Phase diagrams of a ferroelectric superlattice: A Green’s function technique study

A Fermi-type Green’s function method has been used to investigate the phase transition properties of a ferroelectric superlattice with two alternating materials on the basis of the transverse Ising model. By performing a higher-order decoupling to the equations of motion for the Green’s functions, the eigenfrequencies of the infinite ferroelectric superlattice are obtained. Moreover, we discuss the dependence of the phase diagrams on the interface coupling strength, the transverse field, and the thicknesses of two slabs. The comparison between the Green’s function technique and the usual mean-field approximation is illustrated.     

Analytic and numeric Green’s functions for a two-dimensional electron gas in an orthogonal magnetic field

We have derived closed analytic expressions for the Green’s function of an electron in a two-dimensional electron gas threaded by a uniform perpendicular magnetic field, also in the presence of a uniform electric field and of a parabolic spatial confinement. A workable and powerful numerical procedure for the calculation of the Green’s functions for a large infinitely extended quantum wire is considered exploiting a lattice model for the wire, the tight-binding representation for the corresponding matrix Green’s function, and the Peierls phase factor in the Hamiltonian hopping matrix element to account for the magnetic field. The numerical evaluation of the Green’s function has been performed by means of the decimation-renormalization method, and quite satisfactorily compared with the analytic results worked out in this paper. As an example of the versatility of the numerical and analytic tools here presented, the peculiar semilocal character of the magnetic Green’s function is studied in detail because of its basic importance in determining magneto-transport properties in mesoscopic systems.     

Beliaev theory of spinor Bose–Einstein condensates

By generalizing the Green’s function approach developed by Beliaev [S.T. Beliaev, Sov. Phys. JETP 7 (1958) 299; S.T. Beliaev, Sov. Phys. JETP 7 (1958) 289], we study effects of quantum fluctuations on the energy spectra of spin-1 spinor Bose–Einstein condensates, in particular, of a ⁸⁷Rb condensate in the presence of an external magnetic field. We find that due to quantum fluctuations, the effective mass of magnons, which characterizes the quadratic dispersion relation of spin-wave excitations, increases compared with its mean-field value. The enhancement factor turns out to be the same for two distinct quantum phases: the ferromagnetic and polar phases, and it is a function of only the gas parameter. The lifetime of magnons in a spin-1 ⁸⁷Rb spinor condensate is shown to be much longer than that of phonons due to the difference in their dispersion relations. We propose a scheme to measure the effective mass of magnons in a spinor Bose gas by utilizing the effect of magnons’ nonlinear dispersion relation on the time evolution of the distribution of transverse magnetization. This type of measurement can be applied, for example, to precision magnetometry.     

Green’s function-stochastic methods framework for probing nonlinear evolution problems: Burger’s equation, the nonlinear Schrödinger’s equation, and hydrodynamic organization of near-molecular-scale vorticity

A framework which combines Green’s function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green’s function and stochastic representative solutions of linear drift–diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems – Burgers’ equation and the nonlinear Schrödinger’s equation – are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole–Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger’s vortex sheets. Here, the governing vorticity equation corresponds to the Fokker–Planck equation of an Ornstein–Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the motion and spread of single, multiple, and continuous sets of Burger’s vortex sheets, evolving within deterministic and random strain rate fields, under both viscous and inviscid conditions, are obtained. In order to promote application to other nonlinear problems, a tutorial development of the framework is presented. Likewise, time-incremental solution approaches and construction of approximate, though otherwise difficult-to-obtain backward-time GF’s (useful in solution of forward-time evolution problems) are discussed.     

Study of coupling effect on the performance of a spin-current diode: Nonequilibrium Green’s function based model

In this paper, spin-dependent transport through a spin diode composed of a quantum dot coupled to a normal metal and a ferromagnetic lead is studied. The current polarization and the spin accumulation are analyzed using the equations of motion method within the nonequilibrium Green’s function formalism. We present a suitable method for computing Green’s function without carrying out any self-consistent calculation. The influence of coupling strength and magnetic field on the spin current is studied and observed that this device cannot work as a spin diode under certain conditions.     

Electronic transmission through a ladder with a single side-attached impurity

We study the electronic transmission of a model system composed by two coupled chains with an impurity attached to one of them. Analytical espressions for the transmittivity and for the diagonal and the off-diagonal Green’s function matrix elements are derived. Green’s function behaviour as function of the charge carrier energy is exploited to interpret the system transmittivity calculated by the scattering matrix formalism. We find that while a single substitutional impurity in the ladder may generate a Fano resonance in the transmittivity in the lower or in the higher energy part of the spectrum, in the case of a single side-attached impurity to the ladder, a resonance is found in each energy region. We interpret such resonances in terms of local density of states and off-diagonal Green’s function matrix elements.     

A new approach to the calculation of tight-binding surface density of states in thin films

We present a new approach to the calculation of planar electronic density of states of thin films within the tight-binding scheme. The diagonal elements of the electronic Green's function are obtained by an iteration procedure derived from the transfer-matrix approach for semi-infinite crystals. An application is made to the study of the (100) planes of a model transition metal film with a simple cubic structure.     

Scalar Green’s function for penetrable or dielectric scatterers

In a series of former papers, we developed the so-called self-consistent Green’s function formalism (SGFF) for acoustic and light scattering on impenetrable or ideal metallic scatterers. With the paper at hand we will extend the application of this formalism to penetrable or dielectric scatterers. The concept of the Green’s function of the third kind is utilized which was introduced first by Tai. It must be slightly generalized to allow the treatment of nonspherical scatterers. The following considerations reveal the conceptual equivalence between the Green’s function of the third kind and Waterman’s T-matrix method. It is another goal of this paper to demonstrate that the conventional boundary and volume integral equations can be also derived within the developed Green’s function formalism.     

Boundary symmetries in linear differential and integral equation problems applied to the self-consistent Green’s function formalism of acoustic and electromagnetic scattering

Explicit symmetry relations for the Green’s function subject to homogeneous boundary conditions are derived for arbitrary linear differential or integral equation problems in which the boundary surface has a set of symmetry elements. For corresponding homogeneous problems subject to inhomogeneous boundary conditions implicit symmetry relations involving the Green’s function are obtained. The usefulness of these symmetry relations is illustrated by means of a recently developed self-consistent Green’s function formalism of electromagnetic and acoustic scattering problems applied to the exterior scattering problem. One obtains explicit symmetry relations for the volume Green’s function, the surface Green’s function, and the interaction operator, and the respective symmetry relations are shown to be equivalent. This allows us to treat boundary symmetries of volume-integral equation methods, boundary-integral equation methods, and the T matrix formulation of acoustic and electromagnetic scattering under a common theoretical framework. By specifying a specific expansion basis the coordinate-free symmetry relations of, e.g., the surface Green’s function can be brought into the form of explicit symmetry relations of its expansion coefficient matrix. For the specific choice of radiating spherical wave functions the approach is illustrated by deriving unitary reducible representations of non-cubic finite point groups in this basis, and by deriving the corresponding explicit symmetry relations of the coefficient matrix. The reducible representations can be reduced by group-theoretical techniques, thus bringing the coefficient matrix into block-diagonal form, which can greatly reduce ill-conditioning problems in numerical applications.     

A hybrid method for the parallel computation of Green’s functions

Quantum transport models for nanodevices using the non-equilibrium Green’s function method require the repeated calculation of the block tridiagonal part of the Green’s and lesser Green’s function matrices. This problem is related to the calculation of the inverse of a sparse matrix. Because of the large number of times this calculation needs to be performed, this is computationally very expensive even on supercomputers. The classical approach is based on recurrence formulas which cannot be efficiently parallelized. This practically prevents the solution of large problems with hundreds of thousands of atoms. We propose new recurrences for a general class of sparse matrices to calculate Green’s and lesser Green’s function matrices which extend formulas derived by Takahashi and others. We show that these recurrences may lead to a dramatically reduced computational cost because they only require computing a small number of entries of the inverse matrix. Then, we propose a parallelization strategy for block tridiagonal matrices which involves a combination of Schur complement calculations and cyclic reduction. It achieves good scalability even on problems of modest size.     

Effect of boundary treatments on quantum transport current in the Green’s function and Wigner distribution methods for a nano-scale DG-MOSFET

In this paper, we conduct a study of quantum transport models for a two-dimensional nano-size double gate (DG) MOSFET using two approaches: non-equilibrium Green’s function (NEGF) and Wigner distribution. Both methods are implemented in the framework of the mode space methodology where the electron confinements below the gates are pre-calculated to produce subbands along the vertical direction of the device while the transport along the horizontal channel direction is described by either approach. Each approach handles the open quantum system along the transport direction in a different manner. The NEGF treats the open boundaries with boundary self-energy defined by a Dirichlet to Neumann mapping, which ensures non-reflection at the device boundaries for electron waves leaving the quantum device active region. On the other hand, the Wigner equation method imposes an inflow boundary treatment for the Wigner distribution, which in contrast ensures non-reflection at the boundaries for free electron waves entering the device active region. In both cases the space-charge effect is accounted for by a self-consistent coupling with a Poisson equation. Our goals are to study how the device boundaries are treated in both transport models affects the current calculations, and to investigate the performance of both approaches in modeling the DG-MOSFET. Numerical results show mostly consistent quantum transport characteristics of the DG-MOSFET using both methods, though with higher transport current for the Wigner equation method, and also provide the current–voltage (I–V) curve dependence on various physical parameters such as the gate voltage and the oxide thickness.     

量子扩散通道中Wigner算符的演化规律

众所周知,量子态的演化可用与其相应的Wigner函数演化来代替.因为量子态的Wigner函数和量子态的密度矩阵一样,都包含了概率分布和相位等信息,因此对量子态的Wigner函数进行研究,可以更加快速有效地获取量子态在演化过程的重要信息.本文从经典扩散方程出发,利用密度算符的P表示,导出了量子态密度算符的扩散方程.进一步通过引入量子算符的Weyl编序记号,给出了其对应的Weyl量子化方案.另外,借助于密度算符的另一相空间表示-Wigner函数,建立了Wigner算符在扩散通道中演化方程,并给出了其Wigner算符解的形式.本文推导出了Wigner算符在量子扩散通道中的演化规律,即演化过程中任意时刻Wigner算符的形式.在此结论的基础上,讨论了相干态经过量子扩散通道的演化情况.     

A study of the ground-state energy of He with nucleon–nucleon potentials

Ground-state properties are evaluated for the finite nucleus ⁴He starting from realistic nucleon–nucleon interactions within the framework of the Green’s function approach. For the sake of comparison, the same calculations are performed using the Brueckner–Hartree–Fock approximation. For that purpose four high-quality modern nucleon–nucleon interactions represented in momentum space are employed: the Argonne V18, CD-Bonn, Bonn A and N³LO potentials. In these potentials, the effects of charge dependence are taken into account. Additional binding energy is obtained from the inclusion of the hole–hole scattering term within the framework of the Green function approach. It has been shown that the Green function results agree well with the results obtained by accurate methods for few-nucleon systems such as the Faddeev–Yakubovsky calculation. In this study, a comparison of the calculated ground-state energies, obtained by using the Green function approach and different nucleon–nucleon potentials, with experimental values is carried out. The results show good agreement between the calculated values and the experimental ones.     

Spin-flip effects in a parallel-coupled double quantum dot molecule

We investigate theoretically the electronic transport through a parallel-coupled double quantum dot (DQD) molecule attached to metallic electrodes, in which the spin-flip scattering on each quantum dot is considered. Special attention is paid to the effects of the intradot spin-flip processes on the linear conductance by using the equation of motion approach for Green’s functions. When a weak spin-flip scattering on each quantum dot is present, the single Fano peak splits into two Fano peaks, and the Breit–Wigner resonance may be suppressed slightly. When the spin-flip scattering strength on each quantum dot becomes strong, the linear conductance spectrum consists of two Breit–Wigner peaks and two Fano peaks due to the quantum interference effects. The positions and shapes of these resonant peaks can be controlled by using the magnetic flux through the quantum device.     

Splitting of the Fano resonance by Coulomb interactions in a two-dot quantum ring

We study the splitting of the Fano resonance in a Aharonov–Bohm interferometer with a quantum dot in each of its arms. Both intra- and inter-dot Coulomb repulsions are taken into account by employing the Keldysh nonequilibrium Green’s function technique. The single narrow Fano resonance in the noninteracting case is split into two in the presence of either intra- or inter-dot Coulomb interaction. We find that four Fano peaks emerge in the conductance or local density of states spectra when the two kinds of interactions exist simultaneously. Such behavior holds true for the accompanying broad Breit–Wigner type resonance. We also show that the positions of the Fano peaks can be tuned with the aid of the magnetic flux penetrating through the ring, which might have practical applications in device design or quantum computation.     

从离散Wigner函数的角度探讨量子相干性度量

量子相干性是量子信息处理的基本要素,在量子计算中扮演着重要的角色.为了便于讨论量子相干性在量子计算中的作用,本文从离散Wigner函数角度对量子相干性进行了探讨.首先对奇素数维量子系统的离散Wigner函数进行了分析,分离出表征相干性的部分,提出了一种可能的基于离散Wigner函数的量子相干性度量方法,并对其进行了量子相干性度量规范的分析;同时也比较了该度量与l_1范数相干性度量之间的关系.重要的是,这种度量方法能够明确给出量子相干性程度与衡量量子态量子计算加速能力的负性和之间不等式关系,由此可以解析地解释量子相干性仅是量子计算加速的必要条件.     

Self-consistent Green’s function formalism for acoustic and light scattering. Part 3: Unitarity and symmetry

This paper completes two previous papers in which we have developed the self-consistent Green’s function formalism for acoustic and light scattering. It is concerned with the unitarity and symmetry properties of the interaction and far field scattering operator of this formalism. We will show that these are primarily mathematical properties, and that the principles of energy conservation and reciprocity, which express our physical experience, can be modeled by these mathematical properties. For this we have to distinguish two experimental configurations, and only one of these configurations will allow us to relate energy conservation to unitarity. Closely related to this are questions concerning the definition and measurability of the scattering quantities and the importance of the optical and generalized optical theorem. These questions will be also discussed from the point of view of the self-consistent Green’s function formalism.     

Scattering models for ultracold atoms

We present a review of scattering models that can be used to describe the low-energy behavior of identical bosonic atoms. In the simplest models, the only degrees of freedom are atoms in the same spin state. More elaborate models have other degrees of freedom, such as atoms in other spin states or diatomic molecules. The parameters of the scattering models are specified by giving the S-wave phase shifts for scattering of atoms in the spin state of primary interest. The models are formulated as local quantum field theories and the renormalization of their coupling constants is determined. Some of the parameters can be constrained by renormalizability or by the absence of negative-norm states. The Green’s functions that describe the evolution of two-atom states are determined analytically. They are used to determine the T-matrix elements for atom-atom scattering and the binding energies of diatomic molecules. The scattering models all exhibit universal behavior as the scattering length in a specific spin state becomes large.     

Antisymmetrized Green’s function approach to (e, e′) reactions with a realistic nuclear density

A completely antisymmetrized Green’s function approach to the inclusive quasielastic (e, e′) scattering, including a realistic one-body density, is presented. The single-particle Green’s function is expanded in terms of the eigenfunctions of the non-hermitian optical potential. This allows one to treat final state interactions consistently in the inclusive and in the exclusive reactions. Nuclear correlations are included in the one-body density. Numerical results for the response functions of ¹⁶O and ⁴⁰Ca are presented and discussed.     

Decoherence of odd compass states in the phase-sensitive amplifying/dissipating environment

We study the evolution of odd compass states (specific superpositions of four coherent states), governed by the standard master equation with phase-sensitive amplifying/attenuating terms, in the presence of a Hamiltonian describing a parametric degenerate linear amplifier. Explicit expressions for the time-dependent Wigner function are obtained. The time of disappearance of the so called “sub-Planck structures” is calculated using the negative value of the Wigner function at the origin of phase space. It is shown that this value rapidly decreases during a short “conventional interference degradation time” (CIDT), which is inversely proportional to the size of quantum superposition, provided the anti-Hermitian terms in the master equation are of the same order (or stronger) as the Hermitian ones (governing the parametric amplification). The CIDT is compared with the final positivization time (FPT), when the Wigner function becomes positive. It appears that the FPT does not depend on the size of superpositions, moreover, it can be much bigger in the amplifying media than in the attenuating ones. Paradoxically, strengthening the Hamiltonian part results in decreasing the CIDT, so that the CIDT almost does not depend on the size of superpositions in the asymptotical case of very weak reservoir coupling. We also analyze the evolution of the Mandel factor, showing that for some sets of parameters this factor remains significantly negative, even when the Wigner function becomes positive.