Journal of Russian Laser Research, Volume 39, Number 1, January, 2018 A NEW FIRST-ORDER PHASE TRANSITION FOR AN EXTENDED JAYNES–CUMMINGS–DICKE MODEL WITH A HIGH-FINESSE OPTICAL CAVITY IN THE BEC SYSTEM † Ahmed Salah, 1,2* A. S. Abdel-Rady, 1 Abdel-Nasser A. Osman, 1 and Samia S. A. Hassan 2 1 Mathematics Department, Faculty of Science South Valley University Qena, Egypt 2 Mathematics and Theoretical Physics Department Nuclear Research Center (EAEA) Cairo, Egypt * Corresponding author e-mails: asalah3020 @ gmail.com ahmed.salah @ eaea.org.eg Abstract We present a two-level atomic Bose–Einstein condensate (BEC) with dispersion, which is coupled to a high-finesse optical cavity. We call this model the extended Jaynes–Cummings–Dicke (JC-Dicke) model and introduce an effective Hamiltonian for this system. From the direct product of Heisenberg– Weyl (HW) coherent states for the field and U (2) coherent states for the matter, we obtain the potential energy surface of the system. Within the framework of the mean-field approach, we evaluate the variational energy as the expectation value of the Hamiltonian for the considered state. We investigate numerically the quantum phase transition and the Berry phase for this system. We find the influence of the atom–atom interactions on the quantum phase transition point and obtain a new phase transition occurring when the microwave amplitude changes. Furthermore, we observe that the coherent atoms not only shift the phase transition point but also affect the macroscopic excitations in the superradiant phase. Keywords: two-level atomic Bose–Einstein condensate (BEC), quantum phase transitions (QPTs), Berry phase. 1. Introduction Many models are employed to describe the interaction between the matter and light such as the Dicke model and the Jaynes–Cummings model. The Dicke model (DM) was introduced by Robert H. Dicke in 1954 [1]; it describes the interaction of a large number of two-level systems (e.g., atoms) with a single optical mode and is used to illustrate the importance of collective effects of superradiance. The Jaynes– Cummings Model (JCM) was elaborated by E. T. Jaynes and F. W. Cummings in 1963 [2]; it describes the interaction between the coherent coupling of a single two-level atom and a single mode of the quantized † The numerical results are in full agreement with the results of our paper published in [3] where we studied the same model using a different coherent state. 28 Manuscript submitted by the authors in English on December 25, 2017. 1071-2836/18/3901-0028 c  2018 Springer Science+Business Media, LLC DOI 10.1007/s10946-018-9686-4