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arXiv:2512.13462 · Fluctuation and Noise Letters

Wigner function negativity in a classical model of quantum light

Brian R. La Cour

quant-phSubmitted Dec 15, 2025• Announced Dec 16, 2025• 0 citations
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Abstract

The presence of negative values in the Wigner quasiprobability distribution is deemed one of the hallmarks of nonclassical phenomena in quantum systems. Here we demonstrate a classical model of squeezed light that, when combined with post-selection on amplitude threshold-crossing detection events, is capable of reproducing observed behavior of single-photon added coherent states. In particular, a classical model of balanced homodyne detection and standard tomographic techniques are used to infer the density matrix in the Fock basis. The resulting Wigner functions exhibit negatively for photon-added vacuum and weak coherent states.

DOI
10.1142/S021947752540036X
Comments
7 pages, 3 figures

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The presence of negative values in the Wigner quasiprobability distribution is deemed one of the hallmarks of nonclassical phenomena in quantum systems. Here we demonstrate a classical model of squeezed light that, when combined with post-selection on amplitude threshold-crossing detection events, is capable of reproducing observed behavior of single-photon added coherent states. In particular, a classical model of balanced homodyne detection and standard tomographic techniques are used to infer the density matrix in the Fock basis. The resulting Wigner functions exhibit negatively for photon-added vacuum and weak coherent states.
Quantum states of light having a Wigner function with negative values represent a key resource in quantum communication and quantum information processing. Here we present the generation of such a state at the telecommunication wavelength of 1550 nm. The state is generated by means of photon subtraction from a weakly squeezed vacuum state and is heralded by the ``click'' of a single photon counter. Balanced homodyne detection is applied to reconstruct the Wigner function, also yielding the state's photon-number distribution. The heralding photons are frequency up-converted to 532 nm to allow for the use of a room-temperature (silicon) avalanche photodiode. The Wigner function reads $W(0,0)=\ensuremath{-}0.063\ifmmode\pm\else\textpm\fi{}0.004$ at the origin of phase space, which certifies negativity with more than 15 standard deviations.
Based on the expressions of Wigner functions in Fock-spaces,we reconstruct the Wigner functions of photon-added and-subtracted states of squeezed vacuum states.Negativities of these Wigner functions numerically indicate that these states are really non-classical quantum states.We also discuss how these negativities depend on the photon number k added to and subtraced from the squeezed vacuum states.The negativities are greater when k is odd numbers than those when k is even numbers.The result provides a theoretical foundation for measuring these states.
For the visualization of quantum states, the approach based on Wigner functions can be very effective. Homodyne detection has been extensively used to obtain the density matrix, Wigner functions and tomographic reconstructions of optical fields for many thermal, coherent or squeezed states. Here, we use time-domain optical homodyne tomography for the quantum state recognition and reconstruction of the femtosecond optical field from a nonequilibrium superradiant coherent electron-hole state formed in a semiconductor laser structure. We observe severe deviations from the Poissonian statistics of the photons associated with the coherent laser state when the transformation from lasing to superradiance occurs. The reconstructed Wigner functions show large areas of negative values, a characteristic sign of non-classicality, demonstrating the quantum nature of the generated superradiant emission. The photon number distribution and Wigner function of the SR state are very similar to those of the displaced Fock state
Polarization quasiprobability distribution defined in the Stokes space shares many important properties with the Wigner function for position and momentum. Most notably, they both give correct one-dimensional marginal probability distributions and therefore represent the natural choice for the probability distributions in classical hidden-variable models. In this context, negativity of the Wigner function is considered as proof of nonclassicality for a quantum state. On the contrary, the polarization quasiprobability distribution demonstrates negativity for all quantum states. This feature comes from the discrete nature of Stokes variables; however, it was not observed in previous experiments, because they were performed with photon-number averaging detectors. Here we reconstruct the polarization quasiprobability distribution of a coherent state with photon-number resolving detectors, which allows us to directly observe for the first time its negativity. Furthermore we derive a theoretical polarization quasiprobability distribution for any linearly polarized quantum state.
Abstract The Wigner function was introduced as an attempt to describe quantum-mechanical fields with the tools inherited from classical statistical mechanics. In particular, it is widely used to describe the properties of radiation fields. In fact, a useful way to distinguish between classical and nonclassical states of light is to ask whether their Wigner function has a Gaussian profile or not, respectively. In this paper, we use the basis of Fock states to provide the closed-form expression for the Wigner function of an arbitrary quantum state. Thus, we provide the general expression for the Wigner function of a squeezed Fock, coherent and thermal states, with an arbitrary squeezing parameter. Then, we consider the most fundamental quantum system, Resonance Fluorescence, and obtain closed-form expressions for its Wigner function under various excitation regimes. With them, we discuss the conditions for obtaining a negative-valued Wigner function and the relation it has with population inversion. Finally, we address the problem of the observation of the radiation field, introducing physical detectors into the description of the emission. Notably, we show how to expose the quantumness of a radiation field that has been observed with a detector with finite spectral resolution, even if the observed Wigner function is completely positive.
We construct a new type of photon-added squeezed coherent state generated by repeatedly operating the bosonic creation operator on a new type of squeezed coherent state [Fan H Y and Xiao M 1996 Phys. Lett. A 220 81]. We find that its normalization factor is related to single-variable Hermite polynomials. Furthermore, we investigate its statistical properties, such as Mandel’s Q parameter, photon-number distribution, and Wigner function. The nonclassicality is displayed in terms of the intense oscillation of photon-number distribution and the negativity of the Wigner function.