Experimental determination of a nonclassical Glauber-Sudarshan $P$ function

A quantum state is nonclassical if its Glauber-Sudarshan $P$ function fails to be interpreted as a probability density. This quantity is often highly singular, so that its reconstruction is a demanding task. Here we present the experimental determination of a well-behaved $P$ function showing negativities for a single-photon-added thermal state. This is a direct visualization of the original definition of nonclassicality. The method can be useful under conditions for which many other signatures of nonclassicality would not persist.

Figure 1

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

Figure 2

(Color online) Experimental characteristic functions (solid lines) and best fit to theoretical curves (dashed lines): (a) SPATS, with $\overline{n}=1.11$ and $\eta =0.60$, (b) mixture of SPATS with 19% of the thermal background, with $\overline{n}=3.71$ and $\eta =0.62$. The shaded areas show the standard deviations.

Figure 3

(Color online) Experimentally reconstructed $P$ function of a SPATS, as obtained from Fig. 2a.

Figure 4

(Color online) $P$ functions (solid lines) in parts (a) and (b) are obtained from the experimental characteristic functions in Figs. 2a, 2b, respectively. They are compared with the corresponding theoretical fits (dashed curves). The standard deviations (light shaded areas) and the systematic errors (dark shaded areas) are also given.