Probing phonon dynamics with multidimensional high harmonic carrier-envelope-phase spectroscopy

Abstract

We explore pump-probe high harmonic generation (HHG) from monolayer hexagonal-boron-nitride, where a terahertz pump excites coherent optical phonons that are subsequently probed by an intense infrared pulse that drives HHG. We find, through state-of-the-art ab initio calculations, that the structure of the emission spectrum is attenuated by the presence of coherent phonons and no longer comprises discrete harmonic orders, but rather a continuous emission in the plateau region. The HHG yield strongly oscillates as a function of the pump-probe delay, corresponding to ultrafast changes in the lattice such as specific bond compression or stretching dynamics. We further show that in the regime where the excited phonon period and the pulse duration are of the same order of magnitude, the HHG process becomes sensitive to the carrier-envelope phase (CEP) of the driving field, even though the pulse duration is so long that no such sensitivity is observed in the absence of coherent phonons. The degree of CEP sensitivity versus pump-probe delay is shown to be a highly selective measure for instantaneous structural changes in the lattice, providing an approach for ultrafast multidimensional HHG spectroscopy. Remarkably, the obtained temporal resolution for phonon dynamics is ∼1 femtosecond, which is much shorter than the probe pulse duration because of the inherent subcycle contrast mechanism. Our work paves the way toward routes of probing phonons and ultrafast material structural changes with subcycle temporal resolution and provides a mechanism for controlling the HHG spectrum.

Keywords: HHG; nonlinear optics; phonons; pump-robe spectroscopy; ultrafast spectroscopy.

Fig. 1.

Schematic illustration of the pump-probe HHG setup. (A) A THz pulse excites coherent phonon dynamics in hBN, which are subsequently probed by an intense IR pulse that is polarized in the hBN plane and drives HHG. The harmonic yield is measured with respect to the pump-probe delay and optionally also with respect to the IR pulse CEP and polarization. (B) Illustration of hBN structure and explored phononic landscape. Ionic vibrations along the y axis represent LO modes, while motion out of plane represents ZO modes. The HHG yield is modulated due to changes in the lattice structure. For the LO mode, when the B–N bonds along the y axis are compressed, the second set of B–N bonds in the unit cell (that are rotated by 120° from the y axis) are stretched, and vice versa.

Fig. 2.

Effects of coherent phonons on HHG spectra. (A) HHG spectra with and without pre-excited LO phonon motion for IR laser polarized along the x axis (transverse to phonon motion). The HHG spectra in both cases are shifted from one another for clarity. (B) HHG spectra from incoherent LO phonon case compared to the phonon-free response. The incoherent case was calculated by averaging over a full cycle of the pump-probe delay. Gray lines indicate integer harmonic orders of the IR photon energy, and all spectra are presented in log scale. For the calculation, the LO mode was excited with an amplitude of 5.7% of the lattice parameter for bond stretching and 4.95% of the lattice parameter for bond compression.

Fig. 3.

Temporally resolved HHG response with active LO phonons (y-axis motion). (A) HHG spectra versus pump-probe delay for x-polarized IR probe pulse. (B) Same as A, but for y-polarized IR probe pulse. The spectra are presented in log scale, and the HHG cutoff for the first plateau is indicated with dashed black lines. (C) Normalized HHG yield integrated over the plateau region versus the pump-probe delay for IR pulse polarized along the x axis and y axis. For the calculations, the LO mode was excited with an amplitude of 5.7% of the lattice parameter for bond stretching and 4.95% of the lattice parameter for bond compression. Dashed black line in C indicates the correspondence between the enhancement and suppression of the HHG yield, depending on the polarization axis of the IR pulse.

Fig. 4.

Polarization-resolved HHG response in ion-frozen hBN corresponding to the instantaneous lattice structure. (A) HHG plateau-integrated yield versus the laser polarization axis for the static lattice in the equilibrium geometry (due to the 3-fold rotational symmetry, only 120° are presented). (B) Same as A, but where the ions are statically frozen at a position with 4.95% bond compression for the B–N bond along the y axis. (C) Same as B, but where the ions are statically frozen at a position of 5.7% bond stretching for the B–N bond along the y axis. B and C correspond to the maximally displaced lattice structures along the LO phonon dynamics in Fig. 3C, where due to the reduced 2-fold rotational symmetry, only 180° are presented. (Insets) Schematic representations of the hBN structures (displacements of atoms in B and C are exaggerated for clarity). The axis system denotes the angle of the laser in-plane polarization (from the x axis in the geometries plotted). Each plot is normalized to maximal yield for clarity.

Fig. 5.

CEP-dependent HHG response for LO-driven coherent phonon dynamics. (A) CEP-dependent HHG spectra for the phonon-free static lattice in the equilibrium geometry (in log scale). No CEP dependence is present in the phonon-free case. (B) Same as A, but for the LO-pumped system for the pump-probe delay of 10.1 fs, showing onset of CEP sensitivity. (C) Illustration for the origin of the phonon-induced CEP sensitivity: the driving vector potential is illustrated for ϕ_CEP = 0 and ϕ_CEP=π/2 in arbitrary units (in blue) and is presented in the same scale for the laser envelope function (green) and the phonon trajectory (red). A phase shift of π/2 in the CEP creates a temporal window of a quarter of a cycle that determines if the peak electric field is temporally aligned with the moment of maximal bond compression or if it is offset, leading to CEP sensitivity (black dashed line indicates this moment in time). (D) Degree of CEP sensitivity (η) in the phonon-pumped system versus pump-probe delay (blue line), on the same scale as the temporal distance between the moment of phonon-induced maximal B–N bond compression and the (Right) peak of the laser envelope, δt (green, corresponding to the y axis). The IR probe laser polarization is along the x axis, and the LO phonon is excited just as in Fig. 3. The red line indicates η for the phonon-free system as a reference. The dashed black line indicates the correspondence between the moment of maximal bond compression and the peak in η.