Quantum-optical theory of the few femtosecond nonlinear optical response of Drude metals with a non-parabolic conduction band

TL;DR

The paper addresses the few-femtosecond nonlinear optical response of Drude metals with a non-parabolic conduction band, focusing on transparent conducting oxides under broadband excitation. It develops an energy-space density-matrix framework that tracks populations, polarization, and quantum coherences, and maps from $k$-space to $\mathcal{E}$-space using the Kane dispersion $\hbar k(\mathcal{E}) = \sqrt{2m_e^* \mathcal{E}(1+C\mathcal{E})}$ to reduce to a 1D problem while preserving physics. Key findings include clear signatures of coherence in net absorption dynamics, the emergence of strong excited-state absorption yielding superlinear absorption, and pronounced local-field effects such as duration elongation and spectral broadening under intense broadband pulses; spontaneous emission is negligible in this regime. The work provides a computationally efficient, unified framework for nonlinear light-matter interaction in dispersive, low-density electron systems driven far from equilibrium, enabling quantitative interpretation of experiments and facilitating cross-comparisons with hydrodynamic models and interband extensions.

Abstract

We develop an energy-space density matrix framework to investigate the interaction of extremely short optical pulses (ESPs) with transparent conducting oxides (TCOs). This approach captures not only electron populations, material polarization, and the permittivity, but also the quantum coherences between states. Compared to traditional momentum-space models, the energy-space formulation offers substantial computational simplification while retaining accuracy. Building on but going beyond the scope of Ref.~\cite{single_cycle_nlty_Letter}, we focus on dynamical features previously unexplored. Our formulation reveals clear signatures of quantum coherence in the net absorption dynamics and highlights the emergence of strong excited-state absorption under intense excitation. It also clarifies that spontaneous emission can be neglected in this regime. Furthermore, we investigate the influence of pump pulse intensity on the local field's duration, spectral broadening and shift, and phase induced by carrier dynamics, highlighting the absorptive nature of the nonlinear response. Our results provide a unified framework for understanding nonlinear light-matter interaction in dispersive, low-density electron systems driven far from equilibrium by intense broadband excitation.

Figures (11)

• Figure 1: (a) The square of the absolute value of (the angular average of) the energy space transition dipole matrix elements $|D(k(\mathcal{E}),k(\mathcal{E}'),L)|^2$ [C$^2$ m$^2$] calculated using Eq. \ref{['eq:d2D']} for $L = 4$ nm. (b) A cross-section of $|D(k(\mathcal{E}),k(\mathcal{E}'),L)|^2$ (blue solid line, labeled by the blue dashed line in (a)) and $|p(k(\mathcal{E}),k(\mathcal{E}'),L)|^2$ (orange solid line) for $\mathcal{E}' = \mathcal{E}_F$.
• Figure 2: (a) Schematics of the configuration considered here. (b) The real (blue) and imaginary (red) part of the permittivity of the ITO NP, and the absolute value of the prefactor of the linear quasi-static solution for the local field inside a subwavelength ITO sphere (green) at $T_e = 300$ K. The vertical dashed line represents the central frequency ($175$ THz) used in the simulations.
• Figure 3: (a)-(c) The $\textrm{photon-e}$ term (blue), $\textrm{e-e}$ (orange), and $\textrm{e-ph}$ (green) terms for $E_0 = 0.6$ V/nm at $t = -3.2$ fs, $-0.5$ fs, and $2.2$ fs. (d) The color map of the photon-e term (Eq. \ref{['eq:dfdt_exc_E']}). The black dashed lines label the time sections of (a)-(c). The blue and orange dashed lines label the energy sections shown in (e). (e) The electron distribution dynamics at $\mathcal{E} = \mathcal{E}_F \pm \hbar \omega_0/8$ (blue and orange).
• Figure 4: (a)-(c) The Fourier transform of the photon-e term for $E_0 = 0.6$ V/nm, $1.2$ V/nm, and $2.4$ V/nm.
• Figure 5: The imaginary part of the density matrix for $E_0 = 1.2$ V/nm at (a) $t = -2.5$ fs and (b) $2.5$ fs, respectively.