On the similarities between the Wigner distribution function in classical and quantum optics

The structure of the global “quantum phase space” is analyzed for the harmonic oscillator perturbed by a monochromatic wave in the limit when the perturbation amplitude is small. Usually, the phenomenon of quantum resonance was studied in nondegenerate [G. M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, Chur, 1985)] and degenerate [Demikhovskii, Kamenev, and Luna-Acosta, Phys. Rev. E 52, 3351 (1995)] classically chaotic systems only in the particular regions of the classical phase space, such as the center of the resonance or near the separatrix. The system under consideration is degenerate, and even an infinitely small perturbation generates in the classical phase space an infinite number of the resonant cells which are arranged in the pattern with the axial symmetry of the order 2μ (where μ is the resonance number). We show analytically that the Husimi functions of all Floquet states (the quantum phase space) have the same symmetry as the classical phase space. This correspondence is demonstrated numerically for the Husimi functions of the Floquet states corresponding to the motion near the elliptic stable points (centers of the classical resonance cells). The derived results are valid in the resonance approximation when the perturbation amplitude is small enough, and the stochastic layers in the classical phase space are exponentially thin. The developed approach can be used for studying a global symmetry of more complicated quantum systems with chaotic behavior.

Phase space optics allows the four-dimensional simultaneous visualization of both space and spatial frequency optical information. The Wigner distribution function (WDF) is a common characterization of the phase space. Compared with the two-dimensional complex amplitude coherent optical field, the WDF can characterize optical field with arbitrary coherent states due to its higher dimensions. It is especially advantageous for the representation of partially coherent optical fields. The WDF is real and may have negative values, which are the result of phase-space interference. In this paper, an improved phase-space retrieval method is demonstrated. First, capture three-dimensional intensity focal stack. Then, phase space tomography (PST) combined with a non-linear iterative projection algorithm is conducted to reconstruct the whole WDF. We further analyzed the effect of the microscopy imaging system, i.e., the illumination aperture and the aperture of objective lens effect.

We propose to study the L2-norm distance between classical and quantum phase space distributions, where for the latter we choose the Wigner function, as a global phase space indicator of quantum–classical correspondence. For example, this quantity should be the key to understanding the correspondence between quantum and classical Loschmidt echoes. We concentrate on fully chaotic systems with compact (finite) classical phase space. By means of numerical simulations and heuristic arguments we find that the quantum–classical fidelity stays at one up to the Ehrenfest-type time scale, which is proportional to the logarithm of effective Planck constant, and decays exponentially with a maximal classical Lyapunov exponent, after that time.

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Airy beam is a self-sustained light beam that propagates in free space along a parabolic trajectory with a constant lateral acceleration. It is a diffraction-free beam with an electric field pattern described by the Airy function containing a main lobe and side lobes. Unlike Gaussian beam that expands after the focus, a practical finite-energy Airy beam keeps its pattern without spreading over a relatively long distance during its propagation. This unique feature makes the Airy beam a great potential of designing optical systems for three-dimensional (3D) imaging. In this work, we develop a new optical 3D imaging methodology using Airy beam. The proposed imaging method has an advantage of increased depth of field (DOF) that is important in biological optical imaging. In the first step, we perform a fundamental research on specific parameters that can affect the properties of the Airy beams. The finite-energy Airy beams are generated by optical Fourier transform using a lens with the incoming plane wave that is truncated by an aperture. Our simulation and experiments showed that by changing the aperture size and the focal length of the lens, the Airy beam pattern is modified. In addition, the DOF, the beam size and the main lobe energy are changed. As a result, Airy beam can be designed for specific imaging applications. Next, we consider the phase in the cubic phase pattern for the phase modulation to generate the Airy beam. This problem is related to the wavelength dependent phase modulation since the phase in cubic phase pattern is designed for a specific wavelength whereas a broadband light source is normally used in optical coherence tomography (OCT). A liquid crystal display (LCD) panel from a projector is used as the SLM. The cubic phase patterns with different gray values displayed on the computer screen provide different phase modulations in the SLM. The experimental and simulated results show that the phase modulation affects the beam shape of the Airy beam. If the maximum phase modulation is larger than 1.7pi, the Airy beam can keep its pattern and the unmodulated Gaussian beam can be neglected. Further, we design a 3D imaging system using the phase-space method based on the Wigner distribution function (WDF). We theoretically build the WDF for the Airy beam and show that the WDF of the Airy beam is shifted in the phase-space as the transverse position of incoming Airy beam changes, and the WDF is tilted as changing the axial position (along the propagation direction). A larger truncating factor can reduce the energy contribution in WDF for the side lobes and modify the shape of the main lobe close to a straight line. In this case, the WDF of the Airy beam is similar to that of the Gaussian beam. However, the DOF of the Airy beam is greatly improved. We perform experimental measurements of WDF using the basic idea of measuring both the space and spatial frequency information in a 4-f system. Our experimental results match the simulation and validate the 3D imaging using Airy beam in the phase space.

The concept of the phase space plays a key role in the analysis of oscillating signals. For a 1-D signal, the coordinates of the 2-D phase space are the observation time and the instant frequency. For measurements of propagating wave fields, the time and instant frequency are linked to the spatial location and wave normal, defining a ray. In this case, the phase space is also termed the ray space. Distributions in the ray space find important applications in the analysis of radio occultation (RO) data because they allow the separation of interfering rays in multipath zones. Examples of such distributions are the spectrogram, Wigner distribution function (WDF), and Kirkwood distribution function (KDF). In this study, we analyze the application of the fractional Fourier transform (FrFT) to the construction of distributions in the ray space. The FrFT implements the phase space rotation. We consider the KDF averaged over the rotation group and demonstrate that it equals the WDF convolved with a smoothing kernel. We give examples of processing simple test signals, for which we evaluate the FrFT, KDF, WDF, and smoothed WDF (SWDF). We analyze the advantages of the SWDF and show examples of its application to the analysis of real RO observations.

In this paper, we analysis the effect of partial occlusions in scenes captured using digital holography. We reconstruct the scene from different perspectives. These reconstructions are then combined, allowing one to overcome foreground occlusions that are obscuring one's view of the scene. The analysis in this paper is carried out with the aid of the Wigner distribution function, allowing us to visualize the energy of the object wavefield and the occluding object. wavefield in phase space. We show that by iteratively selecting different views, the original scene can be reconstructed efficiently. This technique would be useful in situations where transmission of the whole digital hologram, or exhaustive reconstruction of every perspective, was not feasible. We provide results using optically captured digital holograms of real-world objects, and simulated occlusions.

Based on the Collins integral formula and the Hermite–Gaussian expansion of a Lorentz function, an analytical expression for the Wigner distribution function (WDF) of a Lorentz–Gauss vortex beam passing through a paraxial ABCD optical system is derived. The WDF properties of a Lorentz–Gauss vortex beam propagating in free space are also demonstrated. During propagation, the WDF pattern shrinks in the direction of the spatial frequency variables and elongates in the direction of the transverse spatial coordinates. With given the transverse spatial coordinates or the spatial frequency variables, the evolvement of the WDF of a Lorentz–Gauss vortex beam during propagation is examined in detail. To show the WDF application, the second-order moments of the WDF for a Lorentz–Gauss vortex beam in free space are also presented. This research exhibits the propagation properties of a Lorentz–Gauss vortex beam from the phase space.

Quantum optics and classical optics have coexisted for nearly a century as two distinct, self-consistent descriptions of light. What influences there were between the two domains all tended to go in one direction, as concepts from classical optics were incorporated into quantum theory's early development. But it's becoming increasingly clear that a significant quantum presence exists in classical territory-and, in particular, that the quintessential quantum attribute, entanglement, can be seen, studied and exploited in classical optics. This blurring of the classical-quantum boundary has opened up a potential new direction for frontier work in optics.

States in classical mechanics are probability measures, and their Fourier transforms are continuous functions of positive type. States in the phase-space formulation of quantum mechanics are Wigner distribution functions, and their (symplectic) Fourier transforms have been characterized by Kastler [Commun. Math. Phys. 1, 14 (1965)] and Loupias and Miracle-Sole [Commun. Math. Phys. 2, 31 (1966); Ann. Insto. H. Poincaré 6, 39 (1967)] as being continuous functions of ‘‘ℏ-positive type.’’ In this paper, (Schwartz) distributions of ℏ-positive type, are defined and studied. It is shown that if such distributions are bounded on a certain sequence of test functions, then they are symplectic Fourier transforms of Wigner distribution functions. These results, are applied to a variety of problems ranging from ones involving the quantum Liouville equation to a problem in signal analysis.

We introduce the number-conserving quantum phase space description as a versatile tool to address fundamental aspects of quantum many-body systems. Using phase space methods we prove two alternative versions of the quantum de Finetti theorem for finite-dimensional bosonic quantum systems, which states that a reduced density matrix of a many-body quantum state can be approximated by a convex combination of product states where the error is proportional to the inverse particle number. This theorem provides a formal justification for the mean-field description of many-body quantum systems, as it shows that quantum correlations can be neglected for the calculation of few-body observables when the particle number is large. Furthermore we discuss methods to derive the exact evolution equations for quantum phase space distribution functions as well as upper and lower bounds for the ground state energy. As an important example, we consider the Bose–Hubbard model and show that the mean-field dynamics is given by a classical phase space flow equivalent to the discrete Gross–Pitaevskii equation.

We show that metrological resolution in the detection of small phase shifts provides a suitable generalization of the degrees of coherence and polarization. Resolution is estimated via Fisher information. Besides the standard two-beam Gaussian case, this approach provides also good results for multiple field components and nonGaussian statistics. This works equally well in quantum and classical optics.

Phase-space optics was one of Adolf Lohmann’s favorite research topics and appears frequently in his work. With a unique style he applied the phase-space perspective to a variety of optical problems, with a focus on more comprehensible problem solutions and a particular interest to analyze optical signals and systems in terms of their space–bandwidth product requirements. In his work, optical phase space is promoted as a hybrid of paraxial wave optics and geometrical optics. In contrast to more rigorous notions of phase space, Lohmann preferred a heuristic approach that avoids almost entirely the mathematical formalism associated with the Wigner distribution function. This approach is introduced for the first time as a distinct way to apply phase-space optics. The application of Lohmann’s phase-space optics to paraxial optical cloaking, the Lohmann–Alvarez lens, and holography is not only used to exemplify this unique form of phase-space optics, but also to highlight Lohmann’s unique and creative way of problem solving.

An appropriate design of wavefront will enable light fields propagating along arbitrary trajectories thus forming accelerating beams in free space. Previous ways of designing such accelerating beams mainly rely on caustic methods, which start from diffraction integrals and only deal with two-dimensional fields. Here we introduce a new perspective to construct accelerating beams in phase space by designing the corresponding Wigner distribution function (WDF). We find such a WDF-based method is capable of providing both the initial field distribution and the angular spectrum in need by projecting the WDF into the real space and the Fourier space respectively. Moreover, this approach applies to the construction of both two- and three-dimensional fields, greatly generalizing previous caustic methods. It may therefore open up a new route to construct highly-tailored accelerating beams and facilitate applications ranging from particle manipulation and trapping to optical routing as well as material processing.

Classical and Quantum Optics is the first text to cover classical and quantum optics as a single subject with the premise that genuine understanding of light requires mastery of both fields. It addresses all aspects of light equally, including geometric optics, wave optics, electromagnetic optics, and quantum optics. This timely book supplies the tools needed to fully explore the behavior of light. Classical and Quantum Optics:Approaches the topic from a conceptual perspective without assuming the reader's knowledgeIncludes relatively little math, making it ideal for students in applied sciences and engineeringStarts with well-established hypotheses and builds on them to explain physical effects and experimental observations Classical and Quantum Optics is suitable for emerging researchers and professionals in optics, natural sciences, engineering, and interdisciplinary fields, as well as students in physics, applied sciences, and engineering.

We propose a new formula for the adiabatic Berry phase which is based on phase-space formulation of quantum mechanics. This approach sheds a new light onto the correspondence between classical and quantum adiabatic phases — both phases are related with the averaging procedure: Hannay angle with averaging over the classical torus and Berry phase with averaging over the entire classical phase space with respect to the corresponding Wigner function.