Delta-function Barrier Research Articles

The Gross–Pitaevskii Equation for an Infinite Square Well with a Delta-Function Barrier

The Gross–Pitaevskii Equation for an Infinite Square Well with a Delta-Function Barrier

Scattering of identical particles by a one-dimensional Dirac delta function barrier potential: The role of statistics

Scattering of non-interacting, identical bosons or fermions by a one-dimensional Dirac delta function barrier potential underlines the importance of the role of statistics (that is, whether the particles obey Fermi–Dirac or Bose–Einstein statistics) in the scattering. We consider an initial wave function for the system that corresponds to one particle incident from the left and one from the right of the potential barrier. For bosons, both particles are scattered either to the left or to the right if the intensity reflection coefficient is 1/2, provided the left and right propagating wave packets fully overlap in the scattering region. For fermions, the particles “pass through” one another, provided the left and right propagating wave packets fully overlap in the scattering region, with zero probability that both particles are scattered to the left or right, consistent with the Pauli exclusion principle.

Berry phase for a Bose gas on a one-dimensional ring

We study a system of strongly interacting one-dimensional (1D) bosons on a ring pierced by a synthetic magnetic flux tube. By the Fermi-Bose mapping, this system is related to the system of spin-polarized non-interacting electrons confined on a ring and pierced by a solenoid (magnetic flux tube). On the ring there is an external localized delta-function potential barrier $V(\phi)=g \delta(\phi-\phi_0)$. We study the Berry phase associated to the adiabatic motion of delta-function barrier around the ring as a function of the strength of the potential $g$ and the number of particles $N$. The behavior of the Berry phase can be explained via quantum mechanical reflection and tunneling through the moving barrier which pushes the particles around the ring. The barrier produces a cusp in the density to which one can associate a missing charge $\Delta q$ (missing density) for the case of electrons (bosons, respectively). We show that the Berry phase (i.e., the Aharonov-Bohm phase) cannot be identified with the quantity $\Delta q/\hbar \oint \mathbf{A}\cdot d\mathbf{l}$. This means that the missing charge cannot be identified as a (quasi)hole. We point out to the connection of this result and recent studies of synthetic anyons in noninteracting systems. In addition, for bosons we study the weakly-interacting regime, which is related to the strongly interacting electrons via Fermi-Bose duality in 1D systems.

Current-phase relation in a short clean josephson junction model: application to MgB2

Motivated by recent data on high-quality MgB2 thin films implying that the smaller energy gap has l = 6 (i-wave) symmetry, we consider a simple model for an all-MgB2 symmetric Josephson Junction (JJ). The model assumes an arbitrary-strength delta-function barrier and one-dimensional current conduction. It is shown that in this context a nodal energy gap with i-wave symmetry acts as an isotropic energy gap (s-wave) with an amplitude modified by the energy-gap misalignment-angle with respect to the crystal principal axes. The corresponding exact Green’s function in momentum space is derived employing a novel approach. The ensuing current-phase relations in the strong and weak barrier-strengths limits are calculated and found to confirm known results, e.g., the Ambegaokar-Baratoff current-phase relation. Inspired by an HTS experiment that established the d-wave energy-gap symmetry, we propose a JJ-related experiment with a MgB2 bicrystal to confirm our premise that the smaller energy has i-wave symmetry.

Giant Stark effect in coupled quantum wells: Analytical model

Coupled quantum wells have been proposed as candidates for highly polarizable structures due to their near-degenerate and dipole-coupled electronic states. Hence, many interesting applications in linear and nonlinear optics can be envisioned. We analyze this proposal considering a simple structure with a delta-function barrier separating the wells. While very substantial Stark shifts are certainly predicted for this geometry, perturbative estimates based on polarizabilities (and hyperpolarizabilities) fail beyond a critical field strength that depends inversely on the barrier. Hence, a giant Stark effect due to near-degenerate states is invariably limited by a small critical field. Our analytical (hyper) polarizability expressions are applied to find quantitative Stark shifts for GaAs quantum wells and transition-metal dichalcogenide bilayers. The predicted Stark shifts and critical fields agree with the field dependence observed in a range of available experiments.

Electron transmission through a periodically driven graphene magnetic barrier

Electronic transport through graphene magnetic barriers is studied theoretically in presence of an external time harmonic scalar potential in the framework of non-perturbative Landau–Floquet Formalism. The oscillating field mostly suppresses the transmission for rectangular magnetic barrier structure and exhibits the Fano resonance for multiphoton processes due to the presence of bound state inside the barrier. While, for a pair of delta function barriers of larger separation, the oscillating potential suppresses the usual Fabry–Perot oscillations in the transmission and a new type of asymmetric Fano resonance is noted for smaller separation, occurring due to extended states between the barriers.

Scheme for accelerating quantum tunneling dynamics

We propose a scheme of the exact fast-forwarding of standard quantum dynamics for a charged particle. The present idea allows the acceleration of both the amplitude and phase of the wave function throughout the fast-forward time range and is distinct from that of Masuda-Nakamura (e.g., Proc. R. Soc. A {\bf 466}, 1135 (2010)) which enabled acceleration of only the amplitude of the wave function on the way. We shall apply the proposed method to the quantum tunneling phenomena and obtain the electro-magnetic field to ensure the rapid penetration of wave functions through a tunneling barrier. Typical examples described here are: 1) an exponential wave packet passing through the delta-function barrier; 2) the opened Moshinsky shutter with a delta-function barrier just behind the shutter. We elucidate the tunneling current in the vicinity of the barrier and find the remarkable enhancement of the tunneling rate (: tunneling power) due to the fast-forwarding. In the case of a very high barrier, in particular, we present the asymptotic analysis and exhibit a suitable driving force to recover a recognizable tunneling current. The analysis is also carried out on the exact acceleration of macroscopic quantum tunneling with use of the nonlinear Schr\"odinger equation which accommodates a tunneling barrier.

Resonant peak splitting through magnetic Kronig–Penney superlattices in graphene

Abstract The resonance splitting effect of Dirac electrons through magnetic Kronig–Penney superlattices with delta-function barriers in graphene is studied theoretically. It is found that both transmission probability and conductance present ( n − 1 ) - fold resonant peak splitting in n vector potential barriers, which is the same as that of standard electrons in semiconductor superlattices [R. Tsu, L. Esaki, Appl. Phys. Lett. 22 (1973) 562]. The resonant peak splitting and wave-vector filtering could be controlled by adjusting the transverse wave vector and structural parameters. These properties may be useful for the design of graphene-based electronic devices.

Resonant reflection of a Bose–Einstein condensate by a double barrier within the Gross–Pitaevskii equation

Resonant reflection of a Bose–Einstein condensate by a double delta-function barrier has been considered analytically using the Gross–Pitaevskii approximation for nonlinearity. The reflecting coefficient has been derived taking into account a weak nonlinearity of the Schrödinger equation produced by the interaction between cold alkali atoms. A nonlinear term is given in the Hartree approximation for short-range interaction between atoms. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflecting coefficient by a multiple-scale method in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without a nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflecting coefficient is derived in the first order on the nonlinearity parameter.

Resonant reflection of Bose-Einstein condensate by a double delta-function barrier within the Gross-Pitaevskii equation

Resonant reflection of a Bose-Einstein condensate by a double delta-function barrier has been considered analytically using the Gross-Pitaevskii approximation for nonlinearity. The reflection coefficient has been derived taking into account a weak nonlinearity of the Schrödinger equation produced by the interaction between cold alkali atoms. Nonlinear term is given in the limit of asymptotically weak interaction and zero temperature. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflection coefficient using a multiple-scale analysis in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflection coefficient is derived in the first order on the nonlinearity parameter.

Hill’s equation with random forcing parameters: The limit of delta function barriers

This paper considers random Hill's equations in the limit where the periodic forcing function becomes a Dirac delta function. For this class of equations, the forcing strength $q_k$, the oscillation frequency $\af_k$, and the period are allowed to vary from cycle to cycle. Such equations arise in astrophysical orbital problems in extended mass distributions, in the reheating problem for inflationary cosmologies, and in periodic Schr{\"o}dinger equations. The growth rates for solutions to the periodic differential equation can be described by a matrix transformation, where the matrix elements vary from cycle to cycle. Working in the delta function limit, this paper addresses several coupled issues: We find the growth rates for the $2 \times 2$ matrices that describe the solutions. This analysis is carried out in the limiting regimes of both large $q_k \gg 1$ and small $q_k \ll 1$ forcing strength parameters. For the latter case, we present an alternate treatment of the dynamics in terms of a Fokker-Planck equation, which allows for a comparison of the two approaches. Finally, we elucidate the relationship between the fundamental parameters $(\af_k,q_k)$ appearing in the stochastic differential equation and the matrix elements that specify the corresponding discrete map. This work provides analytic -- and accurate -- expressions for the growth rates of these stochastic differential equations in both the $q_k \gg1 $ and the $q_k \ll 1$ limits.

Bistable transmission of plane waves across two nonlinear delta functions

We examine the transmission coefficient of plane waves across two nonlinear delta-function barriers with positive opacity. It is shown that this simple system is capable of bistable behavior at sufficiently large input intensities. The bistable behavior is centered around the first transmission resonances and is a simple example of a continuum nonlinear system that can display bistability without the presence of a feedback loop.

Exponential and non-exponential decay of an initially localized quantum state subject to two delta-barriers

We study the time dependence of a quantum particle initially localized'behind two delta-function barriers of strengths v 1 and v 2 . We determine the conditions under which the decay is exponential by varying the ratio v 1 /v 2 and the separation between the delta-barriers. We find that exponential decay occurs even when v 1 /v 2 is of order one. We also find cases of non-exponential decay. An interesting example of non-exponential decay is when the two barriers are very close. We show that the exponential decay is due to poles in the complex k-plane, where k denotes the wave numbers of the energy eigenstates, and we find that the decay rates are fully understood in many cases as being due to a single pole.

Spin polarization of the transport current at the free surface of bulkNiMnSb

The point contact Andreev reflection employing niobium tips was used to determine the degree of transport current spin polarization ( Pt ) at the free surface of bulk NiMnSb at 4.2 K . The data was analyzed within the framework of a modified version of the Blonder, Tinkham, and Klapwijk formulism taking into account the two spin polarized channels in the ferromagnet and treating the interface as a planar delta function barrier of height Z between free electron materials. We find that the measured spin polarization is rather insensitive to different surface preparations and magnetic domain structure, and the maximal value of the Pt at Z=0 is 44% , consistent with recent calculations of the surface reconstruction of NiMnSb . Saturation magnetization of the samples was found to be 3.6 iB per formula unit indicative of a small amount of atomic disorder.

A New Matrix Formulation for One-Dimensional Scattering in Dirac Comb (Electromagnetic Waves Approach)

A formulation of the reflection and transmission of electromagnetic waves by a stratified planer structure is demonstrated. This formula uses the so-called elementary symmetric functions that are extensively used in the mathematical theory of polynomials. The reflection and transmission coefficients of any number of quantum wells or barriers can be written in a similar way. Finally, one-dimensional scattering from a series of delta-function barriers (a system that is called Dirac Comb) is investigated. An excellent agreement has been found with the earlier results based on other approaches. This approach is also shown to be a powerful technique and can be easily used for analyzing various complicated mesoscopic structures and devices.

Floquet scattering in parametric electron pumps

A Floquet scattering approach to parametric electron pumps is presented and compared with Brouwer's adiabatic scattering approach [Phys. Rev. B 58, R10135 (1998)] for a simple scattering model with two harmonically oscillating delta-function barriers. For small strength of oscillating potentials these two approaches give exactly equivalent results while for large strength, these clearly deviate from each other. The validity of the adiabatic theory is also discussed by using the Wigner delay time obtained from the Floquet scattering matrix.

Supercurrent flow through an effective double-barrier structure.

Supercurrent flow is studied in a structure that in the Ginzburg-Landau regime can be described in terms of an effective double barrier potential. In the limit of strongly reflecting barriers, the passage of Cooper pairs through such a structure may be viewed as a realization of resonant tunneling with a rigid wave function. For interbarrier distances smaller than $d_0=\pi\xi(T)$ no current-carrying solutions exist. For distances between $d_0$ and $2d_0$, four solutions exist. The two symmetric solutions obey a current-phase relation of $\sin(\Delta\varphi/2)$, while the two asymmetric solutions satisfy $\Delta\varphi=\pi$ for all allowed values of the current. As the distance exceeds $nd_0$, a new group of four solutions appears, each contaning $(n-1)$ soliton-type oscillations between the barriers. We prove the inexistence of a continuous crossover between the physical solutions of the nonlinear Ginzburg-Landau equation and those of the corresponding linearized Schr\"odinger equation. We also show that under certain conditions a repulsive delta function barrier may quantitatively describe a SNS structure. We are thus able to predict that the critical current of a SNSNS structure vanishes as $\sqrt{T'_c-T}$, where $T'_c$ is lower than the bulk critical temperature.

Detailed Study of Quasi-bound Modes in Leaky Structures

Abstract The inconsistency of the progressive wave boundary conditions for the quasi-bound modes in open leaky structures is shown and a new boundary value formulation is developed. The results obtained are tested for leaky waveguides with delta function barriers for which a simple analytical characteristic equation is available. The method is applied for the determination of eigenvalues of prism-loaded multilayer waveguides. The influence of the prism coupler on the waveguide modes is investigated in detail for slab and four-layer waveguides. The method is compared with several approximate methods.

Scattering from a locally periodic potential

One-dimensional scattering from a series of delta-function barriers is considered, in the context of nonrelativistic quantum mechanics. A surprisingly simple closed-form expression for the transmission probability is obtained, and compared with the earlier result of Kiang [Am. J. Phys. 42, 785–787 (1974)]. The band structure characteristic of periodic potentials emerges even when the number of barriers is quite small. However, the ‘‘saturation’’ phenomenon identified by Lee, Zysnarski, and Kerr [Am. J. Phys. 57, 729–734 (1989)] is found to depend on the energy regime in question.

Topological considerations in quantum theory

We consider the most general case of the restricted rigid rotor, controlled by passive mechanical devices located at θ=0 and θ=π. The purpose of these devices is to restrict the particle motion to a domain of a covering space (0,Mπ), whereM is an odd integer. This system, which is not a Hamiltonian one on the physical space (0, 2π), is compared with a Hamiltonian system having delta function barriers at θ=0 and θ=π. The case ofM an even integer is also discussed by using only one mechanical device at θ=0. This non-Hamiltonian system is compared with a Hamiltonian system having a delta function barrier at θ=0. It is shown that many of the wave functions of the non-Hamiltonian systems are the same as those of the Hamiltonian ones, with an average reflection coefficient of 1/(M+1) for oddM and 2/M for evenM, which are the classical values. We show how, in the case of very largeM, the superposition principle leads to de Broglie resonances.