The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the electron transport, in quantum chemistry—to calculate the static and dynamical properties of many-body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally formulated 86 years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.

Recent advances in Wigner function approaches

We demonstrate electromagnetic quantum states of single photons and of correlated photon pairs exhibiting ``hybrid'' entanglement between spin and orbital angular momentum. These states are obtained from entangled photon pairs emitted by spontaneous parametric down conversion by employing a $q$ plate for coupling the spin and orbital degrees of freedom of a photon. Entanglement and contextual quantum behavior (that is also nonlocal, in the case of photon pairs) is demonstrated by the reported violation of the Clauser-Horne-Shimony-Holt inequality. In addition, a classical analog of the hybrid spin-orbit photonic entanglement is reported and discussed.

Spin-orbit hybrid entanglement of photons and quantum contextuality (vol 82, 022115, 2010)

Perfect vortex beams are the orbital angular momentum (OAM)-carrying beams with fixed annular intensities, which provide a better source of OAM than traditional Laguerre-Gaussian beams. However, ordinary schemes to obtain the perfect vortex beams are usually bulky and unstable. We demonstrate here a novel generation scheme by designing planar Pancharatnam-Berry (PB) phase elements to replace all the elements required. Different from the conventional approaches based on reflective or refractive elements, PB phase elements can dramatically reduce the occupying volume of system. Moreover, the PB phase element scheme is easily developed to produce the perfect vector beams. Therefore, our scheme may provide prominent vortex and vector sources for integrated optical communication and micromanipulation systems.

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Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements

We employ a high quantum efficiency photon number counter to determine the photon number distribution of the output field from a parametric down-converter. The raw photocount data directly demonstrates that the source is nonclassical by 40 standard deviations, and correcting for the quantum efficiency yields a direct observation of oscillations in the photon number distribution.

Direct Observation of Nonclassical Photon Statistics in Parametric Down-Conversion

The “perfect” vortex is a new class of optical vortex beam having ring radius independent of its topological charge (order). One of the simplest techniques to generate such beams is the Fourier transformation of the Bessel-Gauss beams. The variation in ring radius of such vortices require Fourier lenses of different focal lengths and or complicated imaging setup. Here we report a novel experimental scheme to generate perfect vortex of any ring radius using a convex lens and an axicon. As a proof of principle, using a lens of focal length f = 200 mm, we have varied the radius of the vortex beam across 0.3–1.18 mm simply by adjusting the separation between the lens and axicon. This is also a simple scheme to measure the apex angle of an axicon with ease. Using such vortices we have studied non-collinear interaction of photons having orbital angular momentum (OAM) in spontaneous parametric down-conversion (SPDC) process and observed that the angular spectrum of the SPDC photons are independent of OAM of the pump photons rather depends on spatial profile of the pump beam. In the presence of spatial walk-off effect in nonlinear crystals, the SPDC photons have asymmetric angular spectrum with reducing asymmetry at increasing vortex radius.

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Generation of "perfect" vortex of variable size and its effect in angular spectrum of the down-converted photons.

We have experimentally generated higher order optical vortices and scattered them through a ground glass plate that results in speckle formation. Intensity autocorrelation measurements of speckles show that their size decreases with an increase in the order of the vortex. It implies an increase in the angular diameter of the vortices with their order. The characterization of vortices in terms of their annular bright ring also helps us to understand these observations. The results may find applications in stellar intensity interferometry and thermal ghost imaging.

Higher order optical vortices and formation of speckles

We have experimentally reproduced ring-shaped beams from the scattered Laguerre–Gaussian and Bessel–Gaussian beams. A rotating ground glass plate is used as a scattering medium, and a plano–convex lens collects the scattered light to generate ring-shaped beams at the Fourier plane. The obtained experimental results are supported with the numerical results and are in good agreement with the theoretical results proposed by Wang et al. [Opt. Express17, 22366 (2009)].

Experimental generation of ring-shaped beams with random sources

We show, both theoretically and experimentally, that the propagation of optical vortices in free space can be analyzed by using the width [w(z)] of the host Gaussian beam and the inner and outer radii of the vortex beam at the source plane (z=0) as defined in [Opt. Lett.39, 4364 (2014)10.1364/OL.39.004364OPLEDP0146-9592]. We also studied the divergence of vortex beams, considered as the rate of change of inner or outer radius with the propagation distance (z), and found that it varies with the order in the same way as that of the inner and outer radii at z=0. These results may be useful in designing optical fibers for orbital angular momentum modes that play a crucial role in quantum communication.

Divergence of optical vortex beams.

We have experimentally generated higher order optical vortices and scattered them through a ground glass plate that results in speckle formation. Intensity autocorrelation measurements of speckles show that their size decreases with increase in the order of the vortex. It implies increase in angular diameter of the vortices with their order. The characterization of vortices in terms of their annular bright ring also helps us to understand these observations. The results may find applications in stellar intensity interferometry and thermal ghost imaging.

We have experimentally reproduced ring shaped beams from the scattered Laguerre-Gaussian and Bessel-Gaussian beams. A rotating ground glass plate is used as a scattering medium and a plano convex lens collects the scattered light to generate ring shaped beams at the Fourier plane. The obtained experimental results are supported with the numerical results and are in good agreement with the theoretical results proposed by Wang, Cai and Korotkova (Opt. Exp. 17, 22366 (2009)).

Salla Gangi Reddy

Papers

Higher order optical vortices and formation of speckles

Experimental generation of ring-shaped beams with random sources

Divergence of optical vortex beams.

Higher order optical vortices and formation of speckles

Experimental generation of ring-shaped beams with random sources