fig. 5c. Note that for fluences above approximately 60 mJ cm−2 the peak amplitudes are no longer linear and tend to be already saturated before they reverse sign. One effect that would produce one such saturation would be a softening of the frequency of the driven mode QIR due to the fourth-order term \(Q_{\mathrm{IR}}^{4}\) found in the potential of equation (1), an effect that reduces its response to the driving electric field. The evolution of the peak amplitudes with τ at maximum excitation fluence is presented in Fig. 5d,e along with calculated values. Importantly, the calculations were performed in the same manner as were those shown in Fig. 4, but with the exception that here the Raman tensors of both modes are assumed to exhibit a transient sign change and subsequent recovery to their equilibrium values.

While the change in sign of all of the Raman tensor elements indicates a reversal of the ferroelectric polarization, this regime cannot be described by the theory of propagating polaritons alone. What can be quantitatively predicted is that if a ferroelectric domain is reversed beneath the surface, as discussed in ref. 8, a domain wall will propagate into the bulk while emitting coherent electromagnetic radiation. Qualitatively, we expect that this effect should cooperatively overcome the local coercive field as the ferroelectric domain-wall polarization propagates throughout the bulk at near-luminal speeds. Such a highly nonlinear process merits a more comprehensive theory and will be a subject of future studies.

Data availability

Source data are provided with this paper. All other source data related to the results discussed in this paper are available upon reasonable request.


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We acknowledge support from the Deutsche Forschungsgemeinschaft via the Cluster of Excellence ‘The Hamburg Centre for Ultrafast Imaging’.


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The experiment was conceived by M.H., M. Först and A.C. M.H. designed and constructed the experimental set-up with input from M. Först. M.H. and E.R. carried out the experiments and collected data. Experimental data were processed by M.H. M. Fechner carried out all first-principles calculations and designed the finite-difference time-domain model of the polariton propagation. Simulations of the FSRS spectra were carried out by M.H. and M. Fechner. All the authors contributed to the discussion and interpretation of results. M.H., M. Först and A.C. wrote the manuscript with contributions from all authors.

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Correspondence to A. Cavalleri.

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Nature Physics thanks Dominik Juraschek, Dmytro Afanasiev and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Information

Supplementary Sections 1–9. Figs. 1–7 and Table 1.

Source data

Source Data Fig. 1

Matrices for 2D plots + x values and time values.

Source Data Fig. 2

Comma-delimited data for spectrum used for Fig. 2.

Source Data Fig. 3

Comma-delimited data for spectra shown in Fig. 3.

Source Data Fig. 4

Comma-delimited data for calculations in Fig. 4.

Source Data Fig. 5

Comma-delimited data for spectra, calculations and data points shown in Fig. 5.

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Henstridge, M., Först, M., Rowe, E. et al. Nonlocal nonlinear phononics. Nat. Phys. 18, 457–461 (2022).

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