Controlling Ultrafast Excitations in Germanium:The Role of Pump-Pulse Parameters and Multi-Photon Resonances

TL;DR

This paper addresses how ultrafast optical pulses populate electronic states in germanium, and how the residual carrier density and absorbed energy depend on pulse duration, amplitude, and photon energy. It introduces and applies the Dynamical Projective Operatorial Approach (DPOA), an efficient operator-based framework that yields band-resolved populations and decomposes excitations into multi-photon channels. The study reveals that two-photon processes typically dominate, with one- and three-photon channels becoming important only in specific parameter regimes, and it exposes non-monotonic, Rabi-like dynamics as pulse duration changes. The results provide practical guidelines for steering ultrafast excitations along selected multi-photon pathways and demonstrate a broadly applicable methodology for realistic multi-band semiconductors.

Abstract

We employ the Dynamical Projective Operatorial Approach (DPOA) to investigate the ultrafast optical excitations of germanium under intense, ultrashort pump pulses. The method has very low resource demand relative to many other available approaches and enables detailed calculation of the residual electron and hole populations induced by the pump pulse. It provides direct access to the energy distribution of excited carriers and to the total energy transferred to the system. By decomposing the response into contributions from different multi-photon resonant processes, we systematically study the dependence of excited-carrier density and absorbed energy on key pump-pulse parameters: duration, amplitude, and photon energy. Our results reveal a complex interplay between these parameters, governed by resonant Rabi-like dynamics and competition between different multi-photon absorption channels. For the studied germanium setup, we find that two-photon processes are generally dominant, while one- and three-photon channels become significant under specific conditions of pump-pulse frequency, duration, and intensity. This comprehensive analysis offers practical insights for optimizing ultrafast optical control in semiconductors by targeting specific multi-photon pathways.

Figures (5)

• Figure 1: Equilibrium band structure of germanium along the main high-symmetry lines.
• Figure 2: Energy distribution of residual excited-carrier density, $\rho^{\mathrm{res}}(\varepsilon)$, for different pump-pulse photon energies $\hbar\omega_{\mathrm{pu}}$. (a) Total distribution. (b), (c), (d) Contributions from one-, two-, and three-photon processes, respectively. Red (blue) color shows positive (negative) excess populations and corresponds to electron (hole) excitations. The vertical spacing between red and blue branches reflects the pump-pulse photon energy multiples, indicating resonant origins.
• Figure 3: Residual excited-carrier density, $\Delta N^{\mathrm{res}}$, (upper row) and residual absorbed energy per unit cell, $\mathcal{E}^{\mathrm{res}}$, (lower row) as functions of pump-pulse duration, $\tau_{\mathrm{pu}}$, and squared amplitude, $A_{0}^{2}$. (a, e) Total quantities. (b, f), (c, g), (d, h) Contributions from one-, two-, and three-photon processes, respectively.
• Figure 4: Residual excited-carrier density, $\Delta N^{\mathrm{res}}$, (upper row) and residual absorbed energy, $\mathcal{E}^{\mathrm{res}}$, (lower row) as functions of pump-pulse duration, $\tau_{\mathrm{pu}}$, and pump-pulse photon energy, $\hbar\omega_{\mathrm{pu}}$. Columns as in Fig. \ref{['fig:A2-tau']}.
• Figure 5: Residual excited-carrier density, $\Delta N^{\mathrm{res}}$, (upper row) and residual absorbed energy per unit cell, $\mathcal{E}^{\mathrm{res}}$, (lower row) as functions of pump-pulse photon energy $\hbar\omega_{\mathrm{pu}}$ and squared amplitude $A_{0}^{2}$. Columns as in Fig. \ref{['fig:A2-tau']}.