Nonclassicality quasiprobability of single-photon-added thermal states

We report the experimental reconstruction of a nonclassicality quasiprobability for a single-photon-added thermal state. This quantity has significant negativities, which is necessary and sufficient for the nonclassicality of the quantum state. Our method exhibits several advantages compared to the reconstruction of the $P$ function, since the nonclassicality filters used in this case can regularize the quasiprobabilities as well as their statistical uncertainties. A priori assumptions about the quantum state are therefore not necessary. We also demonstrate that, in principle, our method is not limited by small quantum efficiencies.

Figure 1

Scheme for the conditional excitation of a thermal light state (denoted by ${\mathrm{\rho ̂}}_{\mathrm{in}}$) by a single photon. A click in the on-off detector D prepares the photon-added thermal state ${\mathrm{\rho ̂}}_{\mathrm{out}}$ and triggers its balanced homodyne detection (BHD).

Figure 2

Characteristic functions of a SPATS with $\mathrm{n̄}=0.49$ and $\eta =0.62$: the experimental result ${\Phi }_{\exp }(\beta )$, the theoretical expectation ${\Phi }_{\mathrm{th}}(\beta )$, and the result ${\Phi }_{\Omega }(\beta )$ of filtering for the filter width $w=1.4$. The shaded area corresponds to one standard deviation.

Figure 3

Nonclassicality quasiprobability of a SPATS for the same parameters as in Fig. 2. The blue shaded area corresponds to one standard deviation, which is almost completely hidden by the line thickness.

Figure 4

$P$ functions regularized with a rectangular filter. The cutoff parameter at the left side of the figure is rather small, leading to large systematic errors (red shaded). For a larger cutoff parameter, the statistical uncertainty (blue shaded) becomes dominant; see the right side of the figure.