Nonperturbative regime of low-order harmonic generation in intense low-frequency laser field

We find the atomic response to the intense femtosecond laser pulse via solving numerically the three-dimensional non-stationary Schr\"odinger equation (TDSE) for a model atom and calculating its dipole moment. For weak quasi-static fields, the response is well described by a perturbation approach, but for intensities higher than about $0.6 \, \, 10^{14}$ W/cm$^2$ the accuracy of this description is unsatisfactory, regardless of the order of non-linearity taken into account. We suggest fitting the numerical TDSE solution results with a Pad\'e expansion, and show that this approximation describes the response well both in the perturbative regime and beyond it for intensities approximately up to $1.4 \, \, 10^{14}$ W/cm$^2$. To consider the non-perturbative nonlinearity beyond the quasi-static limit we use the model of nonlinear oscillator with the restoring force defined by the found Pad\'e expression. Our model fails to predict the behaviour of the nonlinear refractive index in the nonperturbative domain, but it describes well the nonperturbative growth of the efficiency with the laser intensity for other nonlinear optical processes, namely, the third and fifth harmonic generation in the IR field and the optical rectification in a two-color field.