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Jared S. Ginsberg, M. Mehdi Jadidi, Jin Zhang, Cecilia Y. Chen, Nicolas Tancogne-Dejean, Sang Hoon Chae, Gauri N. Patwardhan, Lede Xian, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), James Hone, Angel Rubio, Alexander L. Gaeta

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[Phonon-enhanced nonlinearities in hexagonal boron nitride](https://mdr.nims.go.jp/datasets/6c9836c4-25f9-4c5d-b662-2f78be328651)

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Phonon-enhanced nonlinearities in hexagonal boron nitrideArticle https://doi.org/10.1038/s41467-023-43501-xPhonon-enhanced nonlinearities inhexagonal boron nitrideJared S. Ginsberg1,11 , M. Mehdi Jadidi1,11, Jin Zhang 2,11 , Cecilia Y. Chen3,11,Nicolas Tancogne-Dejean 2, Sang Hoon Chae4,5,6, Gauri N. Patwardhan1,7,Lede Xian 2, Kenji Watanabe 8, Takashi Taniguchi 9, James Hone 4,Angel Rubio 2,10 & Alexander L. Gaeta 1,3Polar crystals can be driven into collective oscillations by optical fields tunedto precise resonance frequencies. As the amplitude of the excited phononmodes increases, novel processes scaling non-linearly with the applied fieldsbegin to contribute to the dynamics of the atomic system. Here we show twosuch optical nonlinearities that are induced and enhanced by the strongphonon resonance in the van derWaals crystal hexagonal boron nitride (hBN).We predict and observe large sub-picosecond duration signals due to four-wave mixing (FWM) during resonant excitation. The resulting FWM signalallows for time-resolved observation of the crystal motion. In addition, weobserve enhancements of third-harmonic generation with resonant pumpingat the hBN transverse optical phonon. Phonon-induced nonlinear enhance-ments are also predicted to yield large increases in high-harmonic efficienciesbeyond the third.Parametric optical processes in solids can provide a window into theoptical susceptibility, band-structure, and underlying symmetries ofcrystals, eachofwhich candramatically affect the nonlinear frequency-conversion process1–3. Symmetries, more so than any other factor,dictate the allowed higher-order processes in a given nonlinearsystem4. These properties become frequency independent far fromany resonances, as is the case in the visible and near-infrared regimewhere many high-order harmonic generation measurements takeplace5. However, in the mid-infrared regime, polar crystals supportlattice collective oscillations that can be resonantly driven by anoptical field. At frequencies near these phonon resonances the linearoptical responseof the crystal is significantlymodified,manifesting forexample as a peak in the real permittivity6,7. These ionic modes canalter the symmetry properties of the crystal, leading to transientnonlinear optical effects such as those observed in SrTiO3, which canbe driven into a metastable non-centrosymmetric state followingprolonged exposure to a phonon-resonant pump8. Under increasedresonant excitation using femtosecond laser pulses, the amplitude ofthe ionic motion can become nonlinear with the incident fieldstrength. For bulkmaterials suchasLiNbO3 andGaAs, phonon-inducedenhancements of optical nonlinearities9–12 occur in this regime.A strong phonon resonance in themid-IR is present in the van derWaals crystal hexagonal boron nitride (hBN), with a transverse optical(TO) phonon mode at 7.3 µm free-space wavelength (170meV)13. Therelatively light constituent atoms of hBN make this one of the mostenergetic TOphonons, accessible by ultrafast table-top lasers. hBN hasReceived: 16 January 2023Accepted: 10 November 2023Check for updates1Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York, NY 10027, USA. 2Max Planck Institute for Structure andDynamics of Matter and Center for Free-Electron Laser Science, Hamburg 22761, Germany. 3Department of Electrical Engineering, Columbia University, NewYork, New York, NY 10027, USA. 4Department of Mechanical Engineering, ColumbiaUniversity, New York, New York, NY 10027, USA. 5School of Electrical andElectronic Engineering, Nanyang Technological University, Singapore 639798, Singapore. 6School of Materials Science and Engineering, Nanyang Tech-nological University, Singapore 639798, Singapore. 7School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA. 8Research Centerfor FunctionalMaterials, National Institute forMaterials Science, 1-1Namiki, Tsukuba 305-0044, Japan. 9International Center forMaterials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 10Center for Computational Quantum Physics, Simons Foundation FlatironInstitute, New York, NY 10010, USA. 11These authors contributed equally: Jared S. Ginsberg, M. Mehdi Jadidi, Jin Zhang, Cecilia Y. Chen.e-mail: jsg2208@columbia.edu; jin.zhang@mpsd.mpg.de; angel.rubio@mpsd.mpg.de; a.gaeta@columbia.eduNature Communications |         (2023) 14:7685 11234567890():,;1234567890():,;http://orcid.org/0000-0001-7830-3464http://orcid.org/0000-0001-7830-3464http://orcid.org/0000-0001-7830-3464http://orcid.org/0000-0001-7830-3464http://orcid.org/0000-0001-7830-3464http://orcid.org/0000-0003-1383-4824http://orcid.org/0000-0003-1383-4824http://orcid.org/0000-0003-1383-4824http://orcid.org/0000-0003-1383-4824http://orcid.org/0000-0003-1383-4824http://orcid.org/0000-0002-9595-2404http://orcid.org/0000-0002-9595-2404http://orcid.org/0000-0002-9595-2404http://orcid.org/0000-0002-9595-2404http://orcid.org/0000-0002-9595-2404http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0003-2060-3151http://orcid.org/0000-0003-2060-3151http://orcid.org/0000-0003-2060-3151http://orcid.org/0000-0003-2060-3151http://orcid.org/0000-0003-2060-3151http://orcid.org/0000-0001-6877-7316http://orcid.org/0000-0001-6877-7316http://orcid.org/0000-0001-6877-7316http://orcid.org/0000-0001-6877-7316http://orcid.org/0000-0001-6877-7316http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43501-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43501-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43501-x&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43501-x&domain=pdfmailto:jsg2208@columbia.edumailto:jin.zhang@mpsd.mpg.demailto:angel.rubio@mpsd.mpg.demailto:a.gaeta@columbia.eduan energetically favorable AA’ stacked lattice in equilibrium, withalternating boron and nitrogen atoms sitting one on top of the other.An illustration of the resonantly driven, in-plane displacement ofatoms for the TO (E1u) mode of hBN14 is presented in Fig. 1a. At thepoint where the photon and phonon dispersion curves meet, an anti-crossing emerges in the hBN band structure, and the crystal hosts newhybridmodes called phonon-polaritons15. These have been the subjectof intense study due to their long-range propagation16.Using time-resolved measurements, we confirm that when TOphonons of hBN undergo oscillations as indicated by transient four-wave mixing (FWM) signals near the second harmonic generation(SHG) wavelength, which is forbidden with a single beam in a bulksample at equilibrium17–19. The FWM signal is studied as a function ofthe power and polarization of both the phonon-inducing pump andharmonic-generating probe, from which preferential symmetry axesare identified. Moreover, the natural hyperbolicity of the hBN TOphonon makes it an attractive platform for tight confinement ofoptical energy, and therefore for enhancing nonlinearities and light-matter interactions within relatively large volumes6. We extend thescope of these light-matter interactions to a higher order in mid-IRpower by exploiting the strong hyperbolic confinement for evengreater electron-phonon coupling. Specifically, in this work we showenhanced emission from the phonon-electron contributions to opticalthird-harmonic generation (THG) in hBN. We theoretically predict anddemonstrate experimentally the nonlinear response of thin hBN crys-tals associated with this TO phonon mode at 7.3 µm. By sweeping asignificant bandwidth of the mid-IR we demonstrate a greatlyenhanced on-resonance phononic contribution to THG when hBN ispumped at its TO phonon.ResultsPhonon-mediated four-wave mixingWe first characterize theoretically the ionic displacements in bulk hBNunder resonant excitationwith 25-fs FWHMpulses by performing timedependent density-functional theory (TDDFT) simulated atomicoscillations spanning 200-fs, or roughly 8 times the theoretical pulseduration (see Fig. 1b). For a modest input intensity of 1.5 × 1011 W/cm2,we estimate that the phonon amplitude is 1% of the equilibrium latticeconstant. While the period of the lattice oscillation is 25-fs, which isconsistent with the expected phonon frequency, the relaxation timecannot be theoretically determined due to a lack of dissipative path-ways. The amplitudes of atomic motion are plotted as a function ofpump intensity in Fig. 1c. The displacements predicted by TDDFTcalculations are fit by I1/2 with deviations appearing at large intensitiesand reach nearly 10% of the equilibrium lattice constant (2.5 Å)20 at10 TW/cm2. The time-dependent electronic current is extracted, andfrom this we generate the theoretical harmonic spectra employedthroughout this work (see Methods).Multilayer hBN has inversion (and 6-fold rotational) symmetrydue to the natural 2H stacking of its van der Waals structure21. Anycontribution at the second harmonicwavelength in few- tomany- layerhBN is therefore restricted only to the broken inversion symmetrycases of interfaces and an odd number of layers and is inherentlyweak17. By conducting the ultrafast pump-probe experiments laid outin Fig. 2, we show that excitation of the IR-active TO (E1u) phononallows for the presence of FWM signals at energies of twice the probephoton plus or minus one phonon. Our simulations reveal the emer-gence of such an ultrafast, transient signal surrounding the secondharmonic of an 800 nm probe pulse, as shown in Fig. 3a. The signal oneither side of harmonic order 2 highlights the shifting of the signalfrequency up and down by the phonon energy in the two variations ofthe FWM process shown in Fig. 3e. We note that the HHG spectraobtained in the presence of the TO excited phonon display additionalsignals along with the odd harmonics. This results mostly from thepresence of phonon-induced sidebands, which are generated byelectronandphonon frequencies (see SupplementaryDiscussion). Thesideband effect also explains the dip at the even harmonic position inour simulations. The energy width between the two split peaks isapproximately twice the energy of the TO mode, indicating that thenonlinearity is predominantly third-order.In our experiments, the measured signal near the second har-monicwavelength of 396 nm is presented in Fig. 3b as a function of thetime delay between 792 nm and 7.3 µm pulses. The probe pulse froman amplified Titanium-Sapphire laser is scanned in time by amechanical delay line relative to the pump pulse from a mid-infraredoptical parametric amplifier and difference-frequency generationmodule. The powers and relative polarizations are set with filters andhalf-wave plates (HWP), and the two beams are then combined on abeam splitter before being focused onto the sample by a reflectiveobjective. (The experimental setup is shown in Fig. 2b, with furtherdetails in Methods.) When the probe pulse precedes the pump pulse,no FWM signal is measured, indicating that the interface SHG and oddlayer-number contributions are below the noise floor. The time-resolved signal displays a strong signal at the zero-timedelay, when theprobe pulse’s arrival coincides with the excitation of the hBN phonon-polariton. The transient signal relaxes back to zero with a time con-stant of 120 fs, which is approximately twice the pump pulse duration.When pumped far off from the phonon resonance, no FWM signal ismeasured. Fast oscillations on the pump-probe trace provide a directmeasurement of the oscillating atomic displacements in time.NBTO (E1u) Phonon(a)Time (fs)Atomic Displacement (a 0)B AtomN Atom(b)Atomic Displacement (a 0)(c)Pump Intensity (1012 W/cm2)Fit P1/2Fig. 1 | Atomic motion and atomic displacement associated with resonantdriving of the TO (E1u) phonon mode. a Honeycomb lattice arrangement ofhexagonal boron nitride. Arrows illustrate the motion of atoms under resonantoptical excitation. The two species move oppositely from each other in plane andacross all layers for the IR-active TO (E1u) mode. b Simulated atomic displacements,in units of Bohr radii a0, of boron and nitrogen ions in TO (E1u)-excited hBN. A 25-fsFWHM, 1 × 1012W/cm2 pulse excites the latticedynamics. TheTDDFT simulations donot include any damping terms through which to estimate the relaxation time.c Peak amplitude of atomic displacements as a function of pump intensity, fit to I1/2with a small linear-in-intensity correction. Displacements nearing 10% of the equi-librium lattice constant are achievable before the onset of damage.Article https://doi.org/10.1038/s41467-023-43501-xNature Communications |         (2023) 14:7685 2Pump(7300 nm)Probe(792 nm)WaveplateWaveplateBeamsplitterFWM (396 nm)hBN on CaF2Pump(3 - 9.5 μm)WaveplatehBN on CaF2THG (1 – 3.2 μm)(a)(b)Fig. 2 | Setup for two experiments demonstrating phonon-enhanced non-linearity in hBN, in transmission geometry. a Experimental setup for THGexperiments. Detection is performed with PbS and MCT detectors, a lock-inamplifier, and boxcar-averaging. b Experimental setup for pump-probe FWMexperiments. The time-delay is controlled by a mechanical delay stage with sub-1 µm step size. The pump and probe are both focused onto the sample with areflective objective with 0.5 numerical aperture. Detection is performed with asilicon photomultiplier tube and lock-in amplifier.FWM Yield (arb. units)Probe Power (μW)Fit Pprobe2MeasuredFit PpumpMeasuredPump Power (mW)Delay (ps)FWM Yield (arb. units)FWM Yield (arb. units)On Resonance(c) (d)(b)(a)Off ResonanceWavelength (μm)FFTLog(Harmonic Intensity)Harmonic OrderNo Phonon0.60.40.20-0.2-0.4-0.6(e)TO ( E1u)Fig. 3 | Four-wave mixing between a probe signal and the mid-IR pump at thephonon frequency. a TDDFT simulations show the emergence of FWM non-linearity during resonant excitation of the TOphononmode.bTime-resolved FWMyield (normalized) of the 792 nm probe pulse. While the pumps are temporallyoverlapped, an ultrafast third-order nonlinearity is measured. The transient signalvanishes following a 200-fs time constant, or about twice the pulse duration. Theappearance of wings in the time-delay scan is a result of a non-perfectly Gaussianpulse, a result of strong atmospheric absorption. Inset: Fourier-Transform of theFWM time-delay. c Dependence of measured FWM yield on probe power.dDependenceofmeasured FWMonpumppower. The FWMyield increases linearlywith the phonon driving intensity. e Two versions of the proposed FWM process.The 2ωphonon energy difference in the emissions is consistent with the splittingobserved in the theoretical spectra. The error bars in (c, d) represent the range ofthe detected signal over the averaging period.Article https://doi.org/10.1038/s41467-023-43501-xNature Communications |         (2023) 14:7685 3A pedestal on that same signal is a consequence of the finite responsetime of each peak being slower than the driving frequency. In Figs. 3cand 3d we show the dependence of the FWM yield on the intensity ofthe probe and pump, respectively. A quadratic dependence of theFWM intensity on the probepower is observed, while a linear scaling ofthe signal with respect to the mid-IR intensity is found. Figure 3dreproduces the expected linear dependenceon themid-IR pumppulsebased on the pair of χ(3) interactions depicted in Fig. 3e. We do notobserve high-order phonon-resonant processes since the strength ofsuch signals are below the sensitivity of the detection system.We determine the dependence of the ultrafast FWM on theorientation of the pump and probe polarizations with respect to thecrystal high-symmetry axes. Figure 4a gives the total normalized FWMyield for 360o rotation of both pulses (180o rotation is measured andthe data is then mirrored). We observe a polarization behavior uniquefromeither the inherent 6-fold χ(2) or isotropic χ(3) symmetries of purelyelectronic hBN nonlinearities22. Specifically, the emission appears toclosely obey the functional form,FWM θ,ϕð Þ= αcos2 3θð Þ+ βsin2 3θð Þh icos2 θ� ϕð Þ ð1Þwhere θ and ϕ are the angles of the pump and probe relative to thezigzag (ZZ) axis of the crystal, respectively, and α and β determine therelative strengths of the emission along the ZZ andArmchair (AC) axes,respectively. The nonlinear yields peak only along ZZ axes that arebeing resonantly driven with a phonon-polariton. This is most clearlyvisible in the linecuts of the probe polarization dependence for pumpfields alignedparallel to theACandZZaxes, given in Fig. 4b. Evenwhenthe pump excitation is aligned with an AC axis, the two adjacent ZZoriented TO (E1u) phonons oscillate with a relatively small amplitude,andweobserve phonon-mediated FWM,whereas the ZZ axis at exactly90o from that excitation shows no emission. From Fig. 4 we determinethat thephonon-mediated FWM is at least 3 times greater parallel to ZZthan to theACdirections. This is supportedby time-dependent densityfunctional theory simulations in Fig. 4c, which identifies newnonlinearity along both symmetry axes, though much greater for theTO (E1u) phonon than the relative π phase LO (E1u).Phonon-enhanced third-harmonic generationWhen driven beyond the previously discussed weak-excitationregime, further enhanced nonlinearities emerge in hBN. We showthe integrated and normalized experimental THG amplitudes for arange of pump wavelengths from 3 μm to 9.5 μm in Fig. 5a as bluedots, which are in excellent agreementwith the calculations discussedin Fig. 1 and plotted as green dots in Fig. 5a. The third-harmonicexhibits a strong peak for pump wavelengths near the TO phononresonance at λ = 7.3 µm, which is far from any electronic or excitonicresonances. We fit the data to a Lorentzian and extract a resonancefull-width at half-maximumof 500 nm. THG yields are below the noiselevel for all λpump <6 µm or >9 µm, compared to that of the resonantsignal which yields at least a 30-fold increase, and thus the phononicenhancement of the THG coefficient at the phonon-polariton wave-length is significantly greater than the purely-electronic componentin this regime. In Fig. 5b we plot the measured intensity dependenceof the THG signal for λpump = 7.3 µm. The fit to a cubic function indi-cates that the measured nonlinearity is third-order and that the scal-ing is perturbative, even at high intensities23. We note that a similareffect has been observed in the phononic second-harmonic genera-tion of LiNbO3, which also remained in the perturbative regime athigher-than-expected intensities. Ultimately, significant enhance-ment of the phonon-induced nonlinearity could be further providedthrough use of subwavelength structures that support confinedphonon-polaritons6,24.We also performed TDDFT simulations of the wavelengthdependence of a higher-order harmonic (HHG) spectra of bulk hBN(see Fig. S1) for two different pump lasers with wavelengths ofλpump = 7.3 µm (polarized parallel to the TO mode) and λpump = 6.2 µm(polarized parallel to the LO mode). Changing the wavelength andpolarization of the pump laser can lead to the excitation of differentphononmodes and lead to significant modulation of the HHG spectra.Log(Harmonic Yield)FWM Intensity (arb. units)Fig. 4 | Polarization dependence of the FWM process. a Pump and probepolarization dependence of total FWM emission (normalized). White dashed linesindicate ZZ axes and are included as a guide for the eye and to emphasize the 60operiodicity. b Linecuts along the ZZ and AC axes from (a). Solid black lines are fitfrom Eq. 1, with α and β as fit parameters. c TDDFT-computed harmonic generationspectra for pump and probe pulses co-polarized along the ZZ (blue) and AC (red)directions. The TO (E1u) phonon present along the ZZ axes leads to the greatestnonlinearity.Article https://doi.org/10.1038/s41467-023-43501-xNature Communications |         (2023) 14:7685 4Excitation of either the TO or LO mode leads to noticeable modifica-tions of the high-harmonic spectra, with the TO (E1u) enhancementbeing one order of magnitude greater than that caused by LO excita-tion. Furthermore, more intense laser pulses introduce larger phononamplitudes and lead to larger nonlinearity. As seen from a pumpintensity of 2.5 × 1011 W/cm2, the high-harmonic yields can be increasedin a wide energy regime, and the high harmonic generation plateau isenhanced (Fig. S2), which is attributed to the increased atomicmovement and enhanced nonlinearity.DiscussionWe have demonstrated greatly enhanced nonlinearities for opticalparametric processes through resonant phonon driving. Furthermore,the appearance of fast oscillations in the pump-probe signal providesthe capability for real-timemonitoring of atomicmotion and evolutiondriven by ultrafast laser pulses. The maximum achievable FWM effi-ciency is highly sensitive to the underlying symmetries of the hex-agonal lattice, peaking along the ZZ axes where the greatest atomicdisplacements are known to occur. We extend the light-matter inter-actions confined by the hyperbolic nature of the hBN phonon disper-sion to a strongly nonlinear regime by demonstrating that the largeelectron-phonon coupling leads to a nearly two order of magnitudeenhancement of THG. We note that the phonon resonance present inthis work is related to Floquet engineering25,26. Floquet engineeringinvolves applying a periodic perturbation to a quantum system,creating a series of states that can be utilized to engineer variousproperties of the system. The effect of driving the phonon in theharmonic spectra can be interpreted as Floquet phonon engineering,where the harmonic oscillation of the phonon is the external drivingfrequency in the Floquet theory26. Efficient coupling of light to hBNphonon-polaritons at normal incidence places stringent requirementson the allowed optical excitation wavelength. For the free-spacewavevector k =0, the required photon wavelength of 7.3 µm is fixed,independent offlake thickness13.Weonly focus on thephononmode atthe Γ point because of energy and momentum conservation. Photonmomentum is negligible compared to the size of the Brillouin zone ofhBN. The lattice dynamics are driven by an external laser with awavelength of 7.3μmpolarized parallel to the atomic displacements ofthe TO (E1u) mode. Under those conditions, the simulated real-timeevolution of the atomic displacements exhibits clear signatures of onlythe TO (E1u)modebeing excited, as shown in Fig. 1b and S1a. Saturationof the THG yield below its perturbative cubic scaling was not observedand is more likely to occur closer to the onset of sample damage. In aseparate theoretical investigation of monolayer hBN, where the TOand LO branches are degenerate at the Γ point, the LO mode wassimilarly found to yield significant nonlinear effects27.MethodsTheoryTime-dependent wave functions and electronic currents were com-puted by propagating the Kohn–Sham equations in real space and realtime for TDDFT simulations, as implemented in the Octopus code28,29.We employed semi-periodic boundary conditions and the adiabaticLDA30 functional. All calculations were performed using fully relati-vistic Hartwigsen, Goedecker, and Hutter (HGH) pseudopotentials31.The real-space cell is sampled with a grid spacing of 0.4 bohr and theBrillouin zone is sampled with a 42 × 42 × 21 K-point grid, which yieldshighly converged results. To model the hBN crystal, the boron nitridebond length is taken here as the experimental value of 1.445 Å. Thelaser pulses are treated in the dipole approximation using the velocitygauge (implies that we impose the induced vector field to be time-dependent but homogeneous in space), and we use a sine-squaredpulse envelope. In all of our calculations, a carrier-envelope phase off =0 is used32. The full harmonic spectra are computed directly fromthe total electronic current j(r, t) asHHGðωÞ= FT∂∂tZd3rjðr,tÞ� ���������2ð2Þwhere FTdenotes the Fourier transform. The calculations arepreparedusing two independent methods: (i) time-evolution of a distortedatomic configuration (by 1% of the bulk hBN lattice) along the phononmodes; (ii) application of pump laser pulses with the same frequenciesand polarizations as the phonon modes. We confirm the two methodsare equivalent in the high-harmonic generation simulations.ExperimentsWe conducted the nonlinear experiments on high-quality flakes ofhBN, 10–50 nmthick andwith sizes on the order of tens ofmicrons.Weselected CaF2 as the preferred substrate for exfoliated flakes based onits high transparency in the visible and mid-infrared ranges and ourneed for a substrate with low nonlinearity. An amplified Titanium-sapphire laser (Coherent Legend Elite) at 1 kHz repetition rate and 6mJpulse energy was used to pump an optical parametric amplifier (OPA,I3ExperimentTheoryLorentzian FitExperimentCubic FitPump Wavelength (μm)THG Yield (arb. units)I (TW/cm2)THG Yield (arb. units)THG Wavelength (μm)(a) (b)Fig. 5 | Wavelength and power dependence of the THG process. a Normalizedthird-harmonic generation yields of 120-fs pulses as a function of pump wave-lengths throughout the mid-IR. THG yields are below the noise level for all wave-lengths <6 µm or >9 µm. Within a roughly 1 µm bandwidth of the phonon-polaritonresonance, a THG enhancement of 30x is observed. The black line is a Lorentzian fitto the data (blue dots) with full-width at half-maximum of 500nm. The green dotswere obtained by integrating the third-harmonic signal in TDDFT simulations andshow excellent agreement with experiments. b Normalized intensity dependenceof THG pumped on-resonance at 7.3 µm and measured at 2.43 µm. The data is aclose fit to I3, indicating that the nonlinear process is scaling perturbatively. Errorbars in both figures represent the range of the detected signal over the averagingperiod.Article https://doi.org/10.1038/s41467-023-43501-xNature Communications |         (2023) 14:7685 5Light Conversion HE TOPAS Prime). The OPA produces 60-fs durationsignal and idler pulses in the near-IR; an additional difference fre-quency generation (DFG) module seeded by the OPA output provideslonger wavelength pulses (λpump = 3–10μm) with durations of 70 to120 fs for all mid-infrared measurements. Pulse intensities were keptbelow the estimated hBN damage threshold of 50 TW/cm2. For THGexperiments, the pump was focused onto a flake with a 2-cm focallength CaF2 lens. The emitted THG signalwas collected in transmissionby an identical lens. After rejecting the residual pump beam with ashort-pass filter, THG was measured on a PbS detector for λTHG below1.7μm and on a liquid nitrogen-cooled MCT detector for λTHGabove 2μm.For visiblewavelengthmeasurements, the setupwasmodified to apump-probe scheme. A 792 nm, 45-fs pulse from the same amplifiedTi-Sapphire laser is utilized as the probe. A variable time delay sepa-rates the 7.3 µmpumppulse (used to excite thephonon) fromthe near-IR probe, producing the FWM signal at 396 nm. The intensity of bothpulses is kept at or below the TW/cm2 range, below the hBN damagethreshold. The time delay was controlled by a sub-1-µm step sizemechanical delay line. Pump and probe beam polarizations are inde-pendently adjusted using zero-order half-wave plates and wire-gridpolarizers before being combined on a beam splitter. The collinearpump and probe are focused onto an hBN flake using a reflectiveobjective (NA =0.5). The choice of reflective optics ensures the samefocal plane for the two beams with very different wavelengths. TheFWM signal from the 792 nmpulse could be collected by the reflectiveobjective in reflection geometry, or by a CaF2 lens in transmissiongeometry. The signal was then filtered with a 10-nm bandwidthbandpassfilter to reject the residual 792 nmand7.3 µmlight. Detectionof the remaining signal was then performed on a fast photomultipliertube (PMT) and lock-in amplifier.Data availabilityThe data generated during the study is available from the corre-sponding author upon request.References1. Boyd, R. Nonlinear Optics (Elsevier, 2008).2. Lanin, A. A., Stepanov, E. A., Fedotov, A. B. & Zheltikov, A. M.Mapping the electron band structure by intraband high-harmonicgeneration in solids. Optica 4, 516 (2017).3. Klemke, N. et al. Polarization-state-resolved high-harmonic spec-troscopy of solids. Nat. Commun. 10, 1319 (2019).4. Neufeld, O., Podolsky, D. & Cohen, O. Floquet group theory and itsapplication to selection rules in harmonic generation. Nat. Com-mun. 10, 405 (2019).5. Ghimire, S. & Reis, D. A. High-harmonic generation from solids.Nat.Phys. 15, 10–16 (2019).6. Caldwell, J. D. et al. Sub-diffractional volume-confined polaritons inthe natural hyperbolic material hexagonal boron nitride. Nat.Commun. 5, 5221 (2014).7. Flytzanis, Chr & Tang, C. L. 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Adv. 4,eaao5207 (2018).AcknowledgementsThis work is supported as part of Programmable Quantum Materials, anEnergy Frontier Research Center funded by the U.S. Department ofEnergy (DOE), Office of Science, Basic Energy Sciences (BES), underaward no. DE-SC0019443. The work of J.Z., LX., N.T.-D., and A.R. wassupported by the European Research Council (ERC-2015-AdG694097),the Cluster of Excellence ‘CUI: Advanced Imaging of Matter’ of theArticle https://doi.org/10.1038/s41467-023-43501-xNature Communications |         (2023) 14:7685 6Deutsche Forschungsgemeinschaft (DFG)—EXC 2056—project ID390715994, Grupos Consolidados (IT1249-19), partially by the FederalMinistry of Education and Research Grant RouTe-13N14839, the SFB925“Light induced dynamics and control of correlated quantum systems,”The Flatiron Institute is a division of the Simons Foundation. Support bythe Max Planck Institute—New York City Center for Non-EquilibriumQuantum Phenomena is acknowledged. J.Z. acknowledges fundingfrom the European Union’s Horizon 2020 research and innovation pro-gram under the Marie Sklodowska-Curie grant agreement No. 886291(PeSD-NeSL). We thank I-Te Lu for the fruitful discussions. K.W. and T.T.acknowledge support from the Elemental Strategy Initiative conductedby the MEXT, Japan (Grant Number JPMXP0112101001) and JSPSKAKENHI (Grant Numbers 19H05790 and JP20H00354). C.Y.C.acknowledges support from the NSF Graduate Research FellowshipProgram DGE 16-44869.Author contributionsJ.S.G., C.Y.C., M.M.J., and G.N.P. performed experiments. J.Z., N.T.-D.,and L.X. performed theory and simulations. Samples from K.W. and T.T.were prepared by S.H.C. Research was supervised by A.L.G., A.R.,and J.H.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-43501-x.Correspondence and requests for materials should be addressed toJared S. Ginsberg, Jin Zhang, Angel Rubio or Alexander L. Gaeta.Peer review information Nature Communications thanks AndreyFedyanin, Haohai Yu and the other, anonymous, reviewer(s) for theircontribution to the peer review of this work. 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