We introduce a group-theoretical extension of the Dicke model which describes an ensemble of two-level atoms interacting with a finite radiation field. The latter is described by a spin model whose main feature is that it possesses a maximum number of excitations. The approach adopted here leads to a nonlinear extension of the Dicke model that takes into account both an intensity dependent coupling between the atoms and the radiation field, and an additional nonlinear Kerr-like or P\"{o}sch-Teller-like oscillator term, depending on the degree of nonlinearity. We use the energy surface minimization method to demonstrate that the extended Dicke model exhibits a quantum phase transition, and we analyze its dependence upon the maximum number of excitations of the model. Our analysis is carried out via three methods: through mean-field analysis (i.e. by using the tensor product of coherent states), by using parity-preserving symmetry-adapted states (using the critical values obtained in the mean-field analysis and numerically minimizing the energy surface) and by means of the exact quantum solution (i.e. by numerically diagonalizing the Hamiltonian). Possible connections with the $qp$-deformed algebras are also discussed.

中文翻译：

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与有限辐射场相互作用的两能级原子的量子相变

我们介绍了 Dicke 模型的群论扩展，该模型描述了与有限辐射场相互作用的两能级原子的系综。后者由自旋模型描述，其主要特征是它拥有最大数量的激发。这里采用的方法导致 Dicke 模型的非线性扩展，该模型考虑了原子和辐射场之间的强度相关耦合，以及额外的非线性 Kerr 类或 P\"{o}sch-Teller 类振荡器项，取决于非线性程度。我们使用能量表面最小化方法来证明扩展的 Dicke 模型表现出量子相变，我们分析了它对模型最大激发数的依赖性。我们的分析是通过三种方法：通过平均场分析（i. e. 通过使用相干态的张量积），通过使用保持奇偶校验的对称适应状态（使用在平均场分析中获得的临界值并在数值上最小化能量表面）和通过精确的量子解（即通过数值计算）对角化哈密顿量）。还讨论了与 $qp$ 变形代数的可能联系。