Direct Measurement of Anharmonic Decay Channels of a Coherent Phonon

We report channel-resolved measurements of the anharmonic coupling of the coherent $A_{1g}$ phonon in photoexcited bismuth to pairs of high wave vector acoustic phonons. The decay of a coherent phonon can be understood as a parametric resonance process whereby the atomic displacement periodically modulates the frequency of a broad continuum of modes. This coupling drives temporal oscillations in the phonon mean-square displacements at the $A_{1g}$ frequency that are observed across the Brillouin zone by femtosecond x-ray diffuse scattering. We extract anharmonic coupling constants between the $A_{1g}$ and several representative decay channels that are within an order of magnitude of density functional perturbation theory calculations.

Figure 1

(a) Phonon dispersion relation in bismuth along the $\mathit{q}=(\frac{1}{3}\xi \xi \xi )$ direction, illustrating a decay channel of an $A_{1g}$ phonon into a pair of LA phonons at $\mathit{q}$ and $-\mathit{q}$. (b) experimental signature of decay of the $A_{1g}$ phonon in bismuth by the channel shown in (a). The lower (blue) curve shows the relative intensity change of the (2 3 2) Bragg peak, which is proportional to the $A_{1g}$ mode amplitude. The upper (orange) curve shows the relative intensity change of the diffuse scattering $\mathrm{\Delta }I/I$ in a region near $\mathit{q}=(0.10.30.3)$ in the (0 1 1) zone (multiplied by 10). The black lines are simulations. The dashed line indicates a $\pi /2$ phase shift between the $A_{1g}$ mode and the target mode.

Figure 2

(a) Intensity map of the Fourier transform of the femtosecond diffuse scattering signal at $f=2.85\mathrm{THz}$. The red line is the path of the lineout shown in Fig. 3. The points labeled (1)–(4) are used to extract the coupling constants shown in Table 1. The blue and orange boxes outline the regions used to extract the time-domain traces in Fig. 1. The four brightest spots in (a) are due to oscillations of the structure factor from the $A_{1g}$ mode near Bragg peaks and are therefore not present in (b). Brillouin zones relevant to the text are labeled in (b). (b) Predicted intensity map using the dispersion relation of bismuth and the parametric resonance condition, with a uniform anharmonic coupling constant $g=1.0$. The red lines in (b) show the Brillouin zone boundaries.

Figure 3

(a) Lineout along the cut shown in red in Fig. 2. The intensity is scaled by ${\omega }^2$. The bright spot near $q=0$ at 2.85 THz is from the $A_{1g}$ mode. The second harmonic of the acoustic dispersion is plotted on top of the lineout. The $A_{1g}$ mode frequency is represented by a white dashed line. The squeezing signal is brightest where the phonon branches intersect the $A_{1g}$ frequency (red arrows). (b) Intensity at the 2.85 THz along the lineout shown in (a). Blue is experiment; red is the simulation as described in the text. Arrows point to the same positions along the cut as in (a). (c) The reciprocal lattice positions $(hkl)$ indices (in r.l.u) along the cut.