In an often-quoted statement in his book The Principles of Quantum Mechanics, Paul Dirac wrote that in an interference experiment involving multiple photons, "Each phot... interferes only with itself." However, it was shown fairly recently that when two indistinguishable single photons are made to overlap at an optical beam splitter, they do in fact interfere with one another: both photons "stick together" upon exiting the beam splitter, proceeding together along one of the two possible output paths as shown in the figure. Remarkably, the outcome where one photon exits each beam splitter output never occurs, due to destructive quantum interference involving the probability amplitudes for this case. The observation of this phenomenon in 1987 by Hong, Ou, and Mandel clearly demonstrated this two-photon interference effect, which may be though of as a consequence of the bosonic nature of photons. It is commonly thought that the incoming photons must be completely indistinguishable in order for them to experience this remarkable quantum effect, but new research performed over the past year is now questioning this assumption.
In this project we will extend the present state of understanding of this two-photon quantum interference effect by considering two fundamental generalizations: First, we will consider two distinguishable input photons with distinct spatial distribution properties, or wave functions, learning how two-photon interference manifests itself in such a case. Second, instead of considering a single optical beam splitter as the interaction medium, we will consider more complicated types of devices with two "inputs" and two "outputs" in which the aforementioned type of interference between two photon's wave functions may occur. These more complicated devices transform the wave functions of photons in nontrivial ways, giving rise to a potentially rich class of two-photon quantum phenomena. As an added benefit, two realizations of such devices are currently being built and tested in my lab, so that the new predictions resulting from this project may be verified. This theoretical project will involve obtaining some working knowledge of quantum optics, and using symbolic matrix techniques in Mathematica for modeling of the various phenomena involved.