Correlation function $g^{(k)}(0)$ and sub-$k$ photon emission

Peter Grünwald

Published 2019-05-25Version 1

We study the relation between the $k$th-order correlation function $g^{(k)}(0)$ at equal time for all operators and the projection of the underlying quantum state of light onto the subspace with less than $k$ photons. It was previously established that for $g^{(2)}(0)$ falling below 1/2, a non-zero projection on zero or single photons follows, as well as a lower bound on the ratio for single-to-multi-photon emission. Here we generalize these results to higher-orders $k$. When $g^{(k)}(0)$ falls below its value for the Fock state $|k\rangle$ a nonzero projection on the subspace with less than $k$ photons can be concluded. In particular a nonzero lower bound is found, when vacuum is included in this space, whereas, when it is left out, the remaining space of $1$ to $k-1$ photons can have arbitrarily low projection. This is due to the influence of vacuum, artificially enhancing the value of $g^{(k)}(0)$. We derive an effective correlation function $\tilde g^{(k)}(0)$, which takes the effect of vacuum into account and yields a nonzero lower bound on the ratio of sub-$k$-photon-to-$k$-or-more-photon emission. We examine these boundaries for different quantum states and derive a large-$k$ limit. Due to the monotone decrease of the boundaries with increasing $k$, this limit serves as a general lower bound for all correlation functions $g^{(k)}(0)$. Finally, we consider measurement of $\tilde g^{(k)}(0)$.