Control of two-phonon correlations and the mechanism of high-wavevector phonon generation by ultrafast light pulses

Impulsive optical excitation can generate both coherent and squeezed phonons through first- and second-order Raman-like processes. The expectation value of the phonon displacement $\langle u_q\rangle$ oscillates at the phonon mode frequency for the coherent state but remains zero for a pure squeezed state. In contrast, both show oscillations in $\langle |u_q{|}^2\rangle$ at twice the phonon mode frequency. Therefore it can be difficult to distinguish them in a second-order measurement of the displacements as is typical in x-ray diffuse scattering. Here we demonstrate a simple method to distinguish the generation mechanism by measurement of the diffuse scattering following double-impulsive excitation. We find in the case of Ge and GaAs that the generation of large wavevector phonons spanning the Brillouin zone is dominated by a second-order process.

Figure 1

Schematic of sudden excitation and deexcitation of coherent and squeezed modes, where $\omega$ is the phonon frequency. (a) Phase-space diagram of a phonon mode subjected to an impulse, generating a coherent state. An identical impulse at time $\pi /\omega$ later returns the mode to its original state. (b) Phonon mode of the same frequency subjected to a displacement-dependent impulse, generating a squeezed state. A second such displacement-dependent impulse at time $\pi /2\omega$ will return the mode to its original state. Note that in both cases, $\langle {\left|u\right|}^2\rangle$ oscillates at twice the phonon frequency. However, the delay between the two impulsive perturbations that will return the mode to its original state is different for the coherent and squeezed cases. The dashed and solid lines show the evolution of $\langle {\left|u\right|}^2\rangle$ in the absence and presence of the second pump pulse, respectively.

Figure 2

Dispersion of $\langle |u_{\mathbf{q},\lambda }{|}^2\rangle$, matching twice the transverse-acoustic phonon branches, obtained by Fourier transform of temporal oscillations in the diffuse x-ray intensity following single and double optical pump excitations for (a)–(c) Ge, (d)–(f) GaAs, and (g)–(i) InSb. (b), (e), and (h) are with a single pump pulse, whereas (c), (f), and (i) are with two pumps where the pump-pump delay is labeled $\tau$. The measured dispersions closely match twice the calculated acoustic phonon dispersions of (a), (d), and (g) using the force constants from Refs. [21, 22, 23] indicating our measurement is second order as expected. Line weights in (a), (d), and (g) indicate the squared structure factor and thus our expected sensitivity to the branch in the chosen geometry. The green horizontal lines in (c), (f), and (i) show frequencies which should be stopped by the second pulse for squeezed phonons. All measured dispersions have an identical color scale, shown in (b). The horizontal axis represents the scattering vector, the three components of which are shown in the top solid blue, dashed red, and dot-dashed green curves (in reciprocal lattice units). Vertical hashed bars are regions with no detector pixels. Note that the frequency never reaches zero because we did not probe the $\mathrm{\Gamma }$ point. This was by design to avoid Bragg peaks whose intensities would exceed the limits of the detector.

Figure 3

Oscillation amplitude of phonon-phonon correlation (which oscillates at twice the phonon frequency, as seen in Fig. 2) following two-pulse excitation. The data is binned according to frequency and normalized to single-pulse excitation amplitude. The pump-pump delay is labeled $\tau$. Solid green and dashed blue curves show expected trends for impulsive excitation for squeezed and coherent states, respectively, with no free parameters.