Solution of a two-dimensional time-dependent Schrödinger equation describing two interacting atoms in an optical trap

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Abstract

We introduce a numerical method to solve the two-dimensional time-dependent Schrödinger equation, which characterizes a system of two atoms with a finite-range interaction potential confined within a harmonic oscillator trap. We choose a Gaussian-shaped potential for the interaction potential. Such a system has been previously studied analytically, except that a zero-range interaction potential was used instead. We observe a strong agreement between the results for the two types of interactions. Also, we investigate the one-dimensional time-dependent Schrödinger equation for the relative motion and compute the ground state energy level as a function of the coupling strength.

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1. Introduction The physics of ultracold atoms has become one of the most intriguing fields of research. Due to the ability to precisely control the interaction between particles, the dimensions of the system, and its quantum state, researchers can explore a diverse range of phenomena with wide-ranging applications [1]. It is now even possible to experimentally realize systems with only a few atoms, or even just two atoms [2, 3]. This opens up a pathway for experimental verification of few-body models, highlighting the significant importance of solving the Schrödinger equation that describes such systems [4-12]. In this study, we present the solution of the two-dimensional time-dependent Schrödinger equation that describes two interacting atoms subject to the external harmonic oscillator confinement. We assume the bosonic symmetry for the particles. We choose the short-range Gaussian interaction potential to describe the interaction between the particles. Similar system, only for the zero-range interaction potential, or pseudopotential, has been studied previously [13-15]. It is a common practice to represent the interaction potential in terms of the coupling strength or scattering length. In this regard, we adjust the parameters of the Gaussian interaction potential to match the ground state energy level of the pseudopotential. This allows us to effectively analyze the system in the context of coupling strength. Because of the harmonic trap potential, the system can be separated into relative and center-ofmass motions. First, we compute the ground state energy for the relative motion as a function of the coupling strength, Gaussian interaction depth and the correlation between these two parameters. Next, we compare the evolution of the wave function with the solutions derived from the pseudopotential, both for the relative motion and when including the center-of-mass motion. Our comparison reveals a strong agreement with the zero-range interaction potential in both of these cases. The presented method has been employed for anharmonic trap potentials and various scenarios in the works [16-20]. In our present study, however, we illustrate that fitting the energy for the short-range Gaussian interaction with that of the zero-range potential yields an almost exact match of both wave functions within the designated range of the Gaussian interaction. Notably, in the vicinity of the origin, where the range is less than that of the Gaussian interaction, discernible distinctions between the two interaction types become evident. This contrast is even more pronounced in the two-dimensional case within the same region. The remainder of this paper is organized as follows. Section 2 describes the Hamiltonian of two atoms in the harmonic oscillator potential. The interaction between the particles is modeled using a Gaussian-shaped function with a fixed interaction width. We also consider the system with zerorange interaction, for which the analytical solution is known. Section 3 presents the ground state energy results as a function of the interaction parameters. We then proceed to study the dynamics of the wave function for both relative motion and the total Hamiltonian. Section 4 provides a summary of the results obtained. 2. Model Hamiltonian and the method 2.1. Gaussian interaction The Hamiltonian describing two interacting atoms in the harmonic trap reads: ℏ2
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About the authors

I. S. Ishmukhamedov

Institute of Nuclear Physics; Al-Farabi Kazakh National University

Author for correspondence.
Email: i.ishmukhamedov@mail.ru
ORCID iD: 0000-0002-7903-3432

Candidate of Physical and Mathematical Sciences

Almaty, 050032, Kazakhstan; Almaty, 050040, Kazakhstann

A. S. Ishmukhamedov

Institute of Nuclear Physics; Al-Farabi Kazakh National University

Email: altaymedoed@gmail.com
ORCID iD: 0000-0001-5248-3022

Researcher

Almaty, 050032, Kazakhstan; Almaty, 050040, Kazakhstann

Zh. E. Jalankuzov

Institute of Nuclear Physics; Al-Farabi Kazakh National University

Email: jalankuzov.zhanibek@gmail.com
ORCID iD: 0009-0003-1962-8834

Researcher

Almaty, 050032, Kazakhstan; Almaty, 050040, Kazakhstann

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Copyright (c) 2024 Ishmukhamedov I.S., Ishmukhamedov A.S., Jalankuzov Z.E.

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