{
  "id": "1905.10557",
  "version": "v1",
  "published": "2019-05-25T08:48:53.000Z",
  "updated": "2019-05-25T08:48:53.000Z",
  "title": "Correlation function $g^{(k)}(0)$ and sub-$k$ photon emission",
  "authors": [
    "Peter Grünwald"
  ],
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "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 $\\tilde g^{(k)}(0)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-25T08:48:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "photon emission",
      "nonzero lower bound",
      "quantum state",
      "th-order correlation function",
      "general lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}