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 abstract 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.
- closed and approximate solutions to the Schrödinger equation
- Ermakov equation
- generalized harmonic oscillators
- Heisenberg equations of motion
- partial differential equations
- photon statistics
ASJC Scopus subject areas