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arXiv:2202.08092 · Acta Physica Polonica A

Factorization with a Logarithmic Energy Spectrum of a Central Potential

F. Gleisberg, Wolfgang P. Schleich

Published Jun 1, 2023• 3 citations
arXivHTMLPDF

Abstract

We propose a method to factor numbers based on two interacting bosonic atoms in a central potential, where the single-particle spectrum depends logarithmically on the radial quantum numbers of states corresponding to zero angular momentum. Bosons initially prepared in the ground state are excited by a sinusoidally time-dependent interaction into a state characterized by quantum numbers, which represent the factors of the number encoded in the frequency of the perturbation. We also discuss the complete single-particle spectrum as well as the limitations of our method caused by decoherence.

DOI
10.12693/aphyspola.143.s112

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We propose a method to factor numbers based on two interacting bosonic atoms in a central potential, where the single-particle spectrum depends logarithmically on the radial quantum numbers of states corresponding to zero angular momentum. Bosons initially prepared in the ground state are excited by a sinusoidally time-dependent interaction into a state characterized by quantum numbers, which represent the factors of the number encoded in the frequency of the perturbation. We also discuss the complete single-particle spectrum as well as the limitations of our method caused by decoherence.
Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
This chapter presents a detailed examination of a number of relatively simple problems pertaining to the scattering and bound states of two structureless particles interacting through a central field. It describes the spectrum of the angular momentum operators by a method that does not refer at all to the structure of the system. The spherical harmonics are the eigenfunctions appropriate to a single particle that can be described by the three classical degrees of freedom, or for a pair of such particles in their center-of-mass frame. The formulation of collision theory that is based on the one-body Schrodinger equation with a Hermitian interaction. The chapter introduces and discusses the observables pertaining to the two-body problem and explores the eigenvalues and eigenfunctions of these observables. Since the degeneracies of the energy spectrum are a consequence of the symmetries of the Hamiltonian, they can be determined experimentally by destroying this symmetry.
We propose a method to factor numbers based on the quantum dynamics of two interacting bosonic atoms where the single-particle energy spectrum depends logarithmically on the quantum number. We show that two atoms initially prepared in the ground state are preferentially excited by a time-dependent interaction into a two-particle energy state characterized by the factors. Hence, a measurement of the energy of one of the two atoms yields the factors. The number to be factored is encoded in the frequency of a sinusoidally modulated interaction. We also discuss the influence of off-resonant transitions and the limitation of the number to be factored imposed by experimental conditions.