Angular Momentum Eigenstates Research Articles

CONSTRUCTING QUANTUM ANGULAR MOMENTUM L3 IN SPECIFIC DIRECTION BY USING U(1) GROUP

The purpose of this work is to investigate the mathematical structure of finite quantum angular momentum in a specific direction L3 which can be constructed from the representation of the U(1) group. The angular momentum eigenstate is invariant under Abelian rotation symmetry. The character group of U(1) is constructed to show that there exists an additive unitary operator for the angular momentum eigenstate, and the rotation eigenstate is invariant under it. The Weyl relation is proved by showing that the angle and angular momentum L3 are the canonical conjugate pair of observables.

Entanglement Generation via Single-Qubit Rotations in a Torn Hilbert Space

We propose an efficient yet simple protocol to generate arbitrary symmetric entangled states with only global single-qubit rotations in a torn Hilbert space. The system is based on spin-1/2 qubits in a resonator such as atoms in an optical cavity or superconducting qubits coupled to a main bus. By sending light or microwave into the resonator, it induces ac Stark shifts on particular angular-momentum eigenstates (Dicke states) of qubits. Then we are able to generate barriers that hinder transitions between adjacent Dicke states and tear the original Hilbert space into pieces. Therefore, a simple global single-qubit rotation becomes highly nontrivial, and thus generates entanglement among the many-body system. By optimal control of energy shifts on Dicke states, we are able to generate arbitrary symmetric entangled states. We also exemplify that we can create varieties of useful states with near-unity fidelities in only one or very few steps, including W states, spin-squeezed states (SSSs), and Greenberger-Horne-Zeilinger states. Particularly, the SSS can be created by only one step with a squeezing parameter ξR2∼1/N0.843 approaching the Heisenberg limit. Our finding establishes a way for universal entanglement generations with only single-qubit drivings where all the multiple-qubit controls are integrated into simply switching on or off microwave. It has direct applications in the variational quantum optimizer, which is available with existing technology. Published by the American Physical Society 2024

Multi‐Concentric Generalized Perfect Vortex Beams by Hybrid Phase Metasurfaces

AbstractMetasurface‐based vortex beams are expected to overcome the current communication challenges of massive data and limited bandwidth by their theoretical infinite and mutually orthogonal states. However, limited by the spatial extension of array vortices, only finite orbital angular momentum (OAM) eigenstates are available, of which the expansion will boost the bandwidth toward an inexhaustible stage. In this manuscript, multi‐concentric generalized perfect vortex beams (G‐PVBs) are experimentally generated by the fabricated hybrid phase metasurfaces integrating generalized focusing phases and circular domain function into vortex phases. Diverse vortex shapes and helical lobes within beam patterns and spiral interferograms are both arranged in a non‐overlapping concentric manner instead of spatial arrays, respectively, breaking the limitation of symmetric shape while avoiding inter‐modal cross‐talk. The vortex shape, topological charge and vortex number are decoupled mutually in orthogonal polarization states, realizing both space and polarization multiplexing. Furthermore, shaped into anomalous profiles with asymmetric states of polarization, multi‐concentric vector G‐PVBs are experimentally generated to transmit parallelly stable order information with high robustness in turbulence environments. The proposed multi‐concentric G‐PVBs simultaneously expanded the novel vortex dimension by anomalous shapes and information channels of multi‐concentric OAM modes, providing potential application in high‐capacity vortex optical communication and information encryption.

Dynamic and static dipole polarizability of an Aharonov–Bohm ring

The properties of nanoscale rings are sensitive to Aharonov–Bohm fluxes threading the ring. In particular, neighboring angular momentum eigenstates become degenerate in cases of half-integer flux. Such degeneracies have a profound effect on the dipole polarizability leading to a divergent response. We analyze circular finite-width nanorings and derive a simple and compact expression for the polarizability valid for arbitrary ring geometry, frequency and magnetic flux. The dipole response is significantly reduced by finite-width effects, yet divergencies at half-integer flux are predicted irrespective of width. Finally, a non-divergent (but large) response is restored through a non-perturbative treatment of both magnetic and electric fields.

Excited hadron channels in hadronization

The proper treatment of hadronic resonances plays an important role for many aspects of heavy ion collisions. We expect this to be the case also for hadronization, due to the large degeneracies of excited states, and the abundant production of hadrons from their decays. We show how a comprehensive treatment of excited meson states can be incorporated into quark recombination, and in extension, into Hybrid Hadronization. We discuss in detail the quantum mechanics of forming excited states, utilizing the Wigner distribution functions of angular momentum eigenstates of isotropic 3-D harmonic oscillators. We describe how resonance decays can be handled, based on a set of minimal assumptions, by creating an extension of hadron decays in PYTHIA 8. Finally, we present a study of hadron production by jets using PYTHIA and Hybrid Hadronization with excited mesons up to orbital angular momentum L = 4. We find that states up to L = 2 are produced profusely by quark recombination.

Angular momentum eigenstates of the isotropic 3-D harmonic oscillator: Phase-space distributions and coalescence probabilities

The isotropic 3-dimensional harmonic oscillator potential can serve as an approximate description of many systems in atomic, solid state, nuclear, and particle physics. In particular, the question of 2 particles binding (or coalescing) into angular momentum eigenstates in such a potential has interesting applications. We compute the probabilities for coalescence of two distinguishable, non-relativistic particles into such a bound state, where the initial particles are represented by generic wave packets of given average positions and momenta. We use a phase-space formulation and hence need the Wigner distribution functions of angular momentum eigenstates in isotropic 3-dimensional harmonic oscillators. These distribution functions have been discussed in the literature before but we utilize an alternative approach to obtain these functions. Along the way, we derive a general formula that expands angular momentum eigenstates in terms of products of 1-dimensional harmonic oscillator eigenstates.

The Spinning Electron

The notion introduced by Ohanian that spin is a wave property is implemented, both in Dirac and in Schrödinger quantum mechanics. We find that half-integer spin is the consequence of azimuthal dependence in two of the four spinor components, relativistically and non-relativistically. In both cases the spinor components are free particle wavepackets; the total wavefunction is an eigenstate of the total angular momentum in the direction of net particle motion. In the non-relativistic case we make use of the Lévy-Leblond result that four coupled non-relativistic wave equations, equivalent to the Pauli-Schrödinger equation, represent particles of half-integer spin, with g-factor 2. An example of an exact Gaussian solution of the non-relativistic equations is illustrated.

Coherent control mechanisms in bichromatic multiphoton ionization

Free electron vortices (FEVs) generated by multiphoton ionization (MPI) with ultrashort laser pulses have attracted significant attention due to their varied symmetries and unusual topological properties. We study two physical mechanisms of coherent control in atomic MPI with bichromatic polarization-shaped femtosecond laser pulses which give rise to the rich variety of FEVs. In the experiments, we combine pulse shaping of a carrier-envelope phase-stable supercontinuum with photoelectron tomography to generate and reconstruct three-dimensional photoelectron momentum distributions (PMDs). Simultaneous measurements of energetically separated photoelectrons from intraband and interband interference in a single PMD allow us to compare phase and polarization control of the angular distributions by both mechanisms. We investigate phase control in three scenarios: first, counterrotating circularly polarized pulses are employed to contrast the phase-insensitive angular momentum eigenstate created by intraband interference via frequency mixing with the phase-sensitive c 7 rotationally symmetric FEV from pure interband interference of two single-color ionization pathways. In the second scenario, we use orthogonal linearly polarized pulses to compare the phase control properties of a six-lobed angular momentum wave packet from intraband interference to those of a complex shaped interband PMD in the presence of phase fluctuations. Finally, we demonstrate phase control of a photoelectron hologram from mixed interband interference. In a (3 + 1) resonance enhanced MPI scheme, the red pump pulse induces a bound electron wave packet probed by the time-delayed blue pulse. The latter simultaneously creates a reference wave packet by three-photon ionization to form the photoelectron hologram. Rotation of the hologram with c 1 or c 5 rotational symmetry maps the time evolution of the bound wave packet. To analyze our results, we develop analytical expressions for the wave functions of intraband and interband interference in perturbative non-resonant MPI. The experiments are complemented with two-dimensional TDSE simulations to follow the FEV formation dynamics and to validate the physical pictures.

Vortex beam generator working in terahertz region based on transmissive metasurfaces

Owing to the eigenstates of orbital angular momentum (OAM) are infinite and orthogonal to each other, OAM can improve communication capacity infinitely. In this paper, we propose a terahertz vortex beam generator on transmissive metasurfaces. When the y-polarized terahertz wave illuminates vertically to the metasurface along the -z direction, the proposed metasurface generates terahertz vortex beams with topological charges l = ±1, l = ±2 and l = ±3 at the frequency ranging from 0.3 THz to 0.9 THz. We study the far-field radiation, electric field phase, amplitude characteristics, and mode purity with different topological charges. The simulation results are in good agreements with the theoretical predictions.

Emergence of Haldane Pseudo-Potentials in Systems with Short-Range Interactions

In the setting of the fractional quantum Hall effect we study the effects of strong, repulsive two-body interaction potentials of short range. We prove that Haldane’s pseudo-potential operators, including their pre-factors, emerge as mathematically rigorous limits of such interactions when the range of the potential tends to zero while its strength tends to infinity. In a common approach the interaction potential is expanded in angular momentum eigenstates in the lowest Landau level, which amounts to taking the pre-factors to be the moments of the potential. Such a procedure is not appropriate for very strong interactions, however, in particular not in the case of hard spheres. We derive the formulas valid in the short-range case, which involve the scattering lengths of the interaction potential in different angular momentum channels rather than its moments. Our results hold for bosons and fermions alike and generalize previous results in [6], which apply to bosons in the lowest angular momentum channel. Our main theorem asserts the convergence in a norm-resolvent sense of the Hamiltonian on the whole Hilbert space, after appropriate energy scalings, to Hamiltonians with contact interactions in the lowest Landau level.

Parameter estimation of orbital angular momentum based continuous-variable quantum key distribution

In principle, the number of orbital angular momentum (OAM) eigenstates is unbounded; therefore, OAM multiplexing could greatly increase the transmission rate. Compared with optical fibers, free-space channels fluctuate randomly over time due to the effects of atmospheric turbulence. Free-space channel estimation is necessary. Here, we study the channel parameter estimation of continuous-variable quantum key distribution (CVQKD) based on OAM, especially the influence of atmospheric turbulence on transmittance and excess noise. By comparing the secret key rate in the case of single link and multiplexing, the impact of the estimated parameters on the performance of the CVQKD system is evaluated. Simulation results show that the value of the estimated parameter will affect the secret key rate.

Angular momentum radiation from current-carrying molecular junctions

We consider the radiation of angular momentum (AM) from current-carrying molecular junctions. Using the nonequilibrium Green's function method, we derive a convenient formula for the AM radiation and apply it to a prototypical benzene molecule junction. We discuss the selection rules for inelastic transitions between the molecular angular momentum eigenstates due to a 6-fold rotational symmetry. Our study provides important insights into the generation of light with AM from DC-biased molecular junctions.

Subradiance of multilevel fermionic atoms in arrays with filling n≥2

We investigate the subradiance properties of $n\geq 2$ multilevel fermionic atoms loaded into the lowest motional level of a single trap (e.g.~a single optical lattice site or an optical tweezer). As pointed out in our previous work [arXiv:1907.05541], perfectly dark subradiant states emerge from the interplay between fermionic statistics and dipolar interactions. While in [arXiv:1907.05541] we focused on the $n=2$ case, here we provide an in-depth analysis of the single-site dark states for generic filling $n$, and show a tight connection between generic dark states and total angular momentum eigenstates. We show how the latter can also be used to understand the full eigenstate structure of the single-site problem, which we analyze numerically. Apart from this, we discuss two possible schemes to coherently prepare dark states using either a Raman transition or an external magnetic field to lift the Zeeman degeneracy. Although the analysis focuses on the single-site problem, we show that multi-site dark states can be trivially constructed in any geometry out of product states of single-site dark states. Finally, we discuss some possible implementations with alkaline-earth(-like) atoms such as $^{171}$Yb or $^{87}$Sr loaded into optical lattices, where they could be used for potential applications in quantum metrology and quantum information.

Signatures of nodeless multiband superconductivity and particle-hole crossover in the vortex cores of FeTe0.55Se0.45

Scanning tunneling experiments on single crystals of superconducting FeTe$_{0.55}$Se$_{0.45}$ have recently provided evidence for discrete energy levels inside vortices. Although predicted long ago, such levels are seldom resolved due to extrinsic (temperature, instrumentation) and intrinsic (quasiparticle scattering) limitations. We study a microscopic multiband model with parameters appropriate for FeTe$_{0.55}$Se$_{0.45}$. We confirm the existence of well-separated bound states and show that the chemical disorder due to random occupation of the chalcogen site does not affect significantly the vortex-core electronic structure. We further analyze the vortex bound states by projecting the local density of states on angular-momentum eigenstates. A rather complex pattern of bound states emerges from the multiband and mixed electron-hole nature of the normal-state carriers. The character of the vortex states changes from hole-like with negative angular momentum at low energy to electron-like with positive angular momentum at higher energy within the superconducting gap. We show that disorder in the arrangement of vortices most likely explains the differences found experimentally when comparing different vortices.

Optical images rotation and reflection with engineered orbital angular momentum spectrum

According to the concept of digital spiral imaging, we reconstruct an arbitrary structured light beam with information based on orbital angular momentum (OAM) eigenstates, i.e., the Laguerre-Gaussian (LG) modes. We propose a method to realize the rotation and reflection of an image by imparting an extra phase factor to each constituent OAM mode. If each OAM mode is added by a time-varying phase, the structured light beams can be manipulated to rotate with a constant speed, similar to a reverse rotational Doppler effect. While making a conjugate transformation to each OAM mode and then adding a constant phase factor to them, the structured light beams can be flipped over along any axis. In experiment, we take two typical patterns, e.g., a smiley face and a letter “e,” to demonstrate the effectiveness of our scheme. The scheme presented in this work may find potential applications in optical micromanipulation and remote sensing.

Calculating spherical harmonics without derivatives

The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom. It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In this work, we show how one can determine spherical harmonics in a more natural way based on operators and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but little used elsewhere). This new strategy follows naturally after one has introduced Dirac notation, computed the angular momentum algebra, and determined the action of the angular momentum raising and lowering operators on the simultaneous angular momentum eigenstates (under $\hat L^2$ and $\hat L_z$).

A large-alphabet three-party quantum key distribution protocol based on orbital and spin angular momenta hybrid entanglement

The orthogonality of the orbital angular momentum (OAM) eigenstates enables a single photon carry an arbitrary number of bits. Moreover, additional degrees of freedom (DOFs) of OAM can span a high-dimensional Hilbert space, which could greatly increase information capacity and security. Moreover, the use of the spin angular momentum---OAM hybrid entangled state can increase Shannon dimensionality, because photons can be hybrid entangled in multiple DOFs. Based on these observations, we develop a hybrid entanglement quantum key distribution (QKD) protocol to achieve three-party quantum key distribution without classical message exchanges. In our proposed protocol, a communicating party uses a spatial light modulator (SLM) and a specific phase hologram to modulate photons' OAM state. Similarly, the other communicating parties use their SLMs and the fixed different phase holograms to modulate the OAM entangled photon pairs, producing the shared key among the parties Alice, Bob and Charlie without classical message exchanges. More importantly, when the same operation is repeated for every party, our protocol could be extended to a multiple-party QKD protocol.

Orbital angular momentum eigenfunctions for fast and numerically stable evaluations of closed-form pseudopotential matrix elements.

The computation of s-type Gaussian pseudopotential matrix elements involving low powers of the distance from the pseudopotential center using Gaussian orbitals can be reduced to familiar integrals. They may be directly expressed as either simple three-center overlap integrals for even powers of the radial distance from the pseudopotential center or related to the three-center nuclear integrals of a Gaussian charge distribution for odd powers. Orbital angular momentum about each atom is added to these integrals by solid-harmonic differentiation with respect to its center. The solid-harmonic addition theorem allows all the integrals to be factored into products of invariant one-dimensional integrals involving the Gaussian exponents and angular factors that contain the azimuthal quantum numbers but are independent of all Gaussian exponents. Precomputing the angular factors allow looping over all Gaussian exponents about the three centers. The fact that solid harmonics are eigenstates of angular momentum removes the singularities seen in previous treatments of pseudopotential matrix elements.

Partial wave decomposition in Friedrichs model with self-interacting continua

We consider the nonrelativistic model of coupling bare discrete states with continuum states in which the continuum states can have interactions among themselves. By partial-wave decomposition and constraint to the conserved angular momentum eigenstates, the model can be reduced to Friedrichs-like model with additional interactions between the continua. If a kind of factorizable form factor is chosen, the model can be exactly solvable, that is, the generalized discrete eigenstates including bound states, virtual states, and resonances can all be represented using the original bare states, and so do the in-state and out-state. The exact S matrix is thus obtained. We then discuss the behaviors of the dynamically generated S-wave and P-wave discrete states as the coupling is varying when there is only one self-interacting bare continuum state. We find that even when the potential is repulsive there could also be resonances and virtual states. In the P-wave cases with attractive interaction, we find that when there is a near-threshold bound state, there will always be an accompanying virtual state and we also give a more general argument of this effect.

Alignment of wave functions for angular momentum projection

Angular momentum projection is used to obtain eigen states of angular momentum from general wave functions. Multi-configuration mixing calculation with angular momentum projection is an important microscopic method in nuclear physics. For accurate multi-configuration mixing calculation with angular momentum projection, concentrated distribution of $z$ components $K$ of angular momentum in the body-fixed frame ($K$-distribution) is favored. Orientation of wave functions strongly affects $K$-distribution. Minimization of variance of $\hat{J}_z$ is proposed as an alignment method to obtain wave functions that have concentrated $K$-distribution. Benchmark calculations are performed for $\alpha$-$^{24}$Mg cluster structure, triaxially superdeformed states in $^{40}$Ar, and Hartree-Fock states of some nuclei. The proposed alignment method is useful and works well for various wave functions to obtain concentrated $K$-distribution.