Time‐dependent photon statistics in variable media

• Published in: Mathematical methods in the applied sciences 2019-10, Vol.42 (15), p.5040-5051
• Main Authors: Kryuchkov, Sergey I., Suazo, Erwin, Suslov, Sergei K.
• Format: Article
• Language: English
• Subjects: Amplitudes; closed and approximate solutions to the Schrödinger equation; Computer algebra; Equations of motion; Ermakov equation; generalized harmonic oscillators; Heisenberg equations of motion; Hypergeometric functions; Operators (mathematics); partial differential equations; photon statistics; Photons; Program verification (computers); Quantum electrodynamics; Time dependence
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.5285

We find explicit solutions of the Heisenberg equations of motion for a quadratic Hamiltonian, which describes a generic model of variable media in the case of multiparameter squeezed input photon configuration. The corresponding probability amplitudes and photon statistics are also derived in the Schrödinger picture in an operator setting of the quantum electrodynamics; a comparison discussion is made in Heisenberg's picture as well. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. The time‐dependent photon probability amplitudes with respect to the Fock basis are indeed derived in an operator form. Further, explicit expressions for the matrix elements of the displacement and squeeze operators are derived in terms of hypergeometric functions and solutions of a certain Ermakov‐type system. In the Supporting Information, we provide a computer algebra verification of the derivation of the Ermakov‐system and of the solutions of the Heisenberg equations.