Complex saddle point analysis of solid high order harmonic generation and reconstruction of dephasing time

The calculation of laser-induced transition amplitudes using asymptotic methods provides an important complement to the numerical solution of the time-dependent Schrödinger equation in the study of light–matter interactions. Within this framework, solving the saddle-point equations enables an analytical decomposition of the quantum–mechanical transition amplitude into a coherent superposition of quantum orbitals. This is one of the key approaches for quantitatively and analytically investigating transition dynamics among different quantum channels. This method has been successfully applied to the study of gas-phase high-order harmonic generation (HHG), above-threshold ionization (ATI), and high-order ATI. However, for solid-state nonlinear optical phenomena such as solid HHG induced by strong laser fields, previous studies have not been able to fully solve the six-dimensional complex saddle-point equations. These works either entirely or partially neglects the imaginary components of the saddle-point solutions. In this work, we solve, for the first time, the full set of six-dimensional complex saddle-point equations that describe the interaction between solids and intense laser fields. In particular, we focus on the influence of the dephasing time T2—a distinguishing feature of solid-state HHG compared to gas-phase HHG—on the saddle-point solutions, especially their imaginary parts. Our model reveals that the imaginary parts of the ionization and photon emission times, which correspond to the light-induced tunneling time delays from different quantum orbitals, exhibit increasing divergence as the dephasing time T2 decreases. Consequently, the dephasing rate is encoded in the HHG spectra, offering a unique opportunity to reconstruct T2 from experimental HHG spectra using machine learning techniques. Our study demonstrates that, for the theoretical investigation of electron dynamics in solids under strong laser fields, both the imaginary parts of the saddle-point solutions and the dephasing time T2 are important—particularly in the study of light-induced quantum tunneling processes in solids, where their contributions cannot be neglected. In addition, our reconstruction algorithm offers a promising approach for analytically studying light-induced carrier transitions between different energy bands.