This paper analyzes the algebraic and physical properties of the spin and orbital angular momenta of light in the quantum mechanical framework. The consequences of the fact that these are not angular momenta in the quantum mechanical sense are worked out in mathematical detail. It turns out that the spin part of the angular momentum has continuous eigenvalues. Particular attention is given to the paraxial limit, and to the definition of Laguerre--Gaussian modes for photons as well as classical light fields taking full account of the polarization degree of freedom.
It is pointed out that the solutions of the Klein-Gordon and the Dirac equation derived in the paper addressed in this Comment (and many more solutions) may be obtained from generating functions.
We demonstrate an angular momentum-enhanced protocol that permits an angular displacement estimation by using tunable squeezed Bell state and parity detection. We consider the resolution and the sensitivity, super-resolution is presented along with Heisenberg scaling sensitivity for arbitrary tunable factor, the tunable factor which can optimize the sensitivity is also discussed. Additionally, we analyze the advantages of using angular momentum via considering and comparing simulation results. Under the situation of the optimal tunable factor, the Heisenberg-limited sensitivity and $2\left(\el…
We present a mathematical proof of the algorithm allowing to generate all - symmetric and non-symmetric - total angular momentum eigenstates in remote matter qubits by projective measurements, proposed in Maser et al. [Phys. Rev. A 79, 033833 (2009)]. By deriving a recursion formula for the algorithm we show that the generated states are equal to the total angular momentum eigenstates obtained via the usual quantum mechanical coupling of angular momenta. In this way we demonstrate that the algorithm is able to simulate the coupling of N spin-1/2 systems, and to implement the required Clebsch-G…
The Berry phase for a variety of systems comprising of two angular momenta is discussed. These include the electron and proton in the ground state of the hydrogen atom (taking into account the hyperfine interaction), the positronium atom, the spin and orbital angular momenta of a single electron, muon capture by the deuteron in an external magnetic field, etc. Though the time scales involved, the underlying intrinsic Hamiltonians are quite different, as also the possible experimental probes, the geometric nature of the results for the Berry phase due to a time varying externally imposed magnet…
A new Bateman-Hillion solution to the Dirac equation for a relativistic Gaussian electron beam taking explicit account of the $4$-position of the beam waist is presented. This solution has a pure Gaussian form in the paraxial limit but beyond it contains higher order Laguerre-Gaussian components attributable to the tighter focusing. One implication of the mixed mode nature of strongly diffracting beams is that the expectation values for spin and orbital angular momenta are fractional and are interrelated to each other by \textit{intrinsic spin-orbit coupling}. Our results for these properties …
The Yukawa potential is often used to compute bound-state normalizations and energy levels of neutral atoms. By using the generalized parametric Nikiforov-Uvarov (NU) method, we have obtained the approximate analytical solutions of the radial Schrödinger equation (SE) for the Yukawa potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed forms. Some numerical results are presented and showed that these results are in good agreement with those are obtained previously by other methods. Also, we found the energy levels of the familiar pure Coulomb potential ene…
A rigorous application of the correspondence rules shows that the operator of the angular momentum of a quantum particle---corresponding to the classical magnitude $\mathbf{l}= m \mathbf{r} \wedge \mathbf{v}$---is given by $\mathbf{\hat{l}}=\mathbf{r}\wedge(-i\hbar\mathbf{\nabla} -\frac{e}{c}\mathbf{A})$ in the presence of an electromagnetic field. Thus, despite the general opinion on the corresponding rules of quantization, the eigenvalues of the angular momentum depend on the configuration of the electromagnetic field. The usual rules of commutation $[{\hat{l}}_i,{\hat{l}}_j]=i\hbarε_{ijk}{\…
We consider a system of two particles, each with large angular momentum $j$, in the singlet state. The probabilities of finding projections of the angular momenta on selected axes are determined. The generalized Bell inequalities involve these probabilities and we study them using statistical methods. We show that most of Bells inequalities cannot be violated, or are violated only marginally, in the limit $j\to \infinity$. The precision required to confirm a violation appears to be difficult to achieve. In practice, the quantum system, in spite of being entangled, becomes indistinguishable fro…
We obtain analytical solutions of the two-body spinless Salpeter (SS) equation with Yukawa potential within the conventional approximation scheme to the centrifugal term for any -state. The semi-relativistic bound state energy spectra and the corresponding normalized wave functions are calculated by means of the Nikiforov-Uvarov (NU) method. We also obtain the numerical energy spectrum of the SS equation without any approximation to centrifugal term for the same potential and compare them with the approximated numerical ones obtained from the analytical expressions. It is found that the exact …
In the study of systems that cannot be described classically, the Wigner inequality, has received only a small amount of attention. In this paper we extend the Wigner inequality - originally derived in 1969 - and show how it may be used to contradict local realism with only coincidence detections in the absence of two-outcome measurements - that is, for any system where only one possible result of a pair of potential outcomes can be registered. It thus encapsulates a much broader class of measurement schemes than could previously violate a local-realistic inequality. This is possible due to an…
We report the experimental preparation of optical superpositions of high orbital angular momenta(OAM). Our method is based on the use of spatial light modulator to modify the standard Laguerre-Gaussian beams to bear excessive phase helices. We demonstrate the surprising performance of a traditional Mach-Zehnder interferometer with one inserted Dove prism to identify these superposed twisted lights, where the high OAM numbers as well as their possible superpositions can be inferred directly from the interfered bright multiring lattices. The possibility of present scheme working at photon-count …
Yu and Littlejohn recently studied in arXiv:1104.1499 some asymptotics of Wigner symbols with some small and large angular momenta. They found that in this regime the essential information is captured by the geometry of a tetrahedron, and gave new formulae for 9j, 12j and 15j-symbols. We present here an alternative derivation which leads to a simpler formula, based on the use of the Ponzano-Regge formula for the relevant tetrahedron. The approach is generalized to Wigner 3nj-symbols with some large and small angular momenta, where more than one tetrahedron is needed, leading to new asymptotics…
We give an exact self-consistent operator description of the spin and orbital angular momenta, position, and spin-orbit interactions of nonparaxial light in free space. Both quantum-operator formalism and classical energy-flow approach are presented. We apply the general theory to symmetric and asymmetric Bessel beams exhibiting spin- and orbital-dependent intensity profiles. The exact wave solutions are clearly interpreted in terms of the Berry phases, quantization of caustics, and Hall effects of light, which can be readily observed experimentally.
In optical experiments involving a single photon that takes alternative paths through an optical system and ultimately interferes with itself (e.g., Young's double-slit experiment, Mach-Zehnder interferometer, Sagnac interferometer), there exist fundamental connections between the linear and angular momenta of the photon on the one hand, and the ability of an observer to determine the photon's path through the system on the other hand. This paper examines the arguments that relate the photon momenta (through the Heisenberg uncertainty principle) to the "which path" (German: welcher Weg) questi…
Light beams carrying orbital angular momentum, such as Laguerre-Gaussian beams, give rise to the violation of the standard dipolar selection rules during the interaction with matter yielding, in general, an exchange of angular momentum larger than hbar per absorbed photon. By means of ab initio 3D numerical simulations, we investigate in detail the interaction of a hydrogen atom with intense Gaussian and Laguerre-Gaussian light pulses. We analyze the dependence of the angular momentum exchange with the polarization, the orbital angular momentum, and the carrier-envelope phase of light, as well…
We report on the storage of Orbital Angular Momentum (OAM) of light via the phenomenon of Coherent Population Oscillation (CPO) in cold cesium atoms. The experiment is performed using a delayed four wave mixing configuration where the transverse optical information of a probe field carrying OAM associated with its azimuthal phase dependence is stored in the CPO of Zeeman sublevels of the hyperfine transition $F=3\rightarrow F^{\prime}=2$ of cesium $D_{2}$ line. We also demonstrate experimentally the simultaneous storage and retrieval of different OAM states propagating along different directio…
We review and re-examine the description and separation of the spin and orbital angular momenta (AM) of an electromagnetic field in free space. While the spin and orbital AM of light are not separately-meaningful physical quantities in orthodox quantum mechanics or classical field theory, these quantities are routinely measured and used for applications in optics. A meaningful quantum description of the spin and orbital AM of light was recently provided by several authors, which describes separately conserved and measurable integral values of these quantities. However, the electromagnetic fiel…
The usual approximation scheme is used to study the solution of the Duffin-Kemmer-Petiau (DKP) equation for a vector Yukawa potential in the framework of the parametric Nikiforov-Uvarov (NU) method. The approximate energy eigenvalue equation and the corresponding wave function spinor components are calculated for arbitrary total angular momentum in closed form. Further, the approximate energy equation and wave function spinor components are also given for case. A set of parameter values is used to obtain the numerical values for the energy states with various values of quantum levels
We use the tridiagonal representation approach to solve the radial Schrödinger equation for an inverse power-law potential of a combined quartic and sextic degrees and for all angular momenta. The amplitude of the quartic singularity is larger than that of the sextic but the signs are negative and positive, respectively. It turns out that the system has a finite number of bound states, which is determined by the larger ratio of the two singularity amplitudes. The solution is written as a finite series of square integrable functions written in terms of the Bessel polynomial.
Single photons with helical phase structures may carry a quantized amount of orbital angular momentum (OAM) and their entanglement is important for quantum information science and fundamental tests of quantum theory. Because there is no theoretical upper limit on how many quanta of OAM a single photon can carry, it is possible to create entanglement between two particles with an arbitrary high difference in the quantum number. By transferring polarization entanglement to OAM with an interferometric scheme, we generate and verify entanglement up to 600 quanta difference in the orbital angular m…
Magnetostatic modes supported by a ferromagnetic sphere have been known as the Walker modes, each of which possesses an orbital angular momentum as well as a spin angular momentum along a static magnetic field. The Walker modes with non-zero orbital angular momenta exhibit topologically non-trivial spin textures, which we call \textit{magnetic quasi-vortices}. Photons in optical whispering gallery modes supported by a dielectric sphere possess orbital and spin angular momenta forming \textit{optical vortices}. Within a ferromagnetic, as well as dielectric, sphere, two forms of vortices interac…
Coupling of quantum angular momenta: an insight into analogic/discrete and local/global models of computation
在过去的几年里，有一个动荡的活动，旨在为量子计算引入新的概念方案。Marzuoli A and Rasetti M 2002，2005 a中提出的方法依赖于SU（2）角动量的（再）耦合理论，可以被视为对基于“量子位”和布尔门的通常量子电路模型的自旋变量的任意值的推广。计算状态属于由离散和连续参数标记的有限维希尔伯特空间，并且幺正门可以依赖于在有限值集以及连续（角）变量上的量子数。这样一个框架是一个理想的游乐场，讨论离散（数字）和模拟计算过程，以及它们之间的关系发生时，一个一致的半经典限制发生在离散量子门。当使用纯离散的幺正门时，模拟器自然地被建模为量子有限状态的家庭-机器，这些机器反过来代表拓扑量子计算模型的离散版本。我们认为，我们的模型体现了一种统一的计算范式的灵感来自自然，甚至更雄心勃勃的，一个通用的设置，适当编码的量子符号操作的组合，拓扑和代数问题可能会找到他们的“自然”的计算参考模型。
There is an interesting but not so popular quantity called pseudo orbital angular momentum (OAM) in the Landau-level system, besides the well-known canonical and mechanical OAMs. The pseudo OAM can be regarded as a gauge-invariant extension of the canonical OAM, which is formally gauge invariant and reduces to the canonical OAM in a certain gauge. Since both of the pseudo OAM and the mechanical OAM are gauge invariant, it is impossible to judge which of those is superior to the other solely from the gauge principle. However, these two OAMs have totally different physical meanings. The mechanic…
We present a quantum mechanical analysis of the orbital angular momentum of a class of recently discovered elliptically-symmetric stable light fields --- the so-called Ince-Gauss modes. We study, in a fully quantum formalism, how the orbital angular momentum of these beams varies with their ellipticity and discover several compelling features, including: non-monotonic behavior, stable beams with real continuous (non-integer) orbital angular momenta, and orthogonal modes with the same orbital angular momenta. We explore, and explain in detail, the reasons for this behavior. These features may h…
We use the mathematical structure of group algebras and $H^{+}$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are ${\rm SU}(2)$ and its quotient ${\rm SO}(3,\mathbb{R})$. The scheme developed is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. The oth…
Many methods now exist to prepare free electrons into orbital angular momentum states, and the predicted applications of these electron states as probes of materials and scattering processes are numerous. The development of electron orbital angular momentum measurement techniques has lagged behind. We show that coupling between electron orbital angular momentum and a spatially varying magnetic field produces an angular momentum-dependent focusing effect. We propose a design for an orbital angular momentum measurement device built on this principle. As the method of measurement is non-interfero…
Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three SU(2) angular momentum operators, can be analyzed on the basis of a discrete Schroedinger-like equation which provides a semiclassical Hamiltonian picture of the evolution of a `quantum of space', as shown by the authors in a recent paper. Emphasis is given here to the formalization in terms of a quadratic symmetry algebra and its automorphism group. This view is related to the Askey scheme, the hierarchical structure which includes all hypergeometric polynomials of one (discrete or continuou…
A ferromagnetic sphere can support \textit{optical vortices} in forms of whispering gallery modes and \textit{magnetic quasi-vortices} in forms of magnetostatic modes with non-trivial spin textures. These vortices can be characterized by their orbital angular momenta. We experimentally investigate Brillouin scattering of photons in the whispering gallery modes by magnons in the magnetostatic modes, zeroing in on the exchange of the orbital angular momenta between the optical vortices and the magnetic quasi-vortices. We find that the conservation of the orbital angular momentum results in diffe…
The Wigner rotation matrix ($d$-function), which appears as a part of the angular-momentum-projection operator, plays a crucial role in modern nuclear-structure models. However, it is a long-standing problem that its numerical evaluation suffers from serious errors and instability, which hinders precise calculations for nuclear high-spin states. Recently, Tajima [Phys. Rev. C 91, 014320 (2015)] has made a significant step toward solving the problem by suggesting the high-precision Fourier method, which however relies on formula-manipulation softwares. In this paper we propose an effective and …
