Zero-range Potential Research Articles

Wigner time delay revisited

The concept of Wigner time delay was introduced in scattering theory to quantify the delay or advance of an incoming particle in its interaction with the scattering potential. The scattered-wave part of the wave packet, which describes the interacting particle, is delayed/advanced with respect to the non-interacting plane-wave packet. It has been widely assumed that this concept can be transferred to ionization considering it as a half-scattering process. In the present work we show, by analyzing the corresponding wave packets, that this assumption is incorrect. For ionization the outgoing wave packet is a superposition of the continuum states ψk(−)(r,t), which satisfy the incoming-wave boundary condition at large distances r from the origin and which for large time t→∞ form a plane-wave packet. The expansion coefficients in this wave packet are the ionization probability amplitudes. If these amplitudes depend on the scattering phase shift, the plane-wave part of the ionized wave packet may be modified, leading to an ionization analog of the Wigner time delay. However, in most cases such a Wigner time delay is absent for ionization. We illustrate this with two examples: (i) photoionization by a weak field of an atom with the electron bound by a zero-range potential and (ii) strong-field ionization.

Quantum Measurements and Delays in Scattering by Zero-Range Potentials.

Eisenbud-Wigner-Smith delay and the Larmor time give different estimates for the duration of a quantum scattering event. The difference is most pronounced in the case where the de Broglie wavelength is large compared to the size of the scatterer. We use the methods of quantum measurement theory to analyse both approaches and to decide which one of them, if any, describes the duration a particle spends in the region that contains the scattering potential. The cases of transmission, reflection, and three-dimensional elastic scattering are discussed in some detail.

Zero sound in a quantum gas of spin-3/2 atoms with multipole exchange interaction

In the context of quantum gases, we obtain a many-body Hamiltonian for spin-3/2 atoms with general multipole (spin, quadrupole, and octupole) exchange interaction by employing the apparatus of irreducible spherical tensor operators. This Hamiltonian implies the finite-range interaction, whereas, for zero-range (contact) potentials parameterized by the s-wave scattering length, the multipole exchange interaction becomes irrelevant. Following the reduced description method for quantum systems, we derive the quantum kinetic equation for spin-3/2 atoms in a magnetic field and apply it to examine the high-frequency oscillations known as zero sound.

About the connection of the electron binding energy of a single carbon anion with binding energies of an electron attached to carbon molecules

We demonstrate that the model of zero-range potentials can be successfully employed for the description of attached electrons in atomic and molecular anions, for example, negatively charged carbon clusters. To illustrate the capability of the model we calculate the energies of the attached electron for the family of carbon cluster anions consisting of two-, three- (equilateral triangle) and four (tetrahedron) carbon atoms equidistant from each other as well as for a C3 molecule having a chain structure. The considered approach can be easily extended to carbon clusters containing an arbitrary number of atoms arranged in an arbitrary configuration.

Complex Contact Interaction for Systems with Short-Range Two-Body Losses.

The concept of contact interaction is fundamental in various areas of physics. It simplifies physical models by replacing the detailed short-range interaction with a zero-range contact potential that reproduces the same low-energy scattering parameter, i.e., the s-wave scattering length. In this Letter, we generalize this concept to open quantum systems with short-range two-body losses. We show that the short-range two-body losses can be effectively described by a complex scattering length. However, in contrast to closed systems, the dynamics of an open quantum system is governed by the Lindblad master equation the includes a non-Hermitian Hamiltonian as well as an extra recycling term. We thus develop proper methods to regularize both terms in the master equation in the contact (zero-range) limit. We then apply our regularized complex contact interaction to study the dynamic problem of a weakly interacting and dissipating Bose-Einstein condensate. It is found that the physics is greatly enriched because the scattering length is continued from the real axis to the complex plane. For example, we show that a strong dissipation may prevent an attractive Bose-Einstein condensate from collapsing. We further calculate the particle decay in this system to the order of (density)^{3/2} which resembles the celebrated Lee-Huang-Yang correction to the ground state energy of interacting Bose gases [Lee and Yang, Phys. Rev. 105, 1119 (1957)PHRVAO0031-899X10.1103/PhysRev.105.1119; Lee, Huang, and Yang, Phys. Rev. 106, 1135 (1957)PHRVAO0031-899X10.1103/PhysRev.106.1135]. Possible methods for tuning the complex scattering length in cold atomic gas experiments are also discussed.

Binding of heavy fermions by a single light atom in one dimension

We consider the problem of $N$ identical fermions interacting via a zero-range attractive potential with a lighter atom in one dimension. Using the few-body approach based on the Skorniakov and Ter-Martirosian equation, we determine the energies and the critical mass ratios for the emergence of the tetramer, pentamer, and hexamer. For large $N$, we solve the problem analytically by using the mean-field theory based on the Thomas-Fermi approximation. The system becomes bound when the heavy-to-light mass ratio exceeds a critical value, which grows as ${N}^{3}$ at large $N$. We also employ a more sophisticated Hartree-Fock approach, which turns out to be equivalent to the Thomas-Fermi approximation for determining the energies, but provides a better description of the microscopic structure of the clusters.

Zero-range potentials for Dirac particles: Bound-state problems

A model in which a massive Dirac particle in R3 is bound by N⩾1 spatially distributed zero-range potentials is presented. Interactions between the particle and the potentials are modeled by subjecting a particle’s bispinor wave function to certain limiting conditions at the potential centers. Each of these conditions is parametrized by a 2 × 2 Hermitian matrix (or, equivalently, a real scalar and a real vector) and mixes the upper and the lower components of the wave function. The problem of determining particle’s bound-state eigenenergies is reduced to the problem of finding real zeros of a determinant of a certain 2N×2N matrix. As the lower component of the particle’s wave function is inverse-square singular at each of the potential centers, the wave function itself is not square-integrable. Nevertheless, one can define a scalar pseudo-product with the property that wave functions belonging to different eigenenergies are orthogonal with respect to it. The wave functions may then be normalized so that their self-pseudo-products are plus one, minus one or zero. An auxiliary set of Sturmian functions is constructed and used to derive an explicit representation of particle’s matrix Green’s function. For illustration purposes, two particular systems are studied in detail: (1) a particle bound in a field of a single zero-range potential, (2) a particle bound in a field of two identical zero-range potentials.

Wigner time delay of particles elastically scattered by a cluster of zero-range potentials

The Wigner time delay of slow particles in the process of their elastic scattering by complex targets formed by several zero-range potentials is investigated. It is shown that at asymptotically large distances from the target, the Huygens-Fresnel interference pattern formed by spherical waves emitted by each of the potentials is transformed into a system of spherical waves generated by the geometric center of the target. These functions determine flows of particles in and out through the surface of the sphere surrounding the target. The energy derivatives of phase shifts of these functions are the partial Wigner time delay. General formulas that connect the s-phase shifts of particle scattering by each of the zero-range potentials with the phases of particle scattering by the potential cluster forming the target are obtained. Model targets consisting of two-, three- and 4-centers are considered. It is assumed that these targets are built from identical delta-potentials with equal distances between their centers. The partial Wigner time delay of slow particles by considered targets are obtained. We apply the derived general formulas to consideration of electron scattering by atomic clusters that trap electron near the target, and by calculating the times delay of mesons scattered by few-nucleons systems.

3D scattering by 2D periodic zero-range potentials: total reflection/transmission and threshold effects

3D scattering by 2D periodic zero-range potentials: total reflection/transmission and threshold effects

Vacuum Polarization in a Zero-Width Potential: Self-Adjoint Extension

The effects of vacuum polarization associated with a massless scalar field near pointlike source with a zero-range potential in three spatial dimensions are analyzed. The “physical” approach consists in the usage of direct delta-potential as a model of pointlike interaction. We use the Perturbation theory in the Fourier space with dimensional regularization of the momentum integrals. In the weak-field approximation, we compute the effects of interest. The “mathematical” approach implies the self-adjoint extension technique. In the Quantum-Field-Theory framework we consider the massless scalar field in a 3-dimensional Euclidean space with an extracted point. With appropriate boundary conditions it is considered an adequate mathematical model for the description of a pointlike source. We compute the renormalized vacuum expectation value ⟨ϕ2(x)⟩ren of the field square and the renormalized vacuum averaged of the scalar-field’s energy-momentum tensor ⟨Tμν(x)⟩ren. For the physical interpretation of the extension parameter we compare these results with those of perturbative computations. In addition, we present some general formulae for vacuum polarization effects at large distances in the presence of an abstract weak potential with finite-sized compact support.

Vacuum Polarization with Zero-Range Potentials on a Hyperplane

The quantum vacuum fluctuations of a neutral scalar field induced by background zero-range potentials concentrated on a flat hyperplane of co-dimension 1 in (d+1)-dimensional Minkowski spacetime are investigated. Perfectly reflecting and semitransparent surfaces are both taken into account, making reference to the most general local, homogeneous and isotropic boundary conditions compatible with the unitarity of the quantum field theory. The renormalized vacuum polarization is computed for both zero and non-zero mass of the field, implementing a local version of the zeta regularization technique. The asymptotic behaviors of the vacuum polarization for small and large distances from the hyperplane are determined to leading order. It is shown that boundary divergences are softened in the specific case of a pure Dirac delta potential.

Universal duality transformations in interacting one-dimensional quantum systems

One-dimensional quantum systems admit duality relations that put hard-core spinless bosons and fermions in one-to-one correspondence via Girardeau's mapping theorem. The simplest models of soft bosons interacting via zero-range potentials can also be mapped onto dual interacting fermions. However, a systematic approach to one-dimensional statistical transmutation for arbitrary low-energy interactions in the spinless and spinful or multicomponent cases has remained elusive. I develop a general theory of local unitary transformations between one-dimensional quantum systems of bosons and fermions with arbitrary spin or internal structure, single-particle dispersion---including nonrelativistic, relativistic, or otherwise---and low-energy interactions in the universal regime. These transformations generate families of duality relations and models that relate the strong- and weak-coupling limits of the respective dual theories.

Entanglement and fermionization of two distinguishable fermions in a strict and non strict one-dimensional space

The fermionization regime and entanglement correlations of two distinguishable harmonically confined fermions interacting via a zero-range potential is addressed. We present two alternative representations of the ground state that we associate with two different types of one-dimensional spaces. These spaces, in turn, induce different correlations between particles and thus require a suitable definition of entanglement. We find that the entanglement of the ground state is strongly conditioned by those one-dimensional space features. We also find that in the strongly attractive regime the relative ground state is a highly localized state leading to maximum entanglement. Our analysis shows that in the strongly repulsive regime the ground state changes smoothly from a superposition of Slater-like states to a finite superposition of Slaters, this lack of accessible states yields to Pauli blocking as a strong signature of fermionization. Our results indicate that entangled states could be obtained in current experiments by reaching the non-interacting regime from the interacting regime. Entangled states could also be obtained when a state is brought from the interacting regime into the strongly repulsive regime by changing the scattering length near the confinement-induced resonance (CIR). Finally, we show that the first excited state obtained in the absence of interactions and the third excited fermionized state are maximally entangled.

Emergence of N-Body Tunable Interactions in Universal Few-Atom Systems

A three-atom molecule AAB, formed by two identical bosons A and a distinct one B, is studied by considering coupled channels close to a Feshbach resonance. It is assumed that the subsystems AB and AA have, respectively, one and two channels, where, in this case, AA has open and closed channels separated by an energy gap. The induced three-body interaction appearing in the single channel description is derived using the Feshbach projection operators for the open and closed channels. An effective three-body interaction is revealed in the limit where the trap setup is tuned to vanishing scattering lengths . The corresponding homogeneous coupled Faddeev integral equations are derived in the unitarity limit. The s-wave transition matrix for the AA subsystem is obtained with a zero-range potential by a subtractive renormalization scheme with the introduction of two finite parameters, besides the energy gap. The effect of the coupling between the channels in the coupled equations is identified with the energy gap, which essentially provides an ultraviolet scale that competes with the van der Waals radius - this sets the short-range physics of the system in the open channel. The competition occurring at short distances exemplifies the violation of the ``van der Waals universality" for narrow Feshbach resonances in cold atomic setups. In this sense, the active role of the energy gap drives the short-range three-body physics.

On spectral asymptotic for the second-derivative operators

In this work we focus on spectral asymptotic for the second derivative operators. Here we study Schrödinger operator with zero-range potentials, because this operator has great importance for understanding the solvable problems in quantum mechanics and atomic physics. It appears in different models such as the mathematical physics, applied mathematics and theoretical physics. We have two objectives in this work. We first demonstrated that this operator has a continuous spectrum contains an infinite number of bands separated by gaps. We then explained that the bands to gaps ratio tends to zero under certain conditions.

Ground state properties of trapped boson system with finite-range Gaussian repulsion: exact diagonalization study

We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum Lz is carried out to obtain the N-body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body (N = 2) to many-body (N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson (N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N. For a given N, there is found an optimal value of the range of Gaussian potential for which the first vortex (with Lz = N) nucleates at a lower value of the rotational angular velocity Ωc1 compared to the zero-range (δ-function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.

Оптические свойства легированных нанопроволок во внешних электрическом и магнитном поляx

The impurity absorption of light in quantum wires in the presence of external electric and magnetic fields directed parallel to each other and perpendicular to the axis of the nanowire is studied theoretically. Impurities doped in a quantum wire are modeled by a zero-range potential. Expressions are obtained for the light absorption coefficient and the features of impurity absorption of light are studied for all possible directions of polarization of the light wave. In particular, in the presence of external fields, some optical transitions, which are forbidden in the absence of fields, become allowed. The frequency dependences of the light absorption coefficient and the influence of electric and magnetic fields on the value of the impurity light absorption coefficient are investigated. The possibility of controlling the optical characteristics of nanowires using an electric field is shown.

Bose-Einstein Condensation on the Surface of a Sphere.

Motivated by the recent achievement of space-based Bose-Einstein condensates (BEC) with ultracold alkali-metal atoms under microgravity and by the proposal of bubble traps which confine atoms on a thin shell, we investigate the BEC thermodynamics on the surface of a sphere. We determine analytically the critical temperature and the condensate fraction of a noninteracting Bose gas. Then we consider the inclusion of a zero-range interatomic potential, extending the noninteracting results at zero and finite temperature. Both in the noninteracting and interacting cases the crucial role of the finite radius of the sphere is emphasized, showing that in the limit of infinite radius one recovers the familiar two-dimensional results. We also investigate the Berezinski-Kosterlitz-Thouless transition driven by vortical configurations on the surface of the sphere, analyzing the interplay of condensation and superfluidity in this finite-size system.

Zero range potential approximation in quantum scattering problems

This article presents the Zero Range Potential (ZRP) approximation as an alternative to treat quantum scattering problems. The main focus is to provide an undergraduate-level derivation of the ZRP boundary condition that describes the interaction between an electron and a potential, using the spherical-potential-well analytical solutions. The validity of the approximation is discussed qualitatively and quantitatively. Finally, the ZRP approximation is applied to the scattering of an electron by an atom, comparing the results obtained with modern experimental measurements.

Adiabatic projection method with Euclidean time subspace projection

Euclidean time projection is a powerful tool that uses exponential decay to extract the low-energy information of quantum systems. The adiabatic projection method, which is based on Euclidean time projection, is a procedure for studying scattering and reactions on the lattice. The method constructs the adiabatic Hamiltonian that gives the low-lying energies and wave functions of two-cluster systems. In this paper we seek the answer to the question whether an adiabatic Hamiltonian constructed in a smaller subspace of the two-cluster state space can still provide information on the low-lying spectrum and the corresponding wave functions. We present the results from our investigations on constructing the adiabatic Hamiltonian using Euclidean time projection and extracting details of the low-energy spectrum and wave functions by diagonalizing it. In our analyses we consider systems of fermion-fermion and fermion-dimer interacting via a zero-range attractive potential in one dimension, and fermion-fermion interacting via an attractive Gaussian potential in three dimensions. The results presented here provide a guide for improving the adiabatic projection method and for reducing the computational costs of large-scale calculations of \emph{ab initio} nuclear scattering and reactions using Monte Carlo methods.