Relativistic Hydrogen-Like Atom on a Noncommutative Phase Space

The energy levels of hydrogen-like atom on a noncommutative phase space were studied in the framework of relativistic quantum mechanics. The leading order corrections to energy levels 2S _1/2, 2P _1/2 and 2P _3/2 were obtained by using the ?? and the \(\bar \theta \) modified Dirac Hamiltonian of hydrogen-like atom on a noncommutative phase space. The degeneracy of the energy levels 2P _1/2 and 2P _3/2 were removed completely by ??-correction. And the \(\bar \theta \)-correction shifts these energy levels.     

The New Multipartite Squeezing Operator and Some of its Properties

In this paper, we introduce a pair of mutually conjugate multipartite entangled state representations for defining the squeezing operator of entangled multipartite S_n(λ) which involves an n-mode bosonic operator realization of the SU(1,1) Lie algebra. This operator squeezes the multipartite entangled state in a natural way. We discuss the transform properties of a_j and \(a_{j}^{\dagger }\) under the operation of S_n(λ) and derive the interaction Hamiltonian which can generate such an evolution. In addition, the corresponding multipartite squeezed vacuum state |λ〉 is obtained. Based on this, the variances of the n-mode quadratures in |λ〉 are evaluated and the violation of the Bell inequality for |λ〉 is examined by using the formalism of Wigner representation.     

Dominant decays of scalar leptoquarks and scalar gluons in the minimal four-color symmetry model

Fermionic and weak decays of the scalar leptoquarks S = S ₁ ⁽⁺⁾ , S ₁ ^(?) , and S _m and the scalar gluons F = F ₁ and F ₂ predicted by the minimal model involving four-color symmetry and the Higgs mechanism of quark-and lepton-mass splitting are considered. The widths and the branching ratios are calculated for these decays, and the results are analyzed versus the couplings and masses of decaying particles. It is shown that, at relatively small mass splittings Δm within scalar doublets (Δm < m _W), the fermionic decays S ₁ ⁽⁺⁾ → tl _j ⁺ , S ₁ ^(?) → v _i \(\tilde b\), S _m → t \(\tilde \nu \) _j, F ₁ → t \(\tilde b\), and F ₂ → t \(\tilde t\), which are characterized by few-GeV widths for m _S, m _F < 1 TeV and decay branching ratios close to unity, are dominant, but that, for Δm > m _W, the weak decays S → S′W and F → F’W compete with the above fermionic decays. In the case of m _S < m _t, the processes S ₁ ⁽⁺⁾ → cl _j ⁺ , S ₁ ^(?) → v _i \(\tilde b\), S _m → bl _j ⁺ , and S _m → c \(\tilde \nu \) _j, whose total branching ratios are Br(S ₁ ⁽⁺⁾ → cl ⁺) ≈ Br(S ₁ ^(?) → v \(\tilde b\)) ≈ 1, Br(S _m → bl ⁺) ≈ 0.9, and Br(S _m → c \(\tilde \nu \)) ≈ 0.1, appear to be dominant decays of scalar leptoquarks. Searches for these decays at LHC and the Tevatron are of interest.     

Entanglement and Berry Phase in a Parameterized Three-Qubit System

In this paper, we construct a parameterized form of unitary \(\breve {R}_{123}(\theta _{1},\theta _{2},\varphi )\) matrix through the Yang-Baxterization method. Acting such matrix on three-qubit natural basis as a quantum gate, we can obtain a set of entangled states, which possess the same entanglement value depending on the parameters ?? ₁ and ?? ₂. Particularly, such entangled states can produce a set of maximally entangled bases Greenberger-Horne-Zeilinger (GHZ) states with respect to ?? ₁ = ?? ₂ = π/2. Choosing a useful Hamiltonian, one can study the evolution of the eigenstates and investigate the result of Berry phase. It is not difficult to find that the Berry phase for this new three-qubit system consistent with the solid angle on the Bloch sphere.     

Emergence and space–time structure of lump solution to the (2$$$$1)-dimensional generalized KP equation

We derive the relativistic Hamiltonian of hydrogen atom in dynamical non-commutative spaces (DNCS or τ-space). Using this Hamiltonian we calculate the energy shift of the ground state as well the 2P _1/2, 2S _1/2 levels. In all the cases, the energy shift depends on the dynamical non-commutative parameter τ. Using the accuracy of the energy measurement, we obtain an upper bound for τ. We also study the Lamb shift in DNCS. Both 2P _1/2 and 2S _1/2 levels receive corrections due to dynamical non-commutativity of space which is in contrast with the non-dynamical non-commutative spaces (NDNCS or ??-space) in which the 2S _1/2 level receives no correction.     

Inverse Scattering at Fixed Energy on Surfaces with Euclidean Ends

On a fixed Riemann surface (M ₀, g ₀) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix S _V (λ) at frequency λ > 0 for the operator Δ+V determines the potential V if \({V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0)}\) for all γ > 0 and for some \({j\in\{1,2\}}\) , where d(z, z ₀) denotes the distance from z to a fixed point \({z_0\in M_0}\) . The topological condition is given by \({N\geq \max(2g+1,2)}\) for j = 1 and by N ≥ g + 1 if j = 2. In \({\mathbb {R}^2}\) this implies that the operator S _V (λ) determines any C ^{1, α} potential V such that \({V(z)=O(e^{-\gamma|z|^2})}\) for all γ > 0.     

Partial virial theorems for atomic-molecular systems

New virial relations for three-and four-particle atomic-molecular systems are proposed. Using operators of extension or squeezing of interparticle distances, it is shown that, for all pairs of j and k particles in S states of these systems, the following partial virial relations are valid: 〈2T _jk 〉+〈 V _jk 〉=0, where 〈V _jk 〉 is the average Coulomb interaction energy for a pair of particles and 〈T _jk 〉 is a part of the average kinetic energy of the system. There are three and six such relations for three-and four-particle systems, respectively. The conventional virial theorem (〈 2T〉+〈V〉=0) for the average total kinetic and potential energies of the system (〈 T〉 and 〈V〉, respectively) corresponds to the summation of partial virial relations over all pairs of particles. It is shown by an example of variational calculations of the helium atom ⁴He²⁺ e ^? e ^? and the helium muon-electron mesoatom ⁴He²⁺μ^? e ^? that partial virial relations are a highly sensitive indicator of the accuracy of wave functions.     

Evolution of an optically pumped ensemble of cold ground-state atoms in weak light fields

The evolution of multipole moments is analyzed for optically pumped cold ground-state atoms in the limit of low saturation of a closed j₀ → j₁ dipole transition. The longest multipole-moment relaxation times are analyzed as functions of ellipticity and frequency detuning from resonance for transitions with j₀ ? 5. The qualitative difference between the evolution toward steady-state Zeeman sublevel populations and dynamics of transient spontaneous emission is demonstrated for transitions of the following types: j → j?1, j → j with integer j, j → j with half-integer j, and j → j + 1.     

On the calculation of the two-point function for the thirring-model by the method of functional integration

By the method of functional integration the two-point functionS _F for the spinor model with the interaction\( - \lambda (\bar \psi \psi )^2 \) is calculated in a two-dimensional space-time. After Fourier-transformationS _F (p) results as a power series with respect to 1/√λ. If we change the order of terms, we get a series in powers of γp. Each coefficient is a series in powers of 1/√λ. The first terms of this series are considered as a good approximation for bigλ. By reasons of convergence of the integrals we must displace the expansion centre of the series in powers ofγ p fromp ²=0 top ²=a ².     

Control of Quantum Echo and Bell State Swapping of Two Atomic Qubits in the Two-Mode Vacuum Field Environment

We investigate quantum echo control and Bell state swapping for two atomic qubits (TAQs) coupling to two-mode vacuum cavity field (TMVCF) environment via two-photon resonance. We discuss the effect of initial entanglement factor ?? and relative coupling strength R=g₁/g₂ on quantum state fidelity of TAQs, and analyze the relation between three kinds of quantum entanglement(C(ρ_a),C(ρ_f),S(ρ_a)) and quantum state fidelity, then reveal physical essence of quantum echo of TAQs. It is shown that in the identical coupling case R=1, periodic quantum echo of TAQs with π cycle is always produced, and the value of fidelity can be controlled by choosing appropriate ?? and atom-filed interaction time. In the non-identical coupling case R≠1, quantum echoes with periods of π, 2π and 4π can be formed respectively by adjusting R. The characteristics of quantum echo results from the non-Markovianity of TMVCF environment, and then we propose Bell state swapping scheme between TAQs and two-mode cavity field.     

The 3–3–1 Model with RH Neutrinos and Associated ZH Production at High-Energy ${\bf \textit{e}}^{+}{\bf \textit{e}}^{-}$ Collider

In the context of SU(3) _C ?? SU(3) _L ?? U(1) _X (3–3–1) model with right-handed neutrinos, we study the Higgsstrahlung process e ^?+? e ^???→ZH and calculate the cross section of this process at leading order. Our numerical results showed that the production cross sections for this process can be significantly large as \(M_{Z'}\approx \sqrt{s}\). With reasonable values of the Z′ mass M _Z′, Z′ exchange can generate large corrections to the cross sections of this process, which might be detected in the future high-energy linear e ^?+? e ^??? collider experiments.     

Symmetries and Boundary Conditions with a Twist

Interest in finite-size systems has risen in the last decades, due to the focus on nanotechnological applications and because they are convenient for numerical treatment that can subsequently be extrapolated to infinite lattices. Independently of the envisioned application, special attention must be given to boundary condition, which may or may not preserve the symmetry of the infinite lattice. Here, we present a detailed study of the compatibility between boundary conditions and conservation laws. The conflict between open boundary conditions and momentum conservation is well understood, but we examine other symmetries, as well: we discuss gauge invariance, inversion, spin, and particle-hole symmetry and their compatibility with open, periodic, and twisted boundary conditions. In the interest of clarity, we develop the reasoning in the framework of the one-dimensional half-filled Hubbard model, whose Hamiltonian displays a variety of symmetries. Our discussion includes analytical and numerical results. Our analytical survey shows that, as a rule, boundary conditions break one or more symmetries of the infinite-lattice Hamiltonian. The exception is twisted boundary condition with the special torsion Θ = πL/2, where L is the lattice size. Our numerical results for the ground-state energy at half-filling and the energy gap for L = 2–7 show how the breaking of symmetry affects the convergence to the L → ∞ limit. We compare the computed energies and gaps with the exact results for the infinite lattice drawn from the Bethe-Ansatz solution. The deviations are boundary-condition dependent. The special torsion yields more rapid convergence than open or periodic boundary conditions. For sizes as small as L = 7, the numerical results for twisted condition are very close to the L → ∞ limit. We also discuss the ground-state electronic density and magnetization at half filling under the three boundary conditions.     

Precision measurements of optical excitation functions for the mercury atom

We describe a computer-based facility for studying the excitation of atoms by ultramonochromatic electrons and give optical excitation functions for the 12 mercury spectral lines that originate from the n ¹ S ₀, n ¹ P ₁, n ¹ D ₂, n ³ S ₁, n ³ P _j , and n ³ D _j levels. We detected about 100 features in the energy dependences measured from the excitation threshold to 15.5 eV. The previously found positions of the features on the energy scale are in good agreement with our results. Most of the resonant features are shown to be mainly attributable to the decay of short-lived states of the negative mercury ion. We detected a postcollision interaction effect in the optical excitation functions of the lines that originate from the n ¹ S ₀ levels at energies of about 11 eV.     

Thermodynamics of the frustrated J<Subscript>1</Subscript>-J<Subscript>2</Subscript> Heisenberg ferromagnet on the body-centered cubic lattice with arbitrary spin

We use the spin-rotation-invariant Green’s function method as well as thehigh-temperature expansion to discuss the thermodynamic properties of the frustratedspin-S J ₁-J ₂ Heisenbergmagnet on the body-centered cubic lattice. We consider ferromagnetic nearest-neighborbonds J ₁<0 and antiferromagnetic next-nearest-neighbor bonds J ₂ ≥ 0 andarbitrary spin S. We find that the transition point\hbox{$J_2^c$}J2cbetween the ferromagnetic ground state and theantiferromagnetic one is nearly independent of the spin S, i.e., it is very closeto the classical transition point\hbox{$J_2^{c,{\rm clas}}= \frac{2}{3}|J_1|$}J2c,clas=23|J1|. At finite temperatures we focus on the parameterregime\hbox{$J_2<J_2^c$}J2<J2cwith a ferromagnetic ground-state. We calculate theCurie temperature T _C (S, J ₂)and derive an empirical formula describing the influence of the frustration parameterJ ₂ and spin S on T _C . We find that theCurie temperature monotonically decreases with increasing frustration J ₂, where veryclose to\hbox{$J_2^{c,{\rm clas}}$}J2c,clasthe T _C (J ₂)-curveexhibits a fast decay which is well described by a logarithmic term\hbox{$1/\textrm{log}(\frac{2}{3}|J_1|-J_{2})$}1/log(23|J1|?J2). To characterize the magnetic ordering below and aboveT _C , we calculate thespin-spin correlation functions ?S ₀ S _R ?, the spontaneous magnetization, the uniform static susceptibilityχ ₀ as well as the correlation lengthξ.Moreover, we discuss the specific heat C _V and the temperaturedependence of the excitation spectrum. As approaching the transition point\hbox{$J_2^c$}J2csome unusual features were found, such as negativespin-spin correlations at temperatures above T _C even though theground state is ferromagnetic or an increase of the spin stiffness with growingtemperature.     

Non-extensivity of the chemical potential of polymer melts

Following Flory’s ideality hypothesis, the chemical potential of a test chain of length n immersed into a dense solution of chemically identical polymers of length distribution P(N) is extensive in n . We argue that an additional contribution \( \delta\) \( \mu_{{{\rm c}}}^{}\)(n) ～ +1/\( \rho\) \( \sqrt{{n}}\) arises (\( \rho\) being the monomer density) for all P(N) if n ? 〈N〉 which can be traced back to the overall incompressibility of the solution leading to a long-range repulsion between monomers. Focusing on Flory-distributed melts, we obtain \( \delta\) \( \mu_{{{\rm c}}}^{}\)(n) \( \approx\) (1 - 2n/〈N〉)/\( \rho\) \( \sqrt{{n}}\) for n ? 〈N〉² , hence, \( \delta\) \( \mu_{{{\rm c}}}^{}\)(n) \( \approx\) -1/\( \rho\) \( \sqrt{{n}}\) if n is similar to the typical length of the bath 〈N〉 . Similar results are obtained for monodisperse solutions. Our perturbation calculations are checked numerically by analyzing the annealed length distribution P(N) of linear equilibrium polymers generated by Monte Carlo simulation of the bond fluctuation model. As predicted we find, e.g., the non-exponentiality parameter K _p \( \equiv\) 1 - 〈N ^p〉/p!〈N〉^p to decay as K _p \( \approx\) 1/\( \sqrt{{\langle N \rangle }}\) for all moments p of the distribution.     

String order and degenerate entanglement spectrum of S = 1 / 2 and S = 1 Heisenberg bond-alternating chains

Generalized string orders and entanglement spectrum of S = 1/2 and S = 1 Heisenberg bond-alternating chains have been investigated by the infinite time-evolving block decimation (iTEBD) method. Generalized string order parameters with appropriate θ are capable of distinguishing all the topological phases. Central charges c ? 1 and critical exponents β ?1/12 indicate all the topological QPTs belong to the Gaussian universality class. Interestingly, odd- and even-fold degeneracies of the entanglement spectrum are observed. Even-fold (doubly) degenerate entanglement spectra and the typical two-fold degenerate lowest-lying level are found to exist in both the spin-1/2 dimer and the S = 1 Haldane phases. However, odd-fold degenerate entanglement spectra with three-fold degenerate lowest-lying level are observed in both the S = 1 dimer and the S = 2 Haldane phase. The degeneracy of the lowest-lying entanglement spectrum level, which can be understood by entanglement spectra in the dimer limit (J ₁ = 0), is adopted to estimate the lowest boundary of the bipartite entanglement. The entanglement spectrum and the generalized string orders are valuable for uncovering the underlying features of these symmetry-protect topological (SPT) states. Similar entanglement spectrum shows that the S = 1 (S = 2) Haldane phase is essentially the same as the S = 1/2 (S = 1) dimer phase.     

<Emphasis Type="Italic">SU</Emphasis>(2) Non-Abelian Photon

In this paper, we have proposed S U(2) non-Abelian electromagnetism gauge theory. In the theory, photon has self-interaction and interaction between them, which can explain photon entanglement phenomenon in quantum information. Otherwise, we find there are three kinds photons γ ⁺, γ ^? and γ ⁰, they have electric charge + e _γ , ? e _γ and 0, respectively, these prediction are accordance with some experiment results.     

On the possibility of laser generation without inversion on the fine-structure levels of a helium atom

It is shown that laser generation is possible in principle upon coherent excitation according to the V-scheme of the (1s2p)³ P _j levels with j = 1, 2 of a helium atom from the (1s2s)³ S ₁ level of the same atom when the latter level is, in turn, populated by electron or proton impact.     

Description of Some Ground States by Puiseux Techniques

Let \((\Sigma^{+}_{G}, \sigma)\) be a one-sided transitive subshift of finite type, where symbols are given by a finite spin set S, and admissible transitions are represented by an irreducible directed graph G?S×S. Let \(H : \Sigma^{+}_{G}\to\mathbb{R}\) be a locally constant function (that corresponds with a local observable which makes finite-range interactions). Given β>0, let μ _βH be the Gibbs-equilibrium probability measure associated with the observable ?βH. It is known, by using abstract considerations, that {μ _βH } _β>0 converges as β→+∞ to a H-minimizing probability measure \(\mu_{\min}^{H}\) called zero-temperature Gibbs measure. For weighted graphs with a small number of vertices, we describe here an algorithm (similar to the Puiseux algorithm) that gives the explicit form of \(\mu_{\min}^{H}\) on the set of ground-state configurations.     

Entanglement in Mixed-Spin (1/2, 3/2) Heisenberg XXZ Model with Dzyaloshinskii-Moriya Interaction

In this paper, the entanglement in a mixed-spin (1/2, 3/2) Heisenberg XXZ model with Dzyaloshinskii-Moriya (DM) interaction in an inhomogeneous external magnetic field is studied. We not only calculate the ground-state entanglement but also investigate the behaviors of quantum phase transition following the changes of DM interaction and nonuniform magnetic field. More importantly, we note that the DM interaction improves the critical magnetic field B _c , the critical temperature T _c and broadens the region of entanglement.