Biorthogonal quantum mechanics for non-Hermitian multimode and multiphoton Jaynes–Cummings models

We develop a biorthogonal formalism for non-Hermitian multimode and multiphoton Jaynes–Cummings models. For these models, we define supersymmetric generators, which are especially convenient for diagonalizing the Hamiltonians. The Hamiltonian and its adjoint are expressed in terms of supersymmetric generators having the Lie superalgebra properties. The method consists in using a similarity dressing operator that maps onto spaces suitable for diagonalizing Hamiltonians even in an infinite-dimensional Hilbert space. We then successfully solve the eigenproblems related to the Hamiltonian and its adjoint. For each model, the eigenvalues are real, while the eigenstates do not form a set of orthogonal vectors. We then introduce the biorthogonality formalism to construct a consistent theory.