QLunch: Alonso Botero

Speaker: Alonso Botero

Title: Classical and quantum multiport Interference and the semiclassical limit of irreducible representations of the unitary group

Abstract: I discuss an intriguing connection between the representation theory of the unitary group and two forms of multiport interferometry in quantum optics: classical (or single-photon) interferometry and quantum (or multiphoton) interferometry. Three key elements underlie this connection: a classical-interferometric map from the torus of relative phases to the simplex of relative intensities; an expansion of multiphoton states in terms of tensor powers of single-photon states; and finally, the Keyl–Werner theorem, which asymptotically connects the typical isotypic components of tensor powers of mixed single-particle states to the spectrum of the single-particle state. Applying these elements, we analyze the statistics of a multiphoton interferometer described by a unitary matrix U in the regime of many partially indistinguishable photons. We show that the semiclassical envelope of the resulting probability distributions—as well as of certain matrix elements of the dominant isotypic component—reproduces the image of the Haar measure on the torus under the classical interferometric map. This correspondence points to two paths to explore representation matrices of high-dimensional irreducible representations of the unitary group: experimentally, via multiphoton interference, and theoretically, via tensor power expansions of bipartite entangled states.