Irregular oscillatory patterns in the early-time region of coherent phonon generation in silicon

Coherent phonon (CP) generation in an undoped Si crystal is theoretically investigated to shed light on unexplored quantum-mechanical effects in the early-time region immediately after the irradiation of ultrashort laser pulses. We examine time signals attributed to an induced charge density of an ionic core, placing the focus on the effects of the Rabi frequency ${\mathrm{\Omega }}_{0cv}$ on the signals; this frequency corresponds to the peak electric-field of the pulse. It is found that at specific ${\mathrm{\Omega }}_{0cv}$'s, where the energy of plasmon caused by photoexcited carriers coincides with the longitudinal-optical phonon energy, the energetically resonant interaction between these two modes leads to striking anticrossings, revealing irregular oscillations with anomalously enhanced amplitudes in the observed time signals. Also, the oscillatory pattern is subject to the Rabi flopping of the excited carrier density that is controlled by ${\mathrm{\Omega }}_{0cv}$. These findings show that the early-time region is enriched with quantum-mechanical effects inherent in the CP generation, though experimental signals are more or less masked by the so-called coherent artifact due to nonlinear optical effects.

Figure 1

The scheme of the CP generation dynamics. $\mathrm{\Delta }={\omega }_0-E_g$ represents the detuning with $E_g$ as the direct band gap of Si at the $\mathrm{\Gamma }$ point. For more detail, consult the text.

Figure 2

(a) ${\mathrm{\Theta }}_{\mathit{q}}\left(\tau \right)$ and (b) $A_{\mathit{q}}\left(\tau \right)$ as a function of ${\mathrm{\Omega }}_{0cv}$ for $\mathrm{\Delta }={\mathrm{\Delta }}_0$ (red squares) and ${\mathrm{\Delta }}_{-}$ (blue diamonds). (c) The real parts of the adiabatic energy $E_{\mathit{q}j}\left(\tau \right)$ as a function of ${\mathrm{\Omega }}_{0cv}$ for $j=ph$ and $pl$; the eigenvalues mostly dominated by the phonon and plasmon modes are represented by solid and open red squares, respectively. ${\omega }_{\mathit{q}pl}\left(\tau \right)$ values for $\mathrm{\Delta }={\mathrm{\Delta }}_0$ (red dashed line) and ${\mathrm{\Delta }}_{-}$ (blue dashed line) are also included alongside ${\omega }_{\mathit{q}}=63$ meV (green dashed line). (d) The enlarged view of panel (c) around $\mathrm{Re}\left[E_{\mathit{q}ph}\left(\tau \right)\right]={\omega }_{\mathit{q}}$. In all panels, $\tau =20$ fs, and the positions of ${\mathrm{\Omega }}_{0cv}^{\left(C1\right)}$ and ${\mathrm{\Omega }}_{0cv}^{\left(C2\right)}$ are denoted by vertical brown dotted lines. The calculated values in panels (a), (b), and (d) are connected by solid and dashed lines in order to aid the presentation.

Figure 3

(a) ${\theta }_{\mathit{q}}$ and (b) $A_{\mathit{q}}^0$ as a function of ${\mathrm{\Omega }}_{0cv}$ for $\mathrm{\Delta }={\mathrm{\Delta }}_0$ (red squares) and ${\mathrm{\Delta }}_{-}$ (blue diamonds) along with the experimental data of ${\theta }_{\mathit{q}}$ (green circles) [9].

Figure 4

$Q_{\mathit{q}}\left(\tau \right)$ as a function of $\tau$ in the ETR at the specific ${\mathrm{\Omega }}_{0cv}$'s given in panels (a)–(d) with $\mathrm{\Delta }={\mathrm{\Delta }}_0$.