The orbital angular momentum quantum number of Laguerre-Gauss beams has received an explosively increasing amount of attention over the past twenty years. However, often overlooked is the so-called radial number of these beams. We present a derivation of the differential operator formalism of this "forgotten" quantum number. We then draw some connections between this new formalism and the effect the radial number has on beam stability with possible application to quantum communication. We also briefly outline how the radial number is tied to other physical aspects of the beam (such as the Gouy phase, and radial confinement). These do not necessarily constitute finished results, but are instead meant to stimulate discussion of this interesting and often overlooked physical parameter.
Photons propagating in Laguerre-Gaussian modes have characteristic orbital angular momentums, which are fundamental optical degrees of freedom. The orbital angular momentum of light has potential application in high capacity optical communication and even in quantum information processing. In this work, we experimentally construct a ring cavity with 4 lenses and 4 mirrors that is completely degenerate for Laguerre-Gaussian modes. By measuring the transmission peaks and patterns of different modes, the ring cavity is shown to supporting more than 31 Laguerre-Gaussian modes. The constructed dege…
We theoretically describe the quantum properties of a large Fresnel number degenerate optical parametric oscillator with spherical mirrors that is pumped by a Gaussian beam. The resonator is tuned so that the resonance frequency of a given transverse mode family coincides with the down-converted frequency. After demonstrating that only the lower orbital angular momentum (OAM) Laguerre-Gauss modes are amplified above threshold, we focus on the quantum properties of the rest of (classically empty) modes. We find that combinations of opposite OAM (Hybrid Laguerre-Gauss modes) can exhibit arbitrar…
By introducing a new entangled state representation, we show that the Laguerre-Gaussian (LG) mode is just the wave function of the common eigenvector of the orbital angular momentum and the total photon number operators of 2-d oscillator, which can be generated by 50:50 beam splitter with the phase difference phi=Pi/2ϕ between the reflected and transmitted fields. Based on this and using the Weyl ordering invariance under similar transforms, the Wigner representation of LG is directly obtained, which can be considered as the generalized Wigner transform of Hermite Gaussian modes.
We introduce an operator linked with the radial index in the Laguerre-Gauss modes of a two-dimensional harmonic oscillator in cylindrical coordinates. We discuss ladder operators for this variable, and confirm that they obey the commutation relations of the su(1,1) algebra. Using this fact, we examine how basic quantum optical concepts can be recast in terms of radial modes.
Transfer mechanism of orbital angular moment(OAM) of light to trapped ground-state atoms under paraxial approximation is well known. Here we show how optical OAM of a Laguerre-Gaussian(LG) beam under paraxial approximation can be transferred to trapped Rydberg atoms. Optical OAM is shown to be transferable to a Rydberg electronic state in dipole transition. The Gaussian part of the profile of the LG beam, which is generally neglected , is found to have an important effect on the OAM transfer to the Rydberg atoms. Numerical calculations are calculated based on this theory for Rubidium Rydberg a…
The analysis of twisted (vortex) paraxial photons and electrons is fulfilled in the framework of relativistic quantum mechanics. The use of the Foldy-Wouthuysen representation radically simplifies a description of relativistic electrons and clarifies fundamental properties of twisted particles. It is demonstrated that the twisted photon defined by the Laguerre-Gauss wave function is luminal. Its apparent subluminality takes place because the photon energy is contributed by the hidden transversal motion while the average transversal momentum vanishes. Nevertheless, the Laguerre-Gauss photon can…
We report a new set of Laguerre-Gaussian wave-packets that propagate with periodical self-focusing and finite beam width in weakly guiding inhomogeneous media. These wave-packets are solutions to the paraxial form of the wave equation for a medium with parabolic refractive index. The beam width is defined as a solution of the Ermakov equation associated to the harmonic oscillator, so its amplitude is modulated by the strength of the medium inhomogeneity. The conventional Laguerre-Gaussian modes, available for homogenous media, are recovered as a particular case.
Gaussian geometric discord, entanglement and EPR-steering of two rotational mirrors in a double Laguerre-Gaussian cavity optomechanics in the presence of YIG sphere
It is well known that photons can carry a spatial structure akin to a "twisted" or "rippled" wavefront. Such structured light fields have sparked significant interest in both classical and quantum physics, with applications ranging from dense communications to light-matter interaction. Harnessing the full advantage of transverse spatial photonic encoding using the Laguerre-Gaussian (LG) basis in the quantum domain requires control over both the azimuthal (twisted) and radial (rippled) components of photons. However, precise measurement of the radial photonic degree-of-freedom has proven to be …
Spatially structured light has opened a wide range of opportunities for enhanced imaging as well as optical manipulation and particle confinement. Here, we show that phase-coherent illumination with superpositions of radial Laguerre-Gauss (LG) beams provides improved localization for bright optical tweezer traps, with narrowed radial and axial intensity distributions. Further, the Gouy phase shifts for sums of tightly focused radial LG fields can be exploited for novel phase-contrast strategies at the wavelength scale. One example developed here is the suppression of interference fringes from …
We consider the excitation of Rydberg states through photons carrying an intrinsic orbital angular momentum degree of freedom. Laguerre-Gauss modes, with a helical wave-front structure, correspond to such a set of laser beams, which carry some units of orbital angular momentum in their propagation direction. We demonstrate that, in a proper geometrical setting, this orbital angular momentum can be transferred to the internal degrees of freedom of the atoms, thus violating the standard dipolar selection rules. Higher orbital angular momentum states become accessible through a single photon exci…
Laguerre-Gauss (LG) modes represent an orthonormal basis set of solutions of the paraxial wave equation. LG are characterized by two integer parameters $n$ and $\ell$ that are related to the radial and azimuthal profile of the beam. The physical dimension of the mode is instead determined by the beam waist parameter $w_0$: only LG modes with the same $w_0$ satisfy the orthogonality relation. Here, we derive the scalar product between two LG modes with different beam waists and show how this result can be exploited to derive different expansions of a generic beam in terms of LG modes. In partic…
The Laguerre-Gauss modes are a class of fundamental and well-studied optical fields. These stable, shape-invariant photons - exhibiting circular-cylindrical symmetry - are familiar from laser optics, micro-mechanical manipulation, quantum optics, communication, and foundational studies in both classical optics and quantum physics. They are characterized, chiefly, by two modes numbers: the azimuthal index indicating the orbital angular momentum of the beam - which itself has spawned a burgeoning and vibrant sub-field - and the radial index, which up until recently, has largely been ignored. In …
We study the directional properties of a radiation field emitted by a geometrically small system composed of two identical two-level emitters located at short distances and driven by an optical vortex beam, a Laguerre-Gaussian beam which possesses a structured phase and amplitude. We find that the system may operate as a nanoantenna for controlled and tunable directional emission. Polar diagrams of the radiation intensity are presented showing that a constant phase or amplitude difference at the positions of the emitters plays an essential role in the directivity of the emission. We find that …
We study the establishment of vortex entanglement in remote and weakly interacting Bose Einstein condensates. We consider a two-mode photonic resource entangled in its orbital angular momentum (OAM) degree of freedom and, by exploiting the process of light-to-BEC OAM transfer, demonstrate that such entanglement can be efficiently passed to the matter-like systems. Our proposal thus represents a building block for novel low-dissipation and long-memory communication channels based on OAM. We discuss issues of practical realizability, stressing the feasibility of our scheme and present an operati…
Recently, the generation and detection of structured light field have drawn a great deal of attention, due to their importance in high-capacity optical data storage and quantum technology. In this letter, we explore the azimuthal modulation of optical transparency in a four-level double-V type quantum system near a plasmonic nanostructure. A Laguerre-Gaussian beam and the interaction of the system with free-space vacuum modes have been employed to create the phase-dependent absorption of a non-vortex probe field. First, we demonstrate how to identify the azimuthal index associated with the con…
We show that propagating optical fields bearing an axial symmetry are not truly hollow in spite of a null electric field on-axis. The result, obtained by general arguments based upon the vectorial nature of electromagnetic fields, is of particular significance in the situation of an extreme focusing, when the paraxial approximation no longer holds. The rapid spatial variations of fields with a "complicated" spatial structure are extensively analyzed in the general case and for a Laguerre-Gauss beam 2 as well, notably for beams bearing a |l| = 2 orbital angular momentum for which a magnetic fie…
The connection between Poincaré spheres for polariz-ation and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic 2-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite-Gaussian, Laguerre-Gaussian and Generalized Hermite-Laguerre Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the 2-dimensional oscillator, which acquire an extra significance as fa…
Counter-propagating co-axial Laguerre-Gaussian (LG) beams are considered, not in the familiar scenario where the focal planes coincide at $z=0$, but when they are separated by a finite axial distance $d$. The simplest case is where both beams are doughnut beams which have the same linear polarisation. The total fields of this system are shown to display novel amplitude and phase distributions and are shown to give rise to a ring or a finite ring lattice composed of double rings and single central ring. When the beams have slightly different frequencies the ring lattice pattern becomes a finite…
This is a collection of notes that are about spectral form factors of standard ensembles in the random matrix theory, written for the practical usage of current study of late time quantum chaos. More precisely, we consider Gaussian Unitary Ensemble (GUE), Gaussian Orthogonal Ensemble (GOE), Gaussian Symplectic Ensemble (GSE), Wishart-Laguerre Unitary Ensemble (LUE), Wishart-Laguerre Orthogonal Ensemble (LOE), and Wishart-Laguerre Symplectic Ensemble (LSE). These results and their physics applications cover a three-fold classification of late time quantum chaos in terms of spectral form factors…
The identification of orbital angular momentum (OAM) as a fundamental property of a beam of light nearly twenty-five years ago has led to an extensive body of research around this topic. The possibility that single photons can carry OAM has made this degree of freedom an ideal candidate for the investigation of complex quantum phenomena and their applications. Research in this direction has ranged from experiments on complex forms of quantum entanglement to the interaction between light and quantum states of matter. Furthermore, the use of OAM in quantum information has generated a lot of exci…
