Capability of the Free-Ion Eigenstates for Crystal-Field Splitting

Any electronic eigenstate of the paramagnetic ion open-shell is characterized by the three independent multipole asphericities for and 6 related to the second moments of the relevant crystal-field splittings by , where . The Ak as the reduced matrix elements can serve as a reliable measure of the state capability for the splitting produced by the k-rank component of the crystal-field Hamiltonian. These multipolar characteristics allow one to verify any fitted crystal-field parameter set by comparing the calculated second moments and the experimental ones of the relevant crystal-field splittings. We present the multipole characteristics Ak for the extensive set of eigenstates from the lower parts of energy spectra of the tripositive 4 f N ions applying in the calculations the improved eigenfunctions of the free lanthanide ions obtained based on the M. Reid f-shell programs. Such amended asphericities are compared with those achieved for the simplified Russell-Saunders states. Next, they are classified with respect to the absolute or relative weight of Ak in the multipole structure of the considered states. For the majority of the analyzed states (about 80%) the Ak variation is of order of only a few percent. Some essential changes are found primarily for several states of Tm3+, Er3+, Nd3+, and Pr3+ ions. The detailed mechanisms of such Ak changes are unveiled. Particularly, certain noteworthy cancelations as well as enhancements of their magnitudes are explained.

CONCENTRATION OF BLOCH EIGENSTATES IN THE PRESENCE OF GAUGE AT THE SEMI-CLASSICAL LIMIT

We prove a concentration result of a Bloch eigenstate in a periodic channel under a constant gauge, In the semi-classical limit h → 0 these eigenstates concentrate near a maximizer of the scalar potential of the associated Schr6dinger operator, provided the constant gauge converges to a critical value from above. This is in contrast with the ground states which concentrate for any gauge in this limit near a minimizer of the scalar potential.

Slater determinant and exact eigenstates of the two-dimensional Fermi–Hubbard model

We consider the construction of exact eigenstates of the two-dimensional Fermi-Hubbard model defined on an L ×L lattice with a periodic condition. Based on the characteristics of Slater determinants, several methods are introduced to construct exact eigenstates of the model. The eigenstates constructed are independent of the on-site electron interaction and some of them can also represent exact eigenstates of the two-dimensional Bose-Hubbard model.

Sensitivity of energy eigenstates to perturbation in quantum integrable and chaotic systems

We study the sensitivity of energy eigenstates to small perturbation in quantum integrable and chaotic systems.It is shown that the distribution of rescaled components of perturbed states in unperturbed basis exhibits qualitative difference in these two types of systems:being close to the Gaussian form in quantum chaotic systems,while,far from the Gaussian form in integrable systems.

Exact eigenstates for a class of models describing multiphoton processes in the presence of seven bosonic modes

We present the analytical expressions of eigenstates and eigenenergies by using a parameter λ without the assumption of Bethe anzatz for three kinds of seven-bosonic-mode mixing models. The parameter is shown to be determined by the roots of a simple polynomial. We also demonstrate the exact analytical expressions of eigenstates and energies without any unknown parameter.

A Method of Estimating the Eigenstates of Density Operator

We describe a mathematical structure which corresponds to the eigenstates of a density operator. For an unknown density operator, we propose an estimating procedure which uses successive ＂yes/no＂ measurements to scan the Bloch sphere and approximately yields the eigenstates. This result is based on the quantum method of types and implies a relationship between the typical subspace and the Young frame.

New representation of the multimode phase shifting operator and its application

Based on the rotation transformation in phase space and the technique of integration within an ordered product of operators, the coherent state representation of the multimode phase shifting operator and one of its new applications in quantum mechanics are given. It is proved that the coherent state is a natural language for describing the phase shifting operator or multimode phase shifting operator. The multimode phase shifting operator is also a useful tool to solve the dynamic problems of the mnltimode coordinate-momentum coupled harmonic oscillators. The exact energy spectra and eigenstates of such multimode coupled harmonic oscillators can be easily obtained by using the rnultimode phase shifting operator.

Boundary Conditions for Sturm-Liouville Equation with Transition Regions and Barriers or Wells

By means of expansions of rapidly in infinity decreasing functions in delta functions and their derivatives, we derive generalized boundary conditions of the Sturm-Liouville equation for transitions and barriers or wells between two asymptotic potentials for which the solutions are supposed as known. We call such expansions “moment series” because the coefficients are determined by moments of the function. An infinite system of boundary conditions is obtained and it is shown how by truncation it can be reduced to approximations of a different order (explicitly made up to third order). Reflection and refraction problems are considered with such approximations and also discrete bound states possible in nonsymmetric and symmetric potential wells are dealt with. This is applicable for large wavelengths compared with characteristic lengths of potential changes. In Appendices we represent the corresponding foundations of Generalized functions and apply them to barriers and wells and to transition functions. The Sturm-Liouville equation is not only interesting because some important second-order differential equations can be reduced to it but also because it is easier to demonstrates some details of the derivations for this one-dimensional equation than for the full three-dimensional vectorial equations of electrodynamics of media. The article continues a paper that was made long ago.

How Does Wave Packet of a Free Particle Yield?

Preparing a particle in a superposition or a wave packet of eigenstates of a physical quantity is to let it interact with a large object. The composite system composed of the particle and the large object evolves into an entangled state. When the state of the large object is considered to be approximately unchanged, the entangled state can be approximately considered as a product state, and then the particle is prepared in an approximate superposed state. We consider the Schrodinger equation for a composite system with interactions between subsystems as a fundamental postulate and a single particle’s Schrodinger equation must be approximately obtained from it. We argue that superposition of states exists only in composite systems. Interaction exchanging some quantities between subsystems makes conservation laws strictly hold, and no wave packet of a free particle yields. With this point, we can also understand the double-slit experiment and the tunnel phenomenon.

Analytical treatment of Anderson localization in a chain of trapped ions experiencing laser Bessel beams

Trapped ions, under electromagnetic confinement and Coulomb repulsion, can behave as non-interacting particles in one-dimensional lattices. Here we explore analytically the possible effects regarding Anderson localization in a chain of trapped ions experiencing laser Bessel beams. Under an experimentally feasible condition, we predict an analytical form of the energy-dependent mobility edges, which is verified to be in good agreement with the exact numerical results except for the top band. Some other important properties regarding the phonon localization in the ion chain are also discussed both analytically and numerically. Our results are relevant to experimental observation of localization–delocalization transition in the ion trap and helpful for deeper understanding of the rich phenomena due to long-range phonon hopping.

A Novel Model of Charged Leptons

A novel model of charged leptons is presented, which contains two basics hypotheses. The first hypothesis is that the Yukawa coupling between Higgs field and charged leptons is the weak interaction, the Higgs field is a scalar intermediate boson which changes the chirality of charged leptons in the weak interaction. The other hypothesis is that the flavor eigenstates of charged leptons are the superposition states of left-handed and right-handed elementary Weyl spinors before the electroweak symmetry breaking. According to this model, the Yukawa coupling constants between Higgs field and three generations of charged leptons are considered to be a universal constant, and the difference of the masses of different charged leptons is due to the different left-right mixing angles of their flavor eigenstates.

Quantum States of Interacting Atoms in Quasi-One-Dimensional Ultracold Trapped Gases

We theoretically investigate the quantum states of a Hamiltonian model for quasi-one-dimensional ultracold trapped gases. From the ansatz given by the numerical solution of the Schrödinger equation of the system, we develop a scattering potential functional form and an approximate solution for the analytical approach of the model. We obtain the set of approximate eigenstates and eigenenergies that can be used in future improvements on the study of atomic scattering in low dimensional ultracold gases. We also show that there is a parity inversion of the ground state of the model as the interaction strength increases.

New equation for deriving pure state density operators by Weyl correspondence and Wigner operator

By virtue of the completeness of Wigner operator and Weyl correspondence we construct a general equation for deriving pure state density operators. Several important examples are considered as the applications of this equation, which shows that our approach is effective and convenient for deducing these entangled state representations.

Research on eigenstate current control technology of Flashbased FPGA

To solve the Flash-based FPGA in the manufacturing process,the ion implantation process will bring electrons into the floating gate of the P-channel Flash cell so that the Flash switch is in a weak conduction state,resulting in the Flash-based FPGA eigenstate current problem.In this paper,the mechanism of its generation is analyzed,and four methods are used includ-ing ultraviolet light erasing,high-temperature baking,X-ray irradiation,and circuit logic control.A comparison of these four methods can identify the circuit design by using circuit logic to control the path of the power supply that is the most suitable and reliable method to solve the Flash-based FPGA eigenstate current problem.By this method,the power-on current of 3.5 mil-lion Flash-based FPGA can be reduced to less than 0.3 A,and the chip can start normally.The function and performance of the chip can then be further tested and evaluated,which is one of the key technologies for developing Flash-based FPGA.

Eigenstate Distribution Fluctuation of a Quenched Disordered Bose–Hubbard System in Thermal-to-Localized Transitions

We study the thermalization of a quenched disordered Bose–Hubbard system. By considering the eigenstate distribution fluctuation, we show that the thermal to many-body localized transition is always connected to a minimum of this distribution fluctuation. We also observe a Mott-localized regime, where the system fails to thermalize due to the strong on-site repulsion. Lastly, we show how to detect this eigenstate distribution fluctuation in a cold atom system, which is equivalent to measure the Loschmidt echo of the system. Our work suggests a way to measure the thermal-to-localized transitions in experiments, especially for a large system.

Entanglement Swapping for Displaced Two-Mode Squeezed States

We discuss the entanglement swapping of two pairs of displaced two-mode squeezed states and find that the squeezing parameter of the outcoming squeezed state is less than that of the originM squeezed states. The calculation is greatly simplified by virtue of the natural expression of the two-mode squeezing operator in EPR eigenstate representation. A protocol for such entanglement swapping is proposed.

Incoherent control of two classes of finite-level quantum systems

The incoherent control of finite-level quantum systems is investigated. Following a brief introduction to coherent control paradigms in quantum control, a control problem that can not be accomplished using only coherent control is presented. For such a control problem, it is proved that it can be accomplished using incoherent control based on projective measurement and coherent control for two classes of finite-level quantum systems, i.e., eigenstate controllable quantum systems and wavefunction controllable quantum systems.

Wave function for the squeezed atomic coherent state in entangled state representation and some of its applications

Based on the Einstein, Podolsky, and Rosen （EPR） entangled state representation, this paper introduces the wave function for the squeezed atomic coherent state （SACS）, which turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j. As important applications of the wave function, the Wigner function of the SACS and its marginal distribution are obtained and the eigenproblems of some Hamiltonians for the generalized angular momentum system are solved.

Spin and Quantization of Gravitational Space

According to the formula of translational motion of vector along an infinitesimal closed curve in gravitational space, this article shows that the space and time both are quantized;the called center singularity of Schwarzschild metric does not exist physically, and Einstein’s theory of gravity is compatible with the traditional quantum theory in essence;the quantized gravitational space is just the spin network which consists of infinite quantized loops linking and intersecting each other, and that whether the particle is in spin eigenstate depends on the translational track of its spin vector in gravitational space.

Steady-state relation of a two-level system strongly coupled to a many-body quantum chaotic environment

We study the long-time average of the reduced density matrix(RDM)of a two-level system as the central system,which is locally coupled to a many-body quantum chaotic system as the environment,under an overall Schr?dinger evolution.A phenomenological relation among elements of the RDM is proposed for a dissipative interaction in the strong coupling regime and is tested numerically with the environment as a defect Ising chain,as well as a mixed-field Ising chain.