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ON PHOTON STATISTICS IN VARIABLE MEDIA

Sergey Kryuchkov, Erwin Suazo, Sergeĭ K. Suslov

Published Jan 13, 2014• 3 citations
OpenAlex

Abstract

We find explicit solutions of the Heisenberg equations of motion for a general quadratic Hamiltonian in a variable medium in the case of multi-parameter squeezed input photons. The corresponding photon statistics are also derived in the Schrodinger picture in an abstract Hilbert space operator settings. Their time evolution is given in terms of solutions of certain Ermakov- type system. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. 1. An Introduction

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We find explicit solutions of the Heisenberg equations of motion for a general quadratic Hamiltonian in a variable medium in the case of multi-parameter squeezed input photons. The corresponding photon statistics are also derived in the Schrodinger picture in an abstract Hilbert space operator settings. Their time evolution is given in terms of solutions of certain Ermakov- type system. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. 1. An Introduction
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.
We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a + , which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.
A consistent quantum formalism based on the localized basis of the Wannier functions in the Heisenberg and Schrödinger pictures to describe propagation of an electromagnetic field in a three-dimensional media including diffraction is presented. In the Schrödinger picture the Fokker-Planck equation for the Glauber-Sudarshan quasi-probability and the corresponding Langevin equations are given. As a result the spacetime description is obtained by a simple change of variables in the temporal master equation of the field. Using this formalism it is shown that the existence of integrals of motion in the propagation of light in a medium under the condition of non-degenerated parametric and two-photon interactions results in amplification of modes when non-classical properties of the light are conserved. Quantum propagation of light in a linear medium taking into account the diffraction is considered and its solution is found.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.