Dirac-Like Equation for Two-Level Atom

Spin dynamics of wave packets evolving with the Dirac Hamiltonian in atoms with high Z

The motion of circular WP for one electron in central Coulomb field with high Z is calculated. The WP is defined in terms of solutions of the Dirac equation in order to take into account all possible relevant effects in particular the spin-orbit potential. A time scale is defined within which spin dynamics must be taken into account mainly in the atoms with high Z. Within this time scale there exists a mechanism of collapses and revivals of the spin already shown by the authors for harmonic oscillator potential and called the 'spin-orbit pendulum'. However this effect has not the exact periodi…

Dynamical symmetry of Dirac hydrogen atom with spin symmetry and its connection with Ginocchio's oscillator

The Dirac hydrogen atom with spin symmetry is shown has a SO(4) symmetry. The generators are derived, and the corresponding Casimir operator leads to the energy spectrum naturally. This type hydrogen atom is connected to a four-dimensional Dirac system with equal scalar and vector harmonic oscillator potential, by the Kustaanheimo-Stiefel transformation with a constraint.

任意离散能态下狄拉克单电子原子磁化率的Gordon分解

Gordon decomposition of the magnetizability of a Dirac one-electron atom in an arbitrary discrete energy state

本研究针对带电量$Ze$的点状静止无自旋核的狄拉克单电子原子，利用广义狄拉克-库仑格林函数的Sturmian展开，解析推导任意离散能态下磁化率的抗磁、顺磁贡献闭式表达式，结果可退化为基态已有公式，为后续类氢离子激发态磁化率数值计算提供基础。

Driven Dirac-like Equation via Mirror Oscillation: Controlled Cold-Atom Zitterbewegung

By considering mirror oscillation in a "tripod-scheme" laser-atom system, we advocate explorative studies of driven Dirac-like equations. Both analytical and numerical studies show that mirror oscillation can be used to drive an effective spin-orbit interaction and hence control the amplitude, the frequency, and the damping of the cold-atom Zitterbewegung oscillation. Our results demonstrate an interesting coupling between the mirror mechanical motion and a fundamental quantum coherent oscillation, opening up new means of matter wave manipulation.

Comparison among Klein-Gordon equation, Dirac equation and Relativistic Stationary Schrodinger equation

A particle is always not pure. It always contains hiding antiparticle ingredient which is the essence of special relativity. Accordingly, the Klein-Gordon (KG) equation and Dirac equation are restudied and compared with the Relativistic Stationary Schrödinger Equation (RSSE). When an electron is bound in a Hydrogenlike atom with pointlike nucleus having charge number $Z$, the critical value of $Z, Z_c$, equals to 137 in Dirac equation whereas $Z_c=\sqrt{M/μ} (137)$ in RSSE with $M$ and $μ$ being the total mass of atom and the reduced mass of the electron.

Simulating Dirac models with ultracold atoms in optical lattices

We present a general model allowing "quantum simulation" of one-dimensional Dirac models with 2- and 4-component spinors using ultracold atoms in driven 1D tilted optical latices. The resulting Dirac physics is illustrated by one of its well-known manifestations, Zitterbewegung. This general model can be extended and applied with great flexibility to more complex situations.

Exact Solution of the Klein-Gordon Equation for the Hydrogen Atom Including Electron Spin

The term describing the coupling between total angular momentum and energy-momentum in the hydrogen atom is isolated from the radial Dirac equation and used to replace the corresponding orbital angular momentum coupling term in the radial K-G equation. The resulting spin-corrected K-G equation is a second order differential equation that contains no matrices. It is solved here to generate the same energy eigenvalues for the hydrogen atom as the Dirac equation.

On the effective Dirac dynamics of ultracold atoms in bichromatic optical lattices

We study the dynamics of ultracold atoms in tailored bichromatic optical lattices. By tuning the lattice parameters, one can readily engineer the band structure and realize a Dirac point, i.e. a true crossing of two Bloch bands. The dynamics in the vicinity of such a crossing is described by the one-dimensional Dirac equation, which is rigorously shown beyond the tight-binding approximation. Within this framework we analyze the effects of an external potential and demonstrate numerically that it is possible to demonstrate Klein tunneling with current experimental setups.

Variational Dirac-Coulomb explicitly correlated computations for atoms and molecules

The Dirac-Coulomb equation with positive-energy projection is solved using explicitly correlated Gaussian functions. The algorithm and computational procedure aims for a parts-per-billion convergence of the energy to provide a starting point for further comparison and further developments in relation with high-resolution atomic and molecular spectroscopy. Besides a detailed discussion of the implementation of the fundamental spinor structure, permutation and point-group symmetries, various options for the positive-energy projection procedure are presented. The no-pair Dirac-Coulomb energy conv…

Dispersion Interaction of Atoms with Single-Walled Carbon Nanotubes described by the Dirac Model

We calculate the interaction energy and force between atoms and molecules and single-walled carbon nanotubes described by the Dirac model of graphene. For this purpose the Lifshitz-type formulas adapted for the case of cylindrical geometry with the help of the proximity force approximation are used. The results obtained are compared with those derived from the hydrodymanic model of graphene. Numerical computations are performed for hydrogen atoms and molecules. It is shown that the Dirac model leads to larger values of the van der Waals force than the hydrodynamic model. For a hydrogen molecul…

狄拉克氢原子的非简并兰姆位移解

Nondegenerate, Lamb-shift Solution of the Dirac Hydrogen Atom

本文提出狄拉克方程的新解，利用兰姆位移消除氢原子所有简并性，以拉盖尔多项式给出径向本征函数并验证其可约化为薛定谔波函数。

带电原子链束缚电子的狄拉克方程通解

General solution of the Dirac equation for electrons bound by a charged atomic chain

本文在狄拉克理论中求解带电原子链束缚电子的狄拉克方程，得到解析解，证明电子完全集体化，能量为占据数函数，并揭示谱带分裂为n个子带的精细结构。

Comparison of the hydrodynamic and Dirac models of the dispersion interaction between graphene and H, He${}^{\ast}$, or Na atoms

The van der Waals and Casimir-Polder interaction of different atoms with graphene is investigated using the Dirac model which assumes that the energy of quasiparticles is linear with respect to the momentum. The obtained results for the van der Waals coefficients of hydrogen atoms and molecules and atoms of metastable He${}^{\ast}$ and Na as a function of separation are compared with respective results found using the hydrodynamic model of graphene. It is shown that, regardless of the value of the gap parameter, the Dirac model leads to much smaller values of the van der Waals coefficients tha…

一维超冷原子的双体Kapitza-Dirac散射

Two-Body Kapitza-Dirac Scattering of One-Dimensional Ultracold Atoms

研究一维谐振子中两原子在光格子中的双体Kapitza-Dirac散射，分析相互作用强度、晶格深度等参数对衍射图案的影响，提供精确的少体基准。

变分R矩阵方法应用于狄拉克方程的铍原子研究

Application of the Variational R-matrix Method for the Dirac Equation to the Be Atom

本文应用变分R矩阵方法研究狄拉克方程，分析了铍原子光电离截面中的费诺线形变化及三重态与单重态系列之间的耦合现象。

冷原子狄拉克晶格中Floquet工程化的奇偶反常阶梯

Floquet-Engineered Parity Anomaly Staircase in a Cold Atom Dirac Lattice

提出利用Floquet工程在二维π通量晶格中实现奇偶反常启发的反常霍尔阶梯，通过选择性地占据不同狄拉克扇区的贝里曲率贡献，得到阶梯状霍尔响应。

Addendum to "The Dirac-Coulomb Sturmians and the series expansion of the Dirac-Coulomb Green function: application to the relativistic polarizability of the hydrogen-like atom" [J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 825-61, (E) 30 (1997) 2747]

Closure relations satisfied by the radial Dirac-Coulomb Sturmians are proved analytically. The Sturmian expansion of the Dirac-Coulomb Green function is transformed to the form containing only series with summations running over non-negative indices. The main paper was published in J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 825-61 [Erratum: 30 (1997) 2747], see also J. Phys. A: Math. Gen. 31 (1998) 4963--90 [Erratum: 31 (1998) 7415-6].

电子束和类氢原子中的分数角动量

Fractional Angular Momenta in Electron Beams and Hydrogen-Like Atoms

研究电子束和类氢原子中分数角动量的产生机制，通过Dirac方程解的分解揭示角动量态的混合效应，包括分数Gouy相位和波粒二象性。

Relativistic quantum effects of Dirac particles simulated by ultracold atoms

Quantum simulation is a powerful tool to study a variety of problems in physics, ranging from high-energy physics to condensed-matter physics. In this article, we review the recent theoretical and experimental progress in quantum simulation of Dirac equation with tunable parameters by using ultracold neutral atoms trapped in optical lattices or subject to light-induced synthetic gauge fields. The effective theories for the quasiparticles become relativistic under certain conditions in these systems, making them ideal platforms for studying the exotic relativistic effects. We focus on the reali…

Scattering of light and atoms in a Fermi-Dirac gas with BCS pairing

We theoretically study the optical properties of a Fermi-Dirac gas in the presence of a superfluid state. We calculate the leading quantum-statistical corrections to the standard column density result of the electric susceptibility. We also consider the Bragg diffraction of atoms by means of light-stimulated transitions of photons between two intersecting laser beams. Bardeen-Cooper-Schrieffer pairing between atoms in different internal levels magnifies incoherent scattering processes. The absorption linewidth of a Fermi-Dirac gas is broadened and shifted. Bardeen-Cooper-Schrieffer pairing int…

使用Rayleigh-Ritz方法求解Dirac-Coulomb方程：类氦原子的结果

Solution of the Dirac-Coulomb equation using the Rayleigh-Ritz method. Results for He-like atoms

本文用Rayleigh-Ritz方法结合Slater型旋量轨道求解类氦离子的Dirac-Coulomb方程，计算至Z≤80的总能量，未发现Brown-Ravenhall病迹象。

Are atoms waves or particles?

It is shown that the Kapitza-Dirac effect with atoms, which has been considered to be evidence of their wavelike character, can be interpreted as a scattering of pointlike objects by the periodic laser field.

Dirac Equation For Cold Atoms In Artificial Curved Spacetimes

We argue that the Fermi-Hubbard Hamiltonian describing the physics of ultracold atoms on optical lattices in the presence of artificial non-Abelian gauge fields, is exactly equivalent to the gauge theory Hamiltonian describing Dirac fermions in the lattice. We show that it is possible to couple the Dirac fermions to an "artificial" gravitational field, i.e. to consider the Dirac physics in a curved spacetime. We identify the special class of spacetime metrics that admit a simple realization in terms of a Fermi-Hubbard model subjected to an artificial SU(2) field, corresponding to position depe…

Using Gaussian Basis-Sets with Gaussian Nuclear Charge Distribution to Solve Dirac-Hartree-Fock Equation for 83 Bi-Atom

In this paper, we consider the Dirac-Hartree-Fock equations for system has many-particles. The difficulties associated with Gaussians model are likely to be more complex in relativistic Dirac-Hartree-Fock calculations. To processing these problem, we use accurate techniques. The four-component spinors will be expanded into a finite basis-set, using Gaussian basis-set type dyall.2zp to describe 4-component wave functions, in order to describe the upper and lower two components of the 4-spinors, respectively. The small component Gaussian basis functions have been generated from large component G…

Spontaneous excitation of a uniformly accelerated atom coupled with vacuum Dirac field fluctuations

We study the spontaneous excitation of a uniformly accelerated two-level atom non-linearly coupled with vacuum Dirac field fluctuations using the formalism proposed by Dalibard, Dupont-Roc and Cohen-Tannoudji (DDC) and generalized to the present case in the current paper. We find that a cross term involving both vacuum fluctuations and radiation reaction appears, which is absent in the linear coupling cases such as an atom interacting with vacuum scalar or electromagnetic fluctuations. Furthermore, the contribution of this term actually dominates over that of radiation reaction. Thus, the mean…

Dirac, Rashba and Weyl type spin-orbit couplings: toward experimental realization in ultracold atoms

We propose a hierarchy set of minimal optical Raman lattice schemes toward the experimental realization of spin-orbit (SO) couplings of various types for ultracold atoms, including two-dimensional (2D) Dirac type of $C_4$ symmetry, 2D Rashba and 3D Weyl types. These schemes are well accessible with current cold-atom technology, and in particular, a long-lived Bose-Einstein condensation of the 2D Dirac SO coupling has been experimentally proved. The generation of 2D Rashba and 3D Weyl types has an exquisite request that two sources of laser beams have distinct frequencies of factor-two differen…

Two-dimensional relativistic hydrogenic atoms: A complete set of constants of motion

The complete set of operators commuting with the Dirac Hamiltonian and exact analytic solution of the Dirac equation for the two-dimensional Coulomb potential is presented. Beyond the eigenvalue $μ$ of the operator $j_{z}$, two quantum numbers $η$ and $κ$ are introduced as eigenvalues of hermitian operators $P=βσ_{z}'$ and $K=β(σ_{z}'l_{z}+1/2)$, respectively. The classification of states according to the full set of constants of motion without referring to the non-relativistic limit is proposed. The linear Paschen-Back effect is analyzed using exact field-free wave-functions as a zero-order a…

旋量不变量的代数与相对论性氢原子

Algebra of the spinor invariants and the relativistic hydrogen atom

本文提出利用Dirac方程的三个旋量不变量代数求解带库仑势的Dirac方程，无需超对称量子力学方法，揭示了SO(3)⊗SU(2)对称性，并计算出相对论性氢原子的本征态与能量。

Spin-orbit coupling in the hydrogen atom, the Thomas precession, and the exact solution of Dirac's equation

Bohr's model of the hydrogen atom can be extended to account for the observed spin-orbit interaction, either with the introduction of the Thomas precession, or with the stipulation that, during a spin-flip transition, the orbital radius remains intact. In other words, if there is a desire to extend Bohr's model to accommodate the spin of the electron, then experimental observations mandate the existence of the Thomas precession, which is a questionable hypothesis, or the existence of artificially robust orbits during spin-flip transitions. This is tantamount to admitting that Bohr's model, whi…

4xU(1)引力规范场中的原子狄拉克能级动力学与红移

Atomic Dirac energy-level dynamics and redshift in the 4xU(1) gravity gauge field

本研究应用标准模型的统一引力扩展，在4xU(1)引力规范场中求解类氢原子狄拉克方程，证明引力规范场引起的能级移动与引力红移一致，并揭示引力势梯度导致简并光谱线分裂，为强引力场量子光子学研究开辟新途径。

Electronic Structure of Superheavy Atoms. Revisited

The electronic structure of an atom with Z <= 137 can be described by the Dirac equation with the Coulomb field of a point charge Ze. It was believed that the Dirac equation with Z > 137 is inconsistent and physically meaningless because the formula for the lower energy level of the Dirac Hamiltonian formally gives imaginary eigenvalues. But a strict mathematical consideration shows that difficulties with the electronic spectrum for Z > 137 do not arise if the Dirac Hamiltonian is correctly defined as a self-adjoint operator, see [1]. In this article, we brie y summarize the main physical resu…

冷原子SO(5)狄拉克场的高阶拓扑扭转

A higher-order topological twist on cold-atom SO(5) Dirac fields

本研究提出利用二维拉曼晶格中的自旋3/2超冷费米气体，作为结合Gross-Neveu、Thirring相互作用与高阶拓扑扭转的狄拉克量子场论的量子模拟器，揭示了受隐藏SO(5)对称性保护的高阶拓扑态及丰富相图，为关联高阶拓扑态的实现提供了新途径。

Dirac equation, hydrogen atom spectrum and the Lamb shift in dynamical noncommutative spaces

We derive the relativistic Hamiltonian of hydrogen atom in dynamical noncommutative spaces (DNCS or τ-space). Using this Hamiltonian we calculate the energy shift of the ground state and as well the [2P]_(1/2), [2S]_(1/2) levels. In all cases the energy shift depend on the dynamical noncommutative 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 the levels [2P]_(1/2) and [2S]_(1/2) receive corrections due to dynamical noncommutativity of space which is in contrast to the non-dynamical noncommutative spaces (NDN…

mudirac: a Dirac equation solver for elemental analysis with muonic X-rays

We present a new open source software for the integration of the radial Dirac equation developed specifically with muonic atoms in mind. The software, called mudirac, is written in C++ and can be used to predict frequencies and probabilities of the transitions between levels of the muonic atom. In this way, it provides an invaluable tool in helping with the interpretation of muonic X-ray spectra for elemental analysis. We introduce the details of the algorithms used by the software, show the interpretation of a few example spectra, and discuss the more complex issues involved with predicting t…

The Nernst heat theorem for an atom interacting with graphene: Dirac model with nonzero energy gap and chemical potential

We derive the low-temperature behavior of the Casimir-Polder free energy for a polarizable atom interacting with graphene sheet which possesses the nonzero energy gap $Δ$ and chemical potential $μ$. The response of graphene to the electromagnetic field is described by means of the polarization tensor in the framework of Dirac model on the basis of first principles of thermal quantum field theory in the Matsubara formulation. It is shown that the thermal correction to the Casimir-Polder energy consists of three contributions. The first of them is determined by the Matsubara summation using the …

Thermal Casimir-Polder interaction of different atoms with graphene

The thermal correction to the energy of Casimir-Polder interaction of atoms with a suspended graphene membrane described by the Dirac model is investigated. We show that a major impact on the thermal correction is made by the size of the gap in the energy spectrum of graphene quasiparticles. Specifically, if the temperature is much smaller than the gap parameter (alternatively, larger or of the order of the gap parameter), the thermal correction is shown to be relatively small (alternatively, large). We have calculated the free energy of the thermal Casimir-Polder interaction of atoms of He, N…

Relativistic effective charge model of a multi-electron atom

A relativistic version of the effective charge model for computation of observable characteristics of multi-electron atoms and ions is developed. A complete and orthogonal Dirac hydrogen basis set, depending on one parameter -- effective nuclear charge $Z^{*}$ -- identical for all single-electron wave functions of a given atom or ion, is employed for the construction of the secondary-quantized representation. The effective charge is uniquely determined by the charge of the nucleus and a set of electron occupation numbers for a given state. We thoroughly study the accuracy of the leading-order …

Second-order Stark effect and polarizability of a relativistic two-dimensional hydrogen-like atom in the ground state

The second-order Stark effect for a planar Dirac one-electron atom in the ground state is analyzed within the framework of the Rayleigh-Schrödinger perturbation theory, with the use of the Sturmian series expansion of the generalized Dirac-Coulomb Green function. A closed-form analytical expression for the static dipole polarizability of that system is found. The formula involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit argument. Numerical values of the polarizabilities for relativistic planar hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided …

Quantum walks and quantum simulations with Bloch oscillating spinor atoms

We propose a scheme for the realization of a quantum walker and a quantum simulator for the Dirac equation with ultracold spinor atoms in driven optical lattices. A precise control of the dynamics of the atomic matter wave can be realized using time-dependent external forces. If the force depends on the spin state of the atoms, the dynamics will entangle the inner and outer degrees of freedom which offers unique opportunities for quantum information and quantum simulation. Here, we introduce a method to realize a quantum walker based on the state-dependent transport of spinor atoms and a coher…

Analog quantum simulation of the spinor-four Dirac equation with an artificial gauge field

A two-dimensional spatially and temporally modulated Wannier-Stark system of ultracold atoms in optical lattices is shown to mimic the behavior of a Dirac particle. Suitable additional modulations generate an artificial gauge field which simulates a magnetic field and imposes the use of the full spinor-four Dirac equation.

Dirac-like Hamiltonians associated to Schrödinger factorizations

In this work, we have extended the factorization method of scalar shape-invariant Schrö\-din\-ger Hamiltonians to a class of Dirac-like matrix Hamiltonians. The intertwining operators of the Schrödinger equations have been implemented in the Dirac-like shape invariant equations. We have considered also another kind of anti-intertwining operators changing the sign of energy. The Dirac-like Hamiltonians can be obtained from reduction of higher dimensional spin systems. Two examples have been worked out, one obtained from the sphere ${\cal S}^2$ and a second one, having a non-Hermitian character,…

静态弯曲时空中 Dirac 方程的代数解

An algebraic solution of Dirac equation on a static curved space-time

本文提出两种代数方法求解静态弯曲时空中的 Dirac 方程，分别利用 $su(1,1)$ 代数与 Schrödinger 因式分解，得到多种势的精确能谱和本征函数。

Synthetic Unruh effect in cold atoms

We propose to simulate a Dirac field near an event horizon using ultracold atoms in an optical lattice. Such a quantum simulator allows for the observation of the celebrated Unruh effect. Our proposal involves three stages: (1) preparation of the ground state of a massless 2D Dirac field in Minkowski spacetime; (2) quench of the optical lattice setup to simulate how an accelerated observer would view that state; (3) measurement of the local quantum fluctuation spectra by one-particle excitation spectroscopy in order to simulate a De Witt detector. According to Unruh's prediction, fluctuations …

Cold atom simulation of interacting relativistic quantum field theories

We demonstrate that Dirac fermions self-interacting or coupled to dynamic scalar fields can emerge in the low energy sector of designed bosonic and fermionic cold atom systems. We illustrate this with two examples defined in two spacetime dimensions. The first one is the self-interacting Thirring model. The second one is a model of Dirac fermions coupled to a dynamic scalar field that gives rise to the Gross-Neveu model. The proposed cold atom experiments can be used to probe spectral or correlation properties of interacting quantum field theories thereby presenting an alternative to lattice g…

Dirac Equation and Planck-Scale Quantities

This work investigates in which form quantities with Planck dimensions occur already in the common quantum theory with local Lorentz symmetry. Since such Planck quantities as Planck length or Planck mass involve the Planck constant h, the velocity of light c and the Newton gravitational constant G, the relativistic Dirac equation (h, c) in the Newtonian gravitational potential (G) can be considered as a test theory. The evaluation of the break-off condition of the power series of the radial energy eigenfunctions of a purely gravitational atom leads to exact terms for the energy eigenvalues E f…

重原子的基态能量：主项与次主项渐近

The Ground State Energy of Heavy Atoms: Leading and Subleading Asymptotics

本文在无对形式体系中研究了核电荷 $Z$ 较大的中性原子基态能量，证明对于一大类定义 Dirac 海的投影，其无对基态能量在次主项阶数上不超过 Furry 能量，关键工具是扩展了 Séré 的 Dirac-Hartree-Fock 理论结果。

The point spectrum of the Dirac Hamiltonian on the zero-gravity Kerr-Newman spacetime

In this short paper, we review the Dirac equation on the zero-gravity Kerr-Newman spacetime. Our main objective is to provide a correspondence between the classification of the bound states for the zGKN spectrum and the usual hydrogenic states $1s_{1/2}$, $2s_{1/2}$, etc. of the Hydrogen atom.