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arXiv:2010.02173 · Physical Review Letters

Gaussian versus Non-Gaussian Filtering of Phase-Insensitive Nonclassicality

B. Kühn, W. Vogel, Valérian Thiel, Sofiane Merkouche, Brian J. Smith

Published Apr 30, 2021• 8 citations

Abstract

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.

DOI: 10.1103/physrevlett.126.173603

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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.

We report the direct continuous-in-phase sampling of a regularized

P

function, the so-called nonclassicality quasiprobability, for squeezed light. Through their negativities, the resulting phase-space representation uncovers the quantum character of the state. In contrast to discrete phase-locked measurements, our approach allows an unconditional verification of nonclassicality by getting rid of interpolation errors due to fixed phases. To realize the equal phase distribution of measured quadratures, a data selection is implemented with quantum random numbers created by measuring the vacuum noise. The continuously measured squeezed field was generated in an optical parametric amplifier. Suitable pattern functions for obtaining the regularized

P

function are investigated. The significance of detecting negativities in our application is determined. The sampling of nonclassicality quasiprobabilities is shown to be a powerful and universal method to visualize quantum effects within arbitrary quantum states.

Systems of on-off detectors are well established for measuring radiation fields in the regime of small photon numbers. We propose to combine these detector systems with unbalanced homodyning with a weak local oscillator. This approach yields phase-space functions, which represent the click counterpart of the s parametrized quasiprobabilities of standard photoelectric detection theory. This introduced class of distributions can be directly sampled from the measured click-counting statistics. Therefore, our technique visualizes nonclassical effects without further data processing. Surprisingly, a small number of on-off diodes can yield more insight than perfect photon number resolution. Quantum signatures in the particle and wave domain of the quantized radiation field, as shown by photon number and squeezed states, respectively, will be uncovered in terms of negativities of the sampled phase-space functions. Application in the vast fields of quantum optics and quantum technology will benefit from our efficient nonclassicality characterization approach.

The algebraic quantification of nonclassicality, which naturally arises from the quantum superposition principle, is related to properties of regular nonclassicality quasiprobabilities. The latter are obtained by non-Gaussian filtering of the Glauber-Sudarshan

P

function. They yield lower bounds for the degree of nonclassicality. We also derive bounds for convex combinations of Gaussian states for certifying quantum non-Gaussianity directly from the experimentally accessible nonclassicality quasiprobabilities. Other quantum-state representations, such as

s

-parametrized quasiprobabilities, insufficiently indicate or even fail to directly uncover detailed information on the properties of quantum states. As an example, our approach is applied to multi-photon-added squeezed vacuum states.

We propose a quantum optical device to experimentally realize quantum processes, which perform the regularization of the---in general highly singular---Glauber--Sudarshan

P

functions of arbitrary quantum states before their application and/or measurement. This allows us to produce a broad class of nonclassical states with regular

P

functions, also called nonclassicality quasiprobabilities. For this purpose, the input states are combined on highly transmissive beam splitters with specific Gaussian or non-Gaussian classical states. We study both balanced and unbalanced homodyne detections for the direct sampling of the output states of the implemented processes, which requires no further regularization or state-reconstruction. By numerical simulations we demonstrate the feasibility of our approach and we outline the generalization to multimode light.

While negativity in phase space is a well-known signature of nonclassicality, a wide variety of nonclassical states require their characterization beyond negativity. We establish a framework of nonclassicality in phase space that addresses nonclassical states comprehensively with a direct experimental evidence. This includes the negativity of phase-space distribution as a special case and further analyzes quantum states with positive distributions effectively. We prove that it detects all nonclassical Gaussian states and all non-Gaussian states of arbitrary dimension remarkably by examining three phase-space points only. Our formalism also provides an experimentally accessible lower bound for a nonclassicality measure based on trace distance. Importantly, this foundational approach can be further adapted to constitute practical tests in two directions looking into particle and wave nature of bosonic systems, via an array of nonideal on-off detectors and coarse-grained homodyne measurement, respectively. All these tests are practically powerful in characterizing nonclasssical states reliably against noise, making a versatile tool for a broad range of quantum systems in quantum technologies.