Photon-matter interactions

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Since the early days of quantum mechanics it has been thought that the photon is unique amongst common particles in having no position operator with commuting components.  In 1999 I constructed such a photon position operator and, in collaboration with William Baylis, later found that there is a whole family of photon position operators depending on the direction and magnitude of the photon's total angular momentum.   In spite of the fact that photons are "counted" every day by experimentalists, there was no precise way to calculate photon number density.  In part because of the perception that there is no photon position operator in that sense that one exists for the electron, it is commonly thought that knowledge of photon energy density is sufficient.  However, in practical books on nonlinear processes and laser theory both photon energy density and number density are considered.  My current research includes an investigation of photon localizability, angular momentum and number density and the application of these concepts to ultrafast laser pulses and focusing of optical beams. In collaboration with Marc Dignam at Queens University I am also working on the description of excitations (called excitons) created in semiconductor nanostructures by optical pulses.  The description of these systems in terms of excitons is close to the physical picture used by experimentalists, and is more practical for numerical calculations than a description based on free electrons and holes.  When excited with two optical pulses, these excitons produce a polarization grating that scatters the excitons into wave vectors that give the four-wave mixing and, in general, the n-wave mixing signals that are commonly measured.  However, there are still logical difficulties with use of the exciton basis and with implementation of the Pauli exclusion principle.  My on-going research in this area will be on the resolution of these issues.