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.

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.

Cavity‐Assisted Atomic External Momenta State Teleportation

AbstractThe quantized momentum eigenstates of neutral atoms offer high resistance to decoherence and thus suggests a viable alternative for quantum informatics tasks including teleportation. In this work, a scheme is suggested to teleport an unknown superposition of two distinct external momentum states of a neutral atom onto the similar momentum states of a distant atom at the receiving end through the off‐resonant atomic Bragg diffraction (ABD). In order to teleport this unknown state from the sender, that is, Alice, to the receiver, that is, Bob, a pre‐existing entangled link formed through a high‐Q cavity at Alice's end and momentum states of a neutral atom at the Bob's end are utilized. Further, while citing the realistic experimental parameters, it is demonstrated that the proposal can easily be implemented experimentally under the prevailing cavity‐QED atom‐field research scenario. The proposal is also generalized to cover the teleportation of the multipartite entangled momenta states and the case of multipartite GHZ entangled states is elucidated in details. However, the procedure is generic and can be extended to cover the teleportation schematics for any arbitrary atomic momenta states.

Scattering on self-dual Taub-NUT

We derive exact solutions of massless free field equations and tree-level two-point amplitudes up to spin 2 on self-dual Taub-NUT (SDTN) spacetime, as well as on its single copy, the self-dual dyon (SDD). We use Killing spinors to build analogues of momentum eigenstates, finding that, in the spirit of color-kinematics duality, those for the SDD lift directly to provide states on the SDTN background if one replaces charge with energy. We discover that they are forced to have faster growth at infinity than in flat space due to the topological non-triviality of these backgrounds. The amplitudes for massless scalars and spinning particles in the and helicity configurations vanish for generic kinematics as a consequence of the integrability of the self-dual sector. The amplitudes are non-vanishing and we compute them exactly in the backgrounds, which are treated non-perturbatively. It is explained how spin is easily introduced via a Newman–Janis imaginary shift along the spin-vector leading directly to the well-known exponential factor in the dot product of the spin with the momenta. We also observe a double copy relation between the gluon amplitude on a SDD and graviton amplitude on a SDTN spacetime.

A reduced inner product for kink states

Solitons in classical field theories correspond to states in quantum field theories. If the spatial dimension is infinite, then momentum eigenstates are not normalizable. This leads to infrared divergences, which are generally regularized via wave packets or by compactification. However, in some applications both possibilities are undesirable. In the present note, we introduce a finite inner product on translation-invariant kink states that allows us to compute probabilities involving these nonnormalizable states. Essentially, it is the quotient of the usual inner product by the translation group. We present a surprisingly simple formula for the reduced inner product, which requires no knowledge of the zero-mode dependence of the states but includes a correction which accounts for the mixing between zero modes and normal modes as the kink moves. As an application, we show that initial and final state corrections to meson multiplication vanish. However, we find that the pole of the subleading term in the initial state requires an infinitesimal imaginary shift.

Infrared finite scattering theory in quantum field theory and quantum gravity

It has been known since the earliest days of quantum field theory (QFT) that infrared divergences arise in scattering theory with massless fields. These infrared divergences are manifestations of the memory effect: At order $1/r$ a massless field generically will not return to the same value at late retarded times ($u\ensuremath{\rightarrow}+\ensuremath{\infty}$) as it had at early retarded times ($u\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$). There is nothing singular about states with memory, but they do not lie in the standard Fock space. Infrared divergences are merely artifacts of trying to represent states with memory in the standard Fock space. If one is interested only in quantities directly relevant to collider physics, infrared divergences can be successfully dealt with by imposing an infrared cutoff, calculating inclusive quantities, and then removing the cutoff. However, this approach does not allow one to treat memory as a quantum observable and is highly unsatisfactory if one wishes to view the $S$-matrix as a fundamental quantity in QFT and quantum gravity, since the $S$-matrix itself is undefined. In order to have a well-defined $S$-matrix, it is necessary to define ``in'' and ``out'' Hilbert spaces that incorporate memory in a satisfactory way. Such a construction was given by Faddeev and Kulish for quantum electrodynamics (QED) with a massive charged field. Their construction can be understood as pairing momentum eigenstates of the charged particles with corresponding memory representations of the electromagnetic field to produce states of vanishing large gauge charges at spatial infinity. (This procedure is usually referred to as ``dressing'' the charged particles.) We investigate this procedure for QED with massless charged particles and show that, as a consequence of collinear divergences, the required dressing in this case has an infinite total energy flux, so that the states obtained in the Faddeev-Kulish construction are unphysical. An additional difficulty arises in Yang-Mills theory, due to the fact that the ``soft Yang-Mills particles'' used for the dressing contribute to the Yang-Mills charge-current flux, thereby invalidating the procedure used to construct eigenstates of large gauge charges at spatial infinity. We show that there are insufficiently many charge eigenstates to accommodate scattering theory. In quantum gravity, the analog of the Faddeev-Kulish construction would attempt to produce a Hilbert space of eigenstates of supertranslation charges at spatial infinity. Again, the Faddeev-Kulish dressing procedure does not produce the desired eigenstates because the dressing contributes to the null memory flux. We prove that there are no eigenstates of supertranslation charges at spatial infinity apart from the vacuum. Thus, analogs of the Faddeev-Kulish construction fail catastrophically in quantum gravity. We investigate some alternatives to Faddeev-Kulish constructions but find that these also do not work. We believe that if one wishes to treat scattering at a fundamental level in quantum gravity---as well as in massless QED and Yang-Mills theory---it is necessary to approach it from an algebraic viewpoint on the ``in'' and ``out'' states, wherein one does not attempt to ``shoehorn'' these states into some prechosen ``in'' and ``out'' Hilbert spaces. We outline the framework of such a scattering theory, which would be manifestly infrared finite.

Kink form factors

We use a recently constructed linearized soliton sector perturbation theory to calculate the form factors relevant to the elastic scattering of ultrarelativistic mesons off of nonrelativistic kinks. Both localized kink wave packets and also delocalized momentum eigenstate kinks are considered. In the delocalized case, the leading term is just the classical kink solution, as was found by Goldstone and Jackiw. The leading delocalized quantum correction agrees with that found by Gervais, Jevicki and Sakita in the ϕ4 model and Weisz in the Sine-Gordon model. In the case of localized kink wave packets, some corrections are found which scale with the wave packet width, and so will be relevant for the coherent scattering of mesons off of kink wave packets.

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.

Spinors and conformal correlators

In conformal field theory, momentum eigenstates can be parameterized by a pair of real spinors, in terms of which special conformal transformations take a simpler form. This well-known fact allows to express 2-point functions of primary operators in the helicity basis, exposing the consequences of unitarity. What is less known is that the same pair of spinors can be used, together with a pair of scalar quantities, to parameterize 3-point functions. We develop this formalism in 3 dimensions and show that it provides a simple realization of the operator product expansion (OPE) for scalar primary operators acting on the vacuum.

Polaron-Depleton Transition in the Yrast Excitations of a One-Dimensional Bose Gas with a Mobile Impurity

We present exact numerical data for the lowest-energy momentum eigenstates (yrast states) of a repulsive spin impurity in a one-dimensional Bose gas using full configuration interaction quantum Monte Carlo (FCIQMC). As a stochastic extension of exact diagonalization, it is well suited for the study of yrast states of a lattice-renormalized model for a quantum gas. Yrast states carry valuable information about the dynamic properties of slow-moving mobile impurities immersed in a many-body system. Based on the energies and the first and second-order correlation functions of yrast states, we identify different dynamical regimes and the transitions between them: The polaron regime, where the impurity’s motion is affected by the Bose gas through a renormalized effective mass; a regime of a gray soliton that is weakly correlated with a stationary impurity, and the depleton regime, where the impurity occupies a dark or gray soliton. Extracting the depleton effective mass reveals a super heavy regime where the magnitude of the (negative) depleton mass exceeds the mass of the finite Bose gas.

Wave packets in QFT: Leading order width corrections to decay rates and clock behavior under Lorentz boosts

Decay rates in quantum field theory (QFT) are typically calculated assuming the particles are represented by momentum eigenstates (i.e., plane waves). However, strictly speaking, localized free particles should be represented by wave packets. This yields width corrections to the decay rate and to the clock behavior under Lorentz boosts. We calculate the decay rate of a particle of mass $M$ modeled as a Gaussian wave packet of width $a$ and centered at zero momentum. We find the decay rate to be ${\mathrm{\ensuremath{\Gamma}}}_{0}[1\ensuremath{-}\frac{3{a}^{2}}{4{M}^{2}}+\mathcal{O}(\frac{{a}^{4}}{{M}^{4}})]$, where ${\mathrm{\ensuremath{\Gamma}}}_{0}$ is the decay rate of the particle at rest treated as a plane wave. The leading correction is then of order $\frac{{a}^{2}}{{M}^{2}}$. We then perform a Lorentz boost of velocity $v$ on the above Gaussian and find that its decay rate does not decrease exactly by the Lorentz factor $\sqrt{1\ensuremath{-}{v}^{2}}$. There is a correction of order $\frac{{a}^{2}{v}^{2}}{{M}^{2}}$. Therefore, the decaying wave packet does not act exactly like a typical clock under Lorentz boosts, and we refer to it is a ``WP clock'' (wave packet clock). A WP clock does not move with a single velocity relative to an observer but has a spread in velocities (more specifically, a spread in momenta). Nonetheless, it is best viewed as a single clock as the wave packet represents a one-particle state in QFT. WP clocks do not violate Lorentz symmetry and are not based on new physics; they are a consequence of the combined requirements of special relativity, quantum mechanics, and localized free particles.

Magnetoelectric torque and edge currents caused by spin-orbit coupling

Using a tight-biding model, we elaborate that the previously discovered out-of-plane polarized helical edge spin current caused by Rashba spin-orbit coupling can be attributed to the fact that in a strip geometry, a positive momentum eigenstate does not always have the same spin polarization at the edge as the corresponding negative momentum eigenstate. In addition, in the presence of a magnetization pointing perpendicular to the edge, an edge charge current is produced, which can be chiral or nonchiral depending on whether the magnetization lies in-plane or out-of-plane. The spin polarization near the edge develops a transverse component orthogonal to the magnetization, which is antisymmetric between the two edges and tends to cause a noncollinear magnetic order between the two edges. If the magnetization only occupies a region near one edge, or in an irregular shaped quantum dot, this transverse component has a nonzero average, rendering a gate voltage-induced magnetoelectric torque without the need of a bias voltage. We also argue that other types of spin-orbit coupling that can be obtained from the Rashba type through a unitary transformation, such as the Dresselhaus spin-orbit coupling, will have similar effects too.

Dressings and Asymptotic Symmetries in Quantum Field Theory

In quantum field theories with massless gauge bosons, the conventional formulation of scattering amplitudes in terms of momentum eigenstates lead to infrared divergences. To resolve this one should dress charged particles with an infinite number of low-energy (soft) gauge bosons. These dressings are closely related to the asymptotic symmetries of the the- ory. The aim of this dissertation is to investigate this relation in detail, and study the implications of dressings in a quantum field theoretical context in asymptotically flat and Schwarzschild spacetimes. In particular, the asymptotic symmetry of interest in gravity is the BMS supertranslation. First, we demonstrate in perturbative quantum gravity that the dressings facilitate charge conservation by carrying a definite supertranslation charge. We then use this property to derive dressed states from the charge conservation laws of asymp- totic symmetries. We develop a unified quantum mechanical framework for the construction of dressings at the null infinity and the horizon of a Schwarzschild black hole, and show that the black hole soft hairs are planted on the horizon by dressed particles falling into the black hole. This framework is then extended to the magnetic parity soft charges of electro- magnetism and gravity. It is shown that one can construct ’t Hooft line dressings at the asymptotic boundaries, which are charged under the magnetic large gauge transformations and the dual supertranslations. Finally, we study the standard and dual supertranslation charges at the black hole horizon. We find central terms in their algebra in the presence of singularities in the parameter functions, which hints at further interesting soft structures yet to be revealed.

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.

Nonleptonic two-body weak decays of Λb in a modified MIT bag model

We study the nonleptonic two-body weak decays of $\Lambda_b$ by modifying the MIT bag model without introducing new parameters to construct the momentum eigenstates of the baryons. We find that the branching ratios of $\Lambda_b^0 \to \Lambda_c^+ \pi^-$, $\Lambda_c^+ K^-$, $p \pi^-$ and $pK^-$ are $(4.5\pm0.2)\times 10^{-3}$, $(3.4\pm 0.1)\times 10^{-4}$, $(5.0\pm 0.5)\times 10^{-6}$ and $(6.0\pm 0.7) \times 10^{-6}$, which are all well consistent with the current experimental data, respectively. We also explore P and CP asymmetries for the decays of $\Lambda_b^0 \to p (\pi^-,K^-)$. In particular, we obtain that the direct CP-violating rate asymmetries in $\Lambda_b^0 \to p \pi^-$ and $\Lambda_b^0 \to p K^-$ are around $-4.4\%$ and $6.7\%$, in comparison with $(-2.5\pm2.9)\%$ and $(-2.5\pm2.2)\%$ from the Particle Data Group in 2020, respectively.

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.