Vacuum-induced \({\mathcal {P}}{\mathcal {T}}\)-symmetry in one- and two-dimensional optical lattices

We investigate a system comprising three-level atoms confined within one- and two-dimensional optical lattices, confined inside the optical cavities. The spatial distribution of atoms follows a Gaussian profile, enabling the realization of parity-time (\({\mathcal {P}}{\mathcal {T}}\))-symmetry through controlled vacuum-induced field modulation. Additionally, we introduce a microwave field to couple the atomic ground states, introducing an additional degree of control over the system dynamics. By carefully tuning the probe field detuning in the presence of both vacuum and microwave fields, we derive the precise conditions necessary for achieving \({\mathcal {P}}{\mathcal {T}}\)-symmetry. Furthermore, we investigate the unique reflection and transmission properties of the system, opening potential applications in advanced optical devices such as non-reciprocal photonic components and quantum light manipulation platforms. Our findings provide a deeper understanding of symmetry-controlled atom-light interactions, paving the way for novel quantum optical technologies.