Complete nonclassicality test with a photon-number-resolving detector

We present a method for the experimental measurement of nonclassicality witnesses and demonstrate its application. Our proposal requires only the coherent displacement of the initial state, which can be achieved by overlapping the latter with a coherent state at a beam splitter, and subsequent photon-number-resolved detection. This setup allows a complete test of nonclassicality of an arbitrary quantum state. The role of the quantum efficiency as well as statistical and systematic uncertainties are discussed. Finally, the scheme is demonstrated for a realistic example.

Figure 1

Experimental scheme for the detection of a nonclassicality witness. The beam splitter displaces the input state $\widehat{\rho }$, and the photodiode (PD) records the photon-number statistics with quantum efficiency ${\eta }_D$. The expectation value of the witness ${\widehat{W}}_{\mathit{\text{w}}}\left(\alpha \right)$ can be calculated from Eq. (11).

Figure 2

Dependence of the expectation value of the witness ${\widehat{W}}_{\mathit{\text{w}}}\left(0\right)$ on the filter width, for a SPATS with mean thermal photon number $\overline{n}=0.8$ and overall efficiency $\eta =0.25$. The blue (bright) shaded area corresponds to a statistical error of one standard deviation; the red (dark) shaded area shows the systematic error due to the truncation of the Fock space.

Figure 3

Dependence of the expectation value of the witness on the coherent displacement $\alpha$, for a SPATS with mean thermal photon number $\overline{n}=0.8$ and overall efficiency $\eta =0.25$, and an optimal filter width $\mathit{\text{w}}=4.2$. The blue shaded area corresponds to a statistical error of one standard deviation; the systematic error is negligible for this width parameter.