The analytically exact solutions to the two-photon Rabi–Stark model are given by using the Bogoliubov operators approach. Transcendental functions responsible for the exact solutions are derived. The zeros of the transcendental functions reproduce completely the regular spectra. The first-order quantum phase transitions are also detected by the pole structure of the derived transcendental functions, similar to the one-photon Rabi–Stark model, different from the two-photon Rabi model where the energy levels for the ground-state and the first excited state do not cross. The spectra collapse characteristics resemble those in two-photon Rabi model. Some low lying levels can split off from the collapse energy, which is different from the one-photon Rabi–Stark model in which all levels collapse.

中文翻译：

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两光子拉比–斯塔克模型

使用Bogoliubov算子方法给出了双光子Rabi–Stark模型的精确解析解。得出负责精确解决方案的先验功能。先验函数的零完全重现了规则光谱。一阶量子相变也可以通过导出的超越函数的极点结构来检测，类似于单光子Rabi–Stark模型，不同于两光子Rabi模型，在该模型中，基态的能级和首先兴奋状态不要交叉。光谱塌陷特征类似于两光子拉比模型中的那些。一些低水平能从崩塌能量中分裂出来，这不同于单光子拉比-斯塔克模型中所有水平都崩塌的模型。