Gaussian versus Non-Gaussian Filtering of Phase-Insensitive Nonclassicality

Measures of quantum properties are essential to understanding the fundamental differences between quantum and classical systems as well as quantifying resources for quantum technologies. Here, two broad classes of bosonic phase-space functions, which are filtered versions of the Glauber-Sudarshan $P$ function, are compared with regard to their ability to uncover nonclassical effects of light through their negativities. Gaussian filtering of the $P$ function yields the family of $s$-parametrized quasiprobabilities, while more powerful regularized nonclassicality quasiprobabilities are obtained by non-Gaussian filtering. A method is proposed to directly sample such phase-space functions for the restricted case of phase-independent quantum states from balanced homodyne measurements. This overcomes difficulties of previous approaches that manually append uniformly distributed optical phases to the measured quadrature data. We experimentally demonstrate this technique for heralded single- and two-photon states using balanced homodyne detection with varying efficiency. The $s$-parametrized quasiprobabilities, which can be directly sampled, are non-negative for detection efficiencies below 0.5. By contrast, we show that significant negativities of non-Gaussian filtered quasiprobabilities uncover nonclassical effects for arbitrarily low efficiencies.

Figure 1

Experimental scheme. PBS: polarizing beam splitter. APD: avalanche photodiode. PZT: piezoelectric actuator. IF: interference filter. BHD: balanced homodyne detection. BS: beam splitter. FBS: fiber beam splitter. Inset: 250 acquisitions of four pulses from the BHD with the residual mean voltage.

Figure 2

The directly sampled $s$-parametrized quasiprobabilities as a function of $|\alpha |$ for $s=-0.04$ and five different quantum efficiencies. (a) Single photon; (b) Two photons. The thin dashed line corresponds to an error of one standard deviation, which is barely noticeable.

Figure 3

Nonclassicality quasiprobability as a function of $|\alpha |$, for the filter parameters $w_{\mathrm{opt}}$ which maximizes the statistical significance of the negativities; see Table I in [26]. They are shown for the two largest quantum efficiencies. The $|\alpha |$ value for which the negativity with the maximal significance is obtained is marked by an arrow, and its position depends on $w_{\mathrm{opt}}$, which is independently optimized for each state. (a) Single photon; (b) two photons. The thin dashed line corresponds to an error of one standard deviation, which is barely noticeable.