Quantum Phase Properties in Collective Three-Level V-Type System with Diamagnetic Term

In this paper, we present a rigorous investigation of the quantum phase transitions (QPTs) for a model which describes the degenerate collective regime of a three-level V-configuration atoms Bose-Einstein condensate (BEC). We consider that the three-level atoms of a BEC coupled to an optical resonator (cavity) with high finesse in the presence of a diamagnetic term, i. e., a system of N-identical three-level atoms interacting with a one-mode quantized electromagnetic cavity field. Also, the different components of this system are labeled by the different phase factors. By using the Glauber’s coherent states for the field, we calculate the free energy and study the finite-temperature phase transition for this model in the thermodynamic limit (N → ∞). Also, the potential energy surfaces constructed by taking the expectation value with respect to the direct product for a test state a direct product of coherent Heisenberg-Weyl HW(1)-states (for the electromagnetic field ), and U(3) (for the atomic field, i. e., matter contribution) coherent states. Moreover, the scaled ground-state energy is obtained by taking the expectation value of an effective Hamiltonian in the framework of the mean-field approach. In the thermodynamic limit, the energy surface takes a simple form for a direct description of the phase transitions. The relation between the QPTs and the symmetry physics are established and the QPTs for this model are investigated numerically. The properties of both stability and the equilibrium are calculated using the catastrophe theory. We notice that the second-order phase transitions (superradiant (SR)) with multi-level systems might be achievable in a wide range of physical systems, especially those where it is possible to engineer the spec-tra and oscillator strengths.