Third-order natural Non-Hermitian quantization of quad-photon Rabi oscillation in an atomic ensemble

•This paper provides an overview of the principles of the four-photon process based on 780 nm, 780 nm, 795 nm, 795 nm and the corresponding energy levels. The effects of single-dressing, parallel double- dressing, cascade double- dressing, and nested double- dressing effects on this nonlinear process are described mathematically, and the patterns of variation of the eigenvalues, the seventh-order nonlinear polarizabilities, and Rcc under the influence of different dressing field strengths are analyzed. On the basis of the analytical results already obtained, the commonalities and differences between the real and imaginary parts in the three cases are discussed. The correspondence between the number of eigenvalues and the number of peaks in the intensity of the seventh-order nonlinear polarizability and the trend of the Rcc oscillation period is compared, thus revealing the influence of the dressing field on the PT system in the time and frequency domains, and in the real and imaginary parts.•For single-dressing and parallel double-dressing, there is no interaction between the dressing fields, the dressing fields correspond to their respective distinct generating fields, and the eigenvalue splitting of the real and imaginary parts shows symmetric breaking; for cascade double-dressing and nested double-dressing, there is interaction between the dressing fields, they correspond to the same generating field, and the two dressing fields of the former have the same relation with respect to the generating field, while the latter have a nested order of precedence and the interaction between the two dressing fields of the nested double-dressing is more obvious. The single dressing effect’s eigenvalue undergoes a second-order splitting near the EP. Among the three double dressing effects, the eigenvalues of parallel double dressing undergo second-order splitting near the EP as in the case of single dressing, while cascade double dressing and nested double dressing show third-order splitting. In the two double-dressing effects with third-order splitting of eigenvalues, there are interactions between the two dressing fields, the modulation effect of the external dressing field is affected by the internal dressing field, and the interaction between the two dressing fields of the nested double- dressing is stronger, which is manifested in the fact that the external dressing field is unable to split the eigenvalues when the internal dressing field’s dressing Rabi frequen