Strong-field Rydberg state excitation (RSE) is a significant physical phenomenon observed when atoms or molecules are exposed to intense laser fields. Numerous theoretical and experimental studies have explored the RSE process to understand its underlying physical mechanisms. However, existing theoretical methods, such as solving the time-dependent Schr?dinger equation (TDSE), while accurately describing the physical process, do not provide a clear physical picture. Consequently, the mechanism of RSE, particularly in elliptically polarized laser fields, remains unclear. To better investigate the RSE in strong fields, we develop a quantum model for resonance excitation based on S-matrix theory. This model describes the electron transition from the ground state to the field-dressed Rydberg state in strong laser fields, allowing for a detailed investigation of the physical characteristics of strong-field RSE.

A quantum model based on S-matrix theory is developed to describe the transition of electrons from the ground state to the Rydberg state in intense laser fields, using the length gauge. Furthermore, we consider the effect of the intense laser field on the Rydberg state, treating the field-dressed Rydberg state as the final state within an elliptically polarized laser field characterized by a sine-square envelope containing 30 optical cycles.

We first use the quantum model to calculate the evolution of the excitation rates of the 4s, 4p, 4d, and 4f states at different laser intensities for laser pulses with an ellipticity of 0.5 (Fig. 1) and provide corresponding explanations. Next, we investigate the ellipticity-dependent excitation rates of different excited states at two laser intensities of 1.4×1012W/cm2 and 6.82×1013W/cm2. When the laser intensity is 1.4×1012W/cm2 (two-photon transition), the excitation rate of the 4s state decreases with increasing ellipticity, while the excitation rate of the 4d state increases (Fig. 2). These findings are consistent with both TDSE results and the perturbation theory presented in a recent paper. When the laser intensity is 6.82×1013W/cm2 (three-photon transition), the excitation rate of the 4p state initially increases, reaching a maximum at an ellipticity of 0.3, and then decreases as ellipticity increases. In contrast, the excitation rate of the 4f state increases with increasing ellipticity [Fig. 2(b)]. Our calculations reveal that at higher laser intensities, the influence of the laser field on the spatial phase distribution of Rydberg states becomes significant, while at lower intensities, this influence can be ignored [Figs. 2(c) and 2(d)]. In addition, we present the excitation rates of excited states with different magnetic quantum numbers at a laser intensity of 6.82×1013W/cm2 (Figs. 3 and 4), which demonstrate the transition selection rules for orbital angular momentum quantum numbers and magnetic quantum numbers. We also provide a theoretical analysis using a circularly polarized laser pulse as an example.

We develop a quantum model describing electron transitions from the ground state to the Rydberg state in intense laser fields based on S-matrix theory. Using this model, we calculate the ellipticity-dependent excitation probabilities of a lithium atom to various Rydberg states in laser fields. For two-photon resonance transitions at relatively low laser intensities, the excitation process described by the quantum model aligns with perturbation theory, where the influence of the laser field on the excited state can be ignored. However, for three-photon resonance transitions at higher intensities, the laser field not only alters the energy levels of the Rydberg state but also affects its spatial phase distribution. Furthermore, theoretical analysis reveals the characteristics of the transition selection rule for angular momentum and magnetic quantum numbers in the multi-photon excitation process in elliptically polarized fields: if the electron absorbs an even (odd) number of photons during resonance excitation, the parity of the orbital angular momentum (magnetic quantum number) of the initial state will match (differ from) that of the Rydberg state. Our work lays the foundation for a deeper understanding of the resonance excitation process.