Nonclassical features of the polarization quasiprobability distribution

Polarization quasiprobability distribution is defined in the space of the Stokes observables. It can be reconstructed with the help of polarization quantum tomography and provides a full description of the so-called polarization sector of quantum states of light. We show here that due to its definition in terms of the discrete-valued Stokes operators, polarization quasiprobability distribution has singularities at integer values of the Stokes observables and takes negative values even for the quantum states typically considered as “classical” ones. In experiments with “bright” multiphoton states, the photon-number resolution is smeared due to the photodetectors’ technical limitations. In this case, nonclassical features of the explored quantum states can be revealed by adding a strong coherent beam into the orthogonal polarization.

Figure 1

The setup for polarization tomography [10, 13]. PBS is the polarizing beam splitter; ${\mathrm{D}}_{\parallel }$ and ${\mathrm{D}}_{\perp }$ are the photodetectors. The signals from the detectors are processed by either digital or analog electronics, after which a computer calculates the probability distributions $W_{\theta \varphi }\left(n\right)$ and performs the Radon transformation.

Figure 2

Left panel: Plot of $w_1\left(S_{23}\right)$. Right panel: Color plot of $w_1(S_2,S_3)$ (blue: negative values; red: positive ones). Integration along the line $S_2=1$ gives infinity; integration along the lines $S_2<1$ gives zero due to the negative-valued areas. All quantities are dimensionless.

Figure 3

Typical probability distributions for (a) a single-photon state and (b) a coherent state with $\alpha =1$ at the input of the polarization tomography setup. The Quantum efficiency of the detectors is $\eta =0.6$ and the angle $\theta$ is chosen to be $\pi /2$. All quantities are dimensionless.

Figure 4

Contour plots of the quasiprobability distribution (55) as a function of the normalized Stokes parameters (52) for the squeezed single-photon state in the absence of losses and photon-number integration (left column) and with ${\epsilon }^2=0.7$ (right column). Top row: no squeezing ($e^r=1$); bottom row: 6 db squeezing ($e^r=2$). The negative-valued areas are encircled by the white lines (the color corresponding to $W_{23}=0$ varies due to the different ratios of the maximal and the minimal values of $W_{23}$). All quantities are dimensionless.

Figure 5

The volume of the negative-valued area of the quasiprobability distribution (55) as a function of the total quantum inefficiency ${\epsilon }^2$. All quantities are dimensionless.