Determination of the Wigner function from photon statistics

I present a simple and robust method of quantum state reconstruction using non-ideal detectors able to distinguish only between presence and absence of photons. Using the scheme, one is able to determine a value of Wigner function in any given point on the phase plane using expectation-maximization estimation technique.

Wigner function tomography is indispensable for characterizing quantum states, but its commonly used version, balanced homodyne detection, suffers from several weaknesses. First, it requires efficient detection, which is critical for measuring fragile non-Gaussian states, especially bright ones. Second, it needs a local oscillator, tailored to match the spatiotemporal properties of the state under test, and fails for multimode and broadband states. Here we propose Wigner func...

The Hong-Ou-Mandel (HOM) experiment was a benchmark in quantum optics, evidencing the quantum nature of the photon. In order to go deeper, and obtain the complete information about the quantum state of a system, for instance, composed by photons, the direct measurement or reconstruction of the Wigner function or other quasi--probability distribution in phase space is necessary. In the present paper, we show that a simple modification in the well-known HOM experiment provides ...

In this tutorial, we introduce the basic concepts and mathematical tools needed for phase-space description of a very common class of states, whose phase properties are described by Gaussian Wigner functions: the Gaussian states. In particular, we address their manipulation, evolution and characterization in view of their application to quantum information.

Experimental reconstructions of photon number distributions of both continuous-wave and pulsed light beams are reported. Our scheme is based on on/off avalanche photodetection assisted by maximum-likelihood estimation and does not involve photon counting. Reconstructions of the distribution for both semiclassical and quantum states of light are reported for single-mode as well as for multimode beams.

We propose a method for characterizing a photodetector by directly reconstructing the Wigner functions of the detector's Positive-Operator-Value-Measure (POVM) elements. This method extends the works of S. Wallentowitz and Vogel [Phys. Rev. A 53, 4528 (1996)] and Banaszek and W\'odkiewicz [Phys. Rev. Lett. 76, 4344 (1996)] for quantum state tomography via weak-field homodyne technique to characterize quantum detectors. The scheme uses displaced thermal mixtures as probes to t...

We investigate exponential phase moments of the s-parametrized quasidistributions (smoothed Wigner functions). We show that the knowledge of these moments as functions of s provides, together with photon-number statistics, a complete description of the quantum state. We demonstrate that the exponential phase moments can be directly sampled from the data recorded in balanced homodyne detection and we present simple expressions for the sampling kernels. The phase moments are Fo...

Most methods for experimentally reconstructing the quantum state of light involve determining a quasiprobability distribution such as the Wigner function. In this paper we present a scheme for measuring individual density matrix elements in the photon number state representation. Remarkably, the scheme is simple, involving two beam splitters and a reference field in a coherent state.

We implement the direct sampling of negative phase-space functions via unbalanced homodyne measurement using click-counting detectors. The negativities significantly certify nonclassical light in the high-loss regime using a small number of detectors which cannot resolve individual photons. We apply our method to heralded single-photon states and experimentally demonstrate the most significant certification of nonclassicality for only two detection bins. By contrast, the freq...

Non-Gaussian correlations in a pure state are inextricably linked with non-classical features, such as a non positive-definite Wigner function. In a commonly used simulation technique in ultracold atoms and quantum optics, known as the truncated Wigner method, the quantum dynamics is mapped to stochastic trajectories in phase space, governed by a positive approximation to the true Wigner distribution. The question thus arises: how accurate is this approach in predicting truly...