The DKP equation for a spin-1 boson in a magnetic field and the two-photon Jaynes–Cummings model

We consider the Duffin–Kemmer–Petiau (DKP) equation for a spin-1 particle moving in the plane under the action of a perpendicular Homogeneous Magnetic Field (HMF) using a fully algebraic method. Interestingly, we show that this quantum relativistic model can be exactly mapped onto the Two-Photon Jaynes-Cummings Model (TPJCM) so widely used in quantum optics to describe two-photon processes during the interaction of a two-level atom with a single-mode radiation field. The Hamiltonian of the equivalent TPJCM we obtain is pseudo-Hermitian, reinforcing the idea that pseudo-Hermitian models are physically relevant. Moreover, using the spectral method, we were able to work out exact closed forms for the associated metric operator and Hermitian Hamiltonian. It turns out that the latter is diagonal and therefore we identified the transformation we applied to calculate it from the TPJCM Hamiltonian as a Foldy-Wouthuysen transformation.