Visualizing nonclassical effects in phase space

Nonclassicality filters provide a universal method to visualize the nonclassicality of arbitrary quantum states of light through negativities of a regularized Glauber-Sudarshan $P$ function, also denoted as nonclassicality quasiprobability. Such filters are introduced and analyzed for optimizing the experimental certification of nonclassical effects. An analytic filter is constructed which preserves the full information on the quantum state. For balanced homodyne detection, the number of data points is analyzed to get the negativities of the nonclassicality quasiprobability with high statistical significance. The method is applied to different scenarios, such as phase-randomized squeezed vacuum states, single-photon-added thermal states, and heralded state engineering with array detectors. The generalization to visualize quantum correlations of multimode radiation fields is also considered.

Figure 1

Filter ${\Omega }_w^{\left(q\right)}\left(\right|\beta \left|\right)$ for different values of the parameter $q$. (top) Vertical axis linear. (bottom) Vertical axis logarithmic.

Figure 2

Filter ${\Omega }_w\left(\right|\beta |;s,C)$ for $C=1.3$ and different values of the parameter $s$.

Figure 3

$P_{\Omega }(\alpha ;w)$ of a squeezed vacuum state with orthogonal quadrature variances $V_p=2.0$ and $V_x=0.5$. The filter ${\Omega }_w(\beta ;s,C)$ is applied, with $w=9.9,C=1.3$, and $s=4$.

Figure 4

Required number $N$ of data pairs $(x_j,{\phi }_j)$ from BHD using the filter ${\Omega }_w^{\left(q\right)}\left(\beta \right)$ with different values of $q$. (top) Fock state $|1\rangle$. (bottom) Fock state $|2\rangle$. The number $N$ is calculated for various quantum efficiencies $\eta$ (dots). The filter width $w$ is optimized for each filter and each considered quantum efficiency.

Figure 5

Required number $N$ of data pairs $(x_j,{\phi }_j)$ from BHD using the filter ${\Omega }_w^{\left(q\right)}\left(\beta \right)$ with different values of $q$. (top) SPATS for $\overline{n}=0.8$. (bottom) 2-PATS for $\overline{n}=0.9$. The number $N$ is calculated for various quantum efficiencies $\eta$ (dots). The filter width $w$ is optimized for each filter and each considered quantum efficiency.

Figure 6

Required number $N$ of data pairs $(x_j,{\phi }_j)$ from BHD using the filter ${\Omega }_w^{\left(q\right)}\left(\beta \right)$ with different values of $q$. The state under study is a completely phase randomized squeezed vacuum state with orthogonal quadrature variances $V_x=0.4$ and $V_p=5.0$. The number $N$ is calculated for various quantum efficiencies $\eta$ (dots). The filter width $w$ is optimized for each filter and each considered quantum efficiency. The line labeled CF shows the number of data points for a statistical significance of five standard deviations for the negativity of $1-\left|\Phi \right(\beta \left)\right|$.

Figure 7

The light source prepares a two-mode squeezed vacuum state. The quantum state of mode $A$ is prepared, conditioned on the events recorded in mode $B$ by an array of $M=8$ on-off detectors. Mode $A$ is subdivided by a beam splitter into modes $A^{\prime }$ and $A^{\prime \prime }$, whose correlations are measured by two BHDs.

Figure 8

Regularized $P$ function $P_{\Omega }({\alpha }^{\prime },{\alpha }^{\prime \prime };w)$ of the two-mode system, modes $A^{\prime }$ and $A^{\prime \prime }$, for (left) $k=1$ and (right) $k=4$ clicks being recorded by the array detector. The filter ${\Omega }_w^{\left(\infty \right)}\left(\beta \right)$ with a filter width $w=1.7$ is applied. Distinct negativities in both cases visualize the nonclassical correlations of the states.