Resonant squeezing and the anharmonic decay of coherent phonons

We show that the anharmonic decay of large-amplitude coherent phonons in a solid generates strongly enhanced squeezing of the phonon modes near points of the Brillouin zone where energy conservation in the three-phonon decay process is satisfied. The squeezing process leads to temporal oscillations of the mean-square displacement of target modes in resonance with the coherent phonon, which are characteristic of coherent phonon decay and do not occur in the decay of a phonon in a well-defined number state. For realistic material parameters of optically excited group-V semimetals, we predict that this squeezing results in strongly enhanced oscillations of the x-ray diffuse scattering intensity at sharply defined values of the x-ray momentum transfer. Numerical simulations of the phonon dynamics and the x-ray diffuse scattering in optically excited bismuth, using harmonic and anharmonic force parameters calculated with constrained density functional theory, demonstrate oscillations of the diffuse scattering intensity of magnitude 10%–20% of the thermal background at points of the Brillouin zone, where resonance occurs. Such oscillations should be observable using time-resolved optical-pump and x-ray-probe facilities available at current x-ray free-electron laser sources.

Figure 1

Calculated x-ray diffuse scattering intensity $I\left(q\right)$ [Eq. (9)] versus time delay after optical pump pulse for selected scattering wave vectors along the $\mathrm{\Sigma }$ line (from $\mathrm{\Gamma }$ through $K$ to $X$) in the Brillouin zone, $\mathrm{\Delta }\mathbf{k}=\mathbf{q}=\xi ({\mathbf{g}}_1-{\mathbf{g}}_3)$, where ${\mathbf{g}}_i$ are the basis vectors of the rhombohedral reciprocal lattice, for values of $\xi =0.05$ (bottom curve), 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, and 0.5 (top curve).

Figure 2

Upper panel: Calculated equilibrium phonon bands in Bi along the $\mathrm{\Sigma }$ line in the Brillouin zone. The solid lines indicate LO and LA branches, for which overtone ($2\omega$) and combination (${\omega }_{\mathrm{LO}}+{\omega }_{\mathrm{LA}}$) frequencies are indicated by dotted lines in the lower panel. Lower panel: Squeezing oscillation spectrum as a function of frequency and momentum along the $\mathrm{\Sigma }$ line, calculated by Fourier transform of the calculated x-ray diffuse scattering for each momentum, $\mathbf{q}=\xi ({\mathbf{g}}_1-{\mathbf{g}}_3)$ (see Fig. 1). Color corresponds to the ${\log }_{10}$ of the intensity, as indicated on the color scale.