A simple framework for Dirac spinors is developed that parametrizes admissible quantum dynamics and also analytically constructs electromagnetic fields, obeying Maxwell's equations, which yield a desired evolution. In particular, we show how to achieve dispersionless rotation and translation of wave packets. Additionally, this formalism can handle control interactions beyond electromagnetic. This work reveals unexpected flexibility of the Dirac equation for control applications, which may open new prospects for quantum technologies.
Semiclassical trajectory-coherent approximation in quantum mechanics: II. High order corrections to the Dirac operators in external electromagnetic field
We introduce an $SU(1,1)$ algebraic approach to study the $(2+1)$-Dirac oscillator in the presence of the Aharonov-Casher effect coupled to an external electromagnetic field in the Minkowski spacetime and the cosmic string spacetime. This approach is based on a quantum mechanics factorization method that allows us to obtain the $su(1,1)$ algebra generators, the energy spectrum and the eigenfunctions. We obtain the coherent states and their temporal evolution for each spinor component of this problem. Finally, for these problems, we calculate some matrix elements and the Schrödinger uncertainty…
We present a self-consistent mean-field model based on a two-component Pauli-like equation that incorporates quantum and relativistic effects (up to second-order in 1/c) for both external and internal electromagnetic fields. By taking the semi-relativistic limit of the Dirac-Maxwell equations in the presence of an external electromagnetic field we obtain an analytical expression of a coherent light-induced mean-field Hamiltonian. The latter exhibits several mechanisms that involve the internal mean fields created by all the electrons and the external electromagnetic field (laser). The role pla…
In the Kapitza-Dirac effect, atoms, molecules, or swift electrons are diffracted off a standing wave grating of the light intensity created by two counter-propagating laser fields. In ultrafast electron optics, such a coherent beam splitter offers interesting perspectives for ultrafast beam shaping. Here, we study, both analytically and numerically, the effect of the inclination angle between two laser fields on the diffraction of pulsed, low-energy electron beams. For sufficiently high light intensities, we observe a rich variety of complex diffraction patterns. These do not only reflect inte…
We construct the Barut-Girardello coherent states for charge carriers in anisotropic 2D-Dirac materials immersed in a constant homogeneous magnetic field which is orthogonal to the sample surface. For that purpose, we solve the anisotropic Dirac equation and identify the appropriate arising and lowering operators. Working in a Landau-like gauge, we explicitly construct nonlinear coherent states as eigenstates of a generalized annihilation operator with complex eigenvalues which depends on an arbitrary function $f$ of the number operator. In order to describe the anisotropy effects on these sta…
Coherent regime of Kapitza-Dirac effect with electrons
Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）；Institute of Physics, Czech Academy of Sciences（捷克科学院物理研究所）；Carl Zeiss Microscopy GmbH（卡尔蔡司显微镜有限公司）
The stationary eigenstates and eigenvalues for the ponderomotive potential of an optical crystal confined in a one-dimensional infinite square well are numerically obtained. The initial states of the incoming particles taken as Gaussian, are expanded in the basis of the stationary eigenstates of the ponderomotive potential, to obtain the subsequent time evolution of the wave function of the particle during the interaction with the optical crystal. From the results of the time evolution of the probability density, it is observed that the particles get localized at equidistant positions in the t…
In this work, we construct coherent states for electrons in anisotropic 2D Dirac materials immersed in a uniform magnetic field perpendicularly oriented to the sample. In order to describe the bidimensional effects on electron dynamics in a semiclassical approach, we adopt the symmetric gauge vector potential to describe the external magnetic field through a vector potential. By solving a Dirac-like equation with an anisotropic Fermi velocity, we identify two sets of scalar ladder operators that allow us to define generalized annihilation operators, which are generators of either the Heisenber…
In this article, we studied the system of (2+1) dimensional Dirac equation in time-dependent noncommutative phase-space. Exactly, we investigated the analytical solution of the corresponding system by the Lewis-Riesenfeld invariant method based on the construction of the Lewis-Riesenfeld invariant. Knowing that we obtained the time-dependent Dirac Hamiltonian of the problem in question from a time-dependent Bopp-Shift translation, then it used to set the Lewis-Riesenfeld invariant operators. Thereafter, the obtained results used to express the eigenfunctions that lead to determining the genera…
Approximate analytical solutions of Dirac Equation with spin and pseudo spin symmetries for the diatomic molecular potentials plus a tensor term with any angular momentum
We study the analytic structure for the eigenvalues of the one-dimensional Dirac oscillator, by analytically continuing its frequency on the complex plane. A twofold Riemann surface is found, connecting the two states of a pair of particle and antiparticle. One can, at least in principle, accomplish the transition from a positive energy state to its antiparticle state by moving the frequency continuously on the complex plane, without changing the Hamiltonian after transition. This result provides a visual explanation for the absence of a negative energy state with the quantum number n=0.
This paper presents new analytic solutions to the Dirac equation employing a recently introduced method that is based on the formulation of spinorial fields and their driving electromagnetic fields in terms of geometric algebras. A first family of solutions describe the shape-preserving translation of a wavepacket along any desired trajectory in the x-y plane. In particular, we show that the dispersionless motion of a Gaussian wavepacket along both elliptical and circular paths can be achieved with rather simple electromagnetic field configurations. A second family of solutions involves a plan…
In this paper, the 2+1 Dirac-Moshinsky oscillator (2+1 DMO) coupled to an external isospin field is mapped onto the Jaynes-Cummings model (JCM), which describes the interaction between two two-level systems and a quantum single-mode field. The time-dependent wave function and the density matrix are obtained in the two cases. In the first case, the quantum number state is considered, while in the second case, the coherent state is considered as an initial state. The effect of both the detuning parameter and the coherence angle are studied on the entanglement and on the population inversion. It …
In our Comment we question the validity of the claim made by the authors of \cite{cc} that their solutions of the Dirac equation in an external {\em time-dependent} electromagnetic field describe beams of electrons. In every time-dependent field, no matter how weak, which has {\em infinite} time duration, there is a continuous electron-positron pair creation and annihilation. Without the proper accounting for these processes, the mathematical solutions of the Dirac equation are not directly applicable to realistic physical situations. In particular, the time evolution of the average values $\l…
We derive new solutions of the Schrödinger, Klein-Gordon and Dirac equations which describe the motion of particles in a uniform magnetic field. In contrast to the well known stationary solutions, our solutions exhibit the behavior of quantum particles which very closely resembles classical helical trajectories. These solutions also serve as an illustration of the meaning of the Ehrenfest theorem in relativistic quantum mechanics.
The analytic specifications of photonic lattices with fractional revival (FR) and perfect state transfer (PST) are reviewed. The approach to their design which is based on orthogonal polynomials is highlighted. A compendium of analytic models with PST is offered. New results on their FR properties are included. The nearest-neighbour approximation is adopted in most of the review; one analytic example with next-to-nearest neighbour interactions is also presented.
Coherent spaces spanned by a finite number of coherent states, are introduced. Their coherence properties are studied, using the Dirac contour representation. It is shown that the corresponding projectors resolve the identity, and that they transform into projectors of the same type, under displacement transformations, and also under time evolution. The set of these spaces, with the logical OR and AND operations is a distributive lattice, and with the logical XOR and AND operations is a Boolean ring (Stone\rq{}s formalism). Applications of this Boolean ring into classical CNOT gates with $n$-a…
Single photons are very useful resources in quantum information science. In real applications it is often required that the photons have a well-defined spectral (or equivalently temporal) modal structure. For example, a rising exponential pulse is able to fully excite a two-level atom while a Gaussian pulse cannot. This motivates the study of continuous-mode single-photon Fock states. Such states are characterized by a spectral (or temporal) pulse shape. In this paper we investigate the statistical property of continuous-mode single-photon Fock states. Instead of the commonly used normal order…
We theoretically demonstrate that non-Abelian braiding operation can be realized through the scattering between chiral Dirac edge modes (CDEMs) in quantum anomalous Hall insulators by analytically deriving its S-matrix. Based on the analytical model, we propose a viable device for the experimental realization and detection of the non-Abelian braiding operations. Through investigating the tunneling conductance in a discretized lattice model, the non-Abelian properties of CDEMs could also be verified in a numerical way. Our proposal for the CDEM-based braiding provides a new avenue for realizing…
In this paper, we will construct the coherent states for a Dirac electron in graphene placed in a constant homogeneous magnetic field which is orthogonal to the graphene surface. First of all, we will identify the appropriate annihilation and creation operators. Then, we will derive the coherent states as eigenstates of the annihilation operator, with complex eigenvalues. Several physical quantities, as the Heisenberg uncertainty product, probability density and mean energy value, will be as well explored.
We examine the bound state solutions of the Dirac equation under the spin and pseudospin symmetries for a new suggested combined potential, Hulten plus a class of Yukawa potential including a Coulomb-like tensor interaction. An improved scheme is employed to deal with the centrifugal (pseudo-centrifugal) term. Using the Nikiforov-Uvarov and SUSYQM methods, we analytically develop the relativistic energy eigenvalues and associated Dirac spinor components of wave functions. We find that both methods give entirely the same results. Modifiable of our results into some particular potential cases, u…
New exact analytical bound-state solutions of the radial Dirac equation in 3+1 dimensions for two sets of couplings and radial potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of equations that can be transformed into polynomial equations. Several analytical results found in the literature, including the Dirac oscillator, are obtained as particular cases of this uni…
We develop an analytic formalism to describe dipole radiation near the Dirac cone of a two-dimensional photonic crystal slab. In contrast to earlier work, we account for all polarization effects and derive a closed-form expression for the dyadic Green's function of the geometry. Using this analytic Green's function, we demonstrate that the dipolar interaction mediated by the slab exhibits winding phases, which are key ingredients for engineering topological systems for quantum emitters, as discussed in detail in J. Perczel et al., (2018), arXiv:1810.12299. As an example, we study the coherent …
