Determination of the Wigner function from photon statistics

We present the first experimental characterization of the azimuthal Wigner distribution of a photon. Our protocol fully characterizes the transverse structure of a photon in conjugate bases of orbital angular momentum (OAM) and azimuthal angle (ANG). We provide a test of our protocol by characterizing pure superpositions and incoherent mixtures of OAM modes in a seven-dimensional space. The time required for performing measurements in our scheme scales only linearly with the ...

In this work, we present an educational activity aimed at measuring the Wigner distribution functions of quantum states of light in the undergraduate laboratory. This project was conceived by students from various courses within the physics undergraduate curriculum, and its outcomes were used in an introductory Quantum Optics course at the Universidad de los Andes in Bogot\'a, Colombia. The activity entails a two-hour laboratory practice in which students engage with a pre-al...

We investigate the capabilities of loss-tolerant quantum state characterization using a photon-number resolving, time-multiplexed detector (TMD). We employ the idea of probing the Wigner function point-by-point in phase space via photon parity measurements and displacement operations, replacing the conventional homodyne tomography. Our emphasis lies on reconstructing the Wigner function of non-Gaussian Fock states with highly negative values in a scheme that is based on a rea...

Despite the indisputable merits of the Wigner phase-space formulation, it has not been widely explored for systems with SU(1,1) symmetry, as a simple operational definition of the Wigner function has proved elusive in this case. We capitalize on the unique properties of the parity operator, to derive in a consistent way a \emph{bona fide} SU(1,1) Wigner function that faithfully parallels the structure of its continuous-variable counterpart. We propose an optical scheme, invol...

We derive a compact expression for the second-order correlation function $g^{(2)} (0)$ of a quantum state in terms of its Wigner function, thereby establishing a direct link between $g^{(2)} (0)$ and the state's shape in phase space. We conduct an experiment that simultaneously measures $g^{(2)} (0)$ through direct photocounting and reconstructs the Wigner function via homodyne tomography. The results confirm our theoretical predictions.

We present a scheme that uses Ramsey interferometry to directly probe the Wigner function of a neutral atom confined in an optical trap. The proposed scheme relies on the well-established fact that the Wigner function at a given point $(x,p)$ in phase space is proportional to the expectation value of the parity operator relative to that point. In this work, we show that parity-even and parity-odd motional states can be mapped to two distinct internal states of the atom by usi...

One of the most prominent quasiprobability functions in quantum mechanics is the Wigner function that gives the right marginal probability functions if integrated over position or momentum. Here we depart from the definition of the position-momentum Wigner function to, in analogy, construct a number-phase Wigner function that, if summed over photon numbers gives the correct phase distribution and integrated over phase gives the right photon distribution.

Photon-number-revolving (PNR) detection allows the direct measurement of the Wigner quasiprobability distribution of an optical mode without the need for numerically processing an inverse Radon transform [K. Banaszek and K. W\'odkiewicz, Phys. Rev. Lett. 76, 4344 (1996)]. In this work, we reproduced the seminal experiment of Banaszek et al. [Phys. Rev. A 60, 674 (1999)] of quantum tomography of a pure coherent state, and of a statistical mixture thereof, and extended it to th...

We report a measurement workflow free of systematic errors consisting of a reconfigurable photon-number-resolving detector, custom electronic circuitry, and faithful data-processing algorithm. We achieve unprecedentedly accurate measurement of various photon-number distributions going beyond the number of detection channels with average fidelity 0.998, where the error is contributed primarily by the sources themselves. Mean numbers of photons cover values up to 20 and faithfu...

The analysis of wave-packet dynamics may be greatly simplified when viewed in phase-space. While harmonic oscillators are often used as a convenient platform to study wave-packets, arbitrary state preparation in these systems is more challenging. Here, we demonstrate a direct measurement of the Wigner distribution of complex photon states in an anharmonic oscillator - a superconducting phase circuit, biased in the small anharmonicity regime. We test our method on both non-cla...