Quantum Jaynes–Cummings model for a two-level system with effects of parametric time- dependences

An approach to exact quantum solutions of the time-dependent two energy level Jaynes–Cummings model with an imaginary photon process is represented in this work. The Lewis–Riesenfeld invariant treatment and the unitary transformation method are used for this purpose. The original Schrödinger equation is reduced to an equivalent solvable one through unitary transformations by using suitable unitary operators. The reduced equation corresponds to a simpler Hamiltonian which is written as a linear combination of the generators of the reduced-dimensional SU(2) algebra. A Hermitian invariant operator is constructed based on the same algebraic formulation and its instantaneous eigenfunctions are obtained. By utilizing such eigenfunctions, the complete quantum wave functions of the system are evaluated. Such wave functions are necessary when we analyze the quantum characteristics of the system.