How to perform the coherent measurement of a curved phase space

IQC-QuICS Math-CS Seminar

Dr. Christopher Sahadev Jackson (Sandia National Laboratories)
Tuesday, September 14, 2021 - 4:00pm
Virtual Via Zoom

In quantum optics, the Hilbert space of a mode of light corresponds to functions on a plane called the phase space (so called because it reminded Boltzmann of oscillators in 2-d real space.)  This correspondence offers three important features:  it can autonomously handle quantum theoretical calculations, it allows for the infinite-dimensional Hilbert space to be easily visualized, and it is intimately related to a basic experimental measurement (the so-called heterodyne detection).  Continuous phase space correspondences exist naturally for many types of Hilbert space besides this particular infinite-dimensional one.  Specifically, the two-sphere is a natural phase space for quantum spin systems.  Although well studied on the theoretical and visualization fronts, the corresponding measurement (theoretically referred to as the spin-coherent-state positive-operator-valued measure or SCS POVM) has yet to find a natural way to be experimentally performed.  In this talk, I will review the history of phase space, it’s connection to representation theory, quantization, coherent states, and continuous measurement.  Finally, I will explain how the SCS POVM can be simply performed, independent of the quantization.  Such a demonstration is a fundamental contribution to the theory of continuous quantum measurement which revives several differential-geometric ideas from the classical and modern theory of complex semisimple Lie groups.