Strong-field laser excitation of solids can produce extremely nonlinear electronic and optical behaviour. As recently demonstrated, this includes the generation of high harmonics extending into the vacuum-ultraviolet and extreme-ultraviolet regions of the electromagnetic spectrum. High harmonic generation is shown to occur fundamentally differently in solids and in dilute atomic gases. How the microscopic mechanisms in the solid and the gas differ remains a topic of intense debate. Here we report a direct comparison of high harmonic generation in the solid and gas phases of argon and krypton. Owing to the weak van der Waals interaction, rare (noble)-gas solids are a near-ideal medium in which to study the role of high density and periodicity in the generation process. We find that the high harmonic generation spectra from the rare-gas solids exhibit multiple plateaus extending well beyond the atomic limit of the corresponding gas-phase harmonics measured under similar conditions. The appearance of multiple plateaus indicates strong interband couplings involving multiple single-particle bands. We also compare the dependence of the solid and gas harmonic yield on laser ellipticity and find that they are similar, suggesting the importance of electron-hole recollision in these solids. This implies that gas-phase methods such as polarization gating for attosecond pulse generation and orbital tomography could be realized in solids.

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The macroscopic characteristics of a material are determined by its elementary excitations, which dictate the response of the system to external stimuli. The spectrum of excitations is related to fluctuations in the density-density correlations and is typically measured through frequency-domain neutron 1 or X-ray [2][3][4] scattering. Time-domain measurements of these correlations could yield a more direct way to investigate the excitations of solids and their couplings both near to and far from equilibrium. Here we show that we can access large portions of the phonon dispersion of germanium by measuring the diffuse scattering from femtosecond X-ray freeelectron laser pulses. A femtosecond optical laser pulse slightly quenches the vibrational frequencies, producing pairs of highwavevector phonons with opposite momenta. These phonons manifest themselves as time-dependent coherences in the displacement correlations 5 probed by the X-ray scattering. As the coherences are preferentially created in regions of strong electron-phonon coupling, the time-resolved approach is a natural spectroscopic tool for probing low-energy collective excitations in solids, and their microscopic interactions.Density fluctuations in nominally periodic media reduce the intensity of the Bragg diffraction peaks and consequently increase the weak diffuse scattering between these peaks, the details of which reflect the amplitudes and spatial frequencies of the fluctuations 6 . The scattered intensity is determined by the dynamic structure factor S(Q, ω) at momentum Q and frequency ω, which is proportional to the Fourier transform of the correlation function of the density-density fluctuations. For phonons, these correlations are u q (0)u −q (t ) , where u q is the phonon amplitude at reduced wavevector q = Q − K Q and K Q is the closest reciprocal lattice vector to Q, and in this context the expectation value is a thermal average 7 . In typical X-ray or neutron scattering experiments the measured diffuse scattering is proportional to the equal-time correlations u q (0)u −q (0) (refs 3,7,8) of the inelastically scattered photons from a highly monochromatic beam. As we demonstrate here in a single crystal of the prototypical semiconductor germanium, a femtosecond laser pulse generates temporal coherences in the equal-time correlation functions g (τ ) = u q u −q parameterized by the pump-probe delay τ between the optical pulse and the X-ray probe. As the X-ray pulse from the free-electron laser (FEL) is short compared with the vibrational motion, we assume that the scattering is effectively instantaneous. Under this approximation we measure g (τ ) stroboscopically, which unlike in the thermal case has an oscillatory contribution from a two-phonon squeezed state generated by the laser pulse, as well as a contribution from incoherent changes in populations 9 . In this paper we focus on the oscillatory component, which yields large portions of the phonon dispersion directly from the measurement without any particular model of the interatomic force...

We review the status of strong-field and attosecond processes in bulk transparent solids near the Keldysh tunneling limit. For high enough fields and low-frequency excitations, the optical and electronic properties of dielectrics can be transiently and reversibly modified within the applied pulse. In Ghimire et al (2011 Phys. Rev. Lett. 107 167407) non-parabolic band effects were seen in photon-assisted tunneling experiments in ZnO crystals in a strong mid-infrared field. Using the same ZnO crystals, Ghimire et al (2011 Nat. Phys. 7 138-41) reported the first observation of non-pertubative high harmonics, extending well above the bandgap into the vacuum ultraviolet. Recent experiments by Schubert et al (2014 Nat. Photonics 8 119-23) showed a carrier envelope phase dependence in the harmonic spectrum in strong-field 30 THz driven GaSe crystals which is the most direct evidence yet of the role of sub-cycle electron dynamics in solid-state harmonic generation. The harmonic generation mechanism is different from the gas phase owing to the high density and periodicity of the crystal. For example, this results in a linear dependence of the high-energy cutoff with the applied field in contrast to the quadratic dependence in the gas phase. Sub-100 attosecond pulses could become possible if the harmonic spectrum can be extended into the extreme ultraviolet (XUV). Here we report harmonics generated in bulk MgO crystals, extending to ∼26 eV when driven by ∼35 fs, 800 nm pulses focused to a ∼1 VÅ −1 peak field. The fundamental strong-field and attosecond response also leads to Wannier-Stark localization and reversible semimetallization as seen in the sub-optical cycle behavior of XUV absorption and photocurrent experiments on fused silica by Schiffrin et al (2013 Nature 493 70-4) and Schultze et al (2013 Nature 493 75-8). These studies are advancing our understanding of fundamental strong-field and attosecond physics in solids with potential applications for compact coherent short-wavelength sources and ultra-high speed optoelectronics.

X-ray scattering is typically used as a weak linear atomic-scale probe of matter. At high intensities, such as produced at free-electron lasers, nonlinearities can become important, and the probe may no longer be considered weak. Here we report the observation of one of the most fundamental nonlinear X-ray-matter interactions: the concerted nonlinear Compton scattering of two identical hard X-ray photons producing a single higher-energy photon. The X-ray intensity reached 4 × 10 20 W cm −2 , corresponding to an electric field well above the atomic unit of strength and within almost four orders of magnitude of the quantum-electrodynamic critical field. We measure a signal from solid beryllium that scales quadratically in intensity, consistent with simultaneous non-resonant two-photon scattering from nearly-free electrons. The high-energy photons show an anomalously large redshift that is incompatible with a free-electron approximation for the ground-state electron distribution, suggesting an enhanced nonlinearity for scattering at large momentum transfer.