Structure of the Harmonic Oscillator in the space of $n$-particle Glauber correlators

Abstract: We map the Hilbert space of the quantum Harmonic oscillator to the space of Glauber's $n$th-order intensity correlators, in effect showing "the correlations between the correlators" for a random sampling of the quantum states. In particular, we show how the popular $g^{(2)}$ function is correlated to the mean population and how a recurrent criterion to identify single-particle states or emitters, namely $g^{(2)}<1/2$, is incorrect as states exist that satisfy this condition with average population larger than one. Our charting of the Hilbert space allows to capture its structure in a simpler and physically more intuitive way that can be used to classify quantum sources by surveying which territory they can access.