Sub-Binomial Light

The click statistics from on-off detector systems is quite different from the counting statistics of the more traditional detectors. This necessitates the introduction of new parameters to characterize the nonclassicality of fields from measurements using on-off detectors. To properly replace the Mandel $Q_{\mathrm{M}}$ parameter, we introduce a parameter $Q_{\mathrm{B}}$. A negative value represents a sub-binomial statistics. This is possible only for quantum fields, even for super-Poisson light. It eliminates the problems encountered in discerning nonclassicality using Mandel’s $Q_{\mathrm{M}}$ for on-off data.

Figure 1

Different $Q$ parameters are plotted for a classical, thermal state, with $⟨n⟩=\eta \overline{n}=2$ and $\nu =0$. The true Mandel parameter is $Q_{\mathrm{M}}$, the binomial one is $Q_{\mathrm{B}}$. Using the definition in Mandel’s form for click statistics yields $Q_{\mathrm{F}}$.

Figure 2

The plot shows the binomial $Q_{\mathrm{B}}$ parameter for the click statistics measured by $N$ on-off detectors, with $2\le N\le 12$. The computed example shows the sub-binomial statistics of $m$ photon Fock states ($1\le m\le 5$).

Figure 3

The plot shows the binomial $Q_{\mathrm{B}}$ parameter for the counting statistics measured by a PNR detector with $N=2$, 3, 4, 5 on-off detectors. The computed example is a SPATS with a mean thermal photon number $\sqrt{0.5}\le \overline{n}\le 4\sqrt{0.5}$. The individual plots have a quantum efficiency $\eta =0.5$, 0.7, 0.9.