Quantum Entanglement in the Rabi Model with the Presence of the \(A^{2}\) Term 

The quantum Rabi model (QRM) is used to describe the light-matter interaction at the quantum level in Cavity Quantum Electrodynamics (Cavity QED). It consists of a two-level system (atom or qubit) coupled to a single-mode quantum field. Introducing an atom into a cavity alters the electromagnetic mode configuration within it. In the realm of Cavity QED, a notable consequence of this alteration is the emergence of a gauge-dependent diamagnetic term referred to as the \(A^{2}\) $A^2$ contribution. In this study, we comparatively analyze the behaviors of the QRM and the influence of the \(A^{2}\) $A^2$ term in the light-matter quantum Hamiltonian by examining the energy spectrum properties in the strong-coupling regime. We then investigate the ground state of the system, measuring its nonclassical properties via the Wigner distribution function for different photon number distributions in Fock space. Finally, we calculate the quantum entanglement in the ground state using the Von Neumann entropy. Our findings reveal that the \(A^{2}\) $A^2$ term and the number of cavity Fock states, N, significantly impact the amount of the quantum entanglement, highlighting their pivotal role.