Вестник Российского университета дружбы народов. Серия: Математика, информатика, физика

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

I. S. Ishmukhamedov, Altay Ishmukhamedov, Zh. E. Jalankuzov

Published Oct 15, 2024• 0 citations

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.

DOI: 10.22363/2658-4670-2024-32-2-172-180

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Two-body atomic system in a one-dimensional anharmonic trap: The energy spectrum

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We numerically investigate the following two-body stationary Schrödinger equation (SE): $$\left\{ { - \frac{{{{\rlap{--} h}^2}}}{{2m}}\frac{\partial }{{x_1^2}} - \frac{{{{\rlap{--} h}^2}}}{{2m}}\frac{\partial }{{x_2^2}} + V\left( {{x_2}} \right) + g\partial \left( {{x_1} - {x_2}} \right)} \right\}x\Psi \left( {{x_1},{x_2}} \right) = E\Psi \left( {{x_1},{x_2}} \right),$$ where $${x_1},{x_2} \in \mathbb{R},V\left( {{x_1}} \right) = {V_0}{\sin ^2}\left( {{k_x}{x_i}} \right)$$ is the potential describing the interaction of atoms with a trap and gδ(x 1–x 2) is the interatomic potential. Previously, a similar problem has been solved analytically for the harmonic interaction $${V_h}\left( {{x_i}} \right) = \frac{1}{2}m\omega x_i^2$$ with the trap, which leads to the separation of coordinates $$y = \frac{{{x_1} + {x_2}}}{{\sqrt 2 }},{\kern 1pt} x = \frac{{{x_1} - {x_2}}}{{\sqrt 2 }}$$ for the center-of-mass and relative motion. The anharmonicity of the trap couples these motions and, there-fore, the problem becomes significantly more complicated. In previous works, the anharmonicity V a = V–V h of the trap has been taken into account in the framework of perturbation theory. In this work, the energy level shifts of a two-body atomic system are calculated beyond the perturbation theory for different magnitudes of parameter g of the interatomic interaction. The results are compared to those computed using the perturbation theory.

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Вестник Российского университета дружбы народов. Серия: Математика, информатика, физика

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

I. S. Ishmukhamedov, Altay Ishmukhamedov, Zh. E. Jalankuzov

Published Oct 15, 2024• 0 citations

Publisher

Abstract

DOI: 10.22363/2658-4670-2024-32-2-172-180

Related papers

Description of ultracold atoms in a one-dimensional geometry of a harmonic trap with a realistic interaction

I. S. Ishmukhamedov, D. S. Valiolda, S. A. Zhaugasheva

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We numerically investigate the following two-body stationary Schrödinger equation (SE):

\left\{ { - \frac{{{{\rlap{--} h}^2}}}{{2m}}\frac{\partial }{{x_1^2}} - \frac{{{{\rlap{--} h}^2}}}{{2m}}\frac{\partial }{{x_2^2}} + V\left( {{x_2}} \right) + g\partial \left( {{x_1} - {x_2}} \right)} \right\}x\Psi \left( {{x_1},{x_2}} \right) = E\Psi \left( {{x_1},{x_2}} \right),

where

{x_1},{x_2} \in \mathbb{R},V\left( {{x_1}} \right) = {V_0}{\sin ^2}\left( {{k_x}{x_i}} \right)

is the potential describing the interaction of atoms with a trap and gδ(x 1–x 2) is the interatomic potential. Previously, a similar problem has been solved analytically for the harmonic interaction

{V_h}\left( {{x_i}} \right) = \frac{1}{2}m\omega x_i^2

with the trap, which leads to the separation of coordinates

y = \frac{{{x_1} + {x_2}}}{{\sqrt 2 }},{\kern 1pt} x = \frac{{{x_1} - {x_2}}}{{\sqrt 2 }}

for the center-of-mass and relative motion. The anharmonicity of the trap couples these motions and, there-fore, the problem becomes significantly more complicated. In previous works, the anharmonicity V a = V–V h of the trap has been taken into account in the framework of perturbation theory. In this work, the energy level shifts of a two-body atomic system are calculated beyond the perturbation theory for different magnitudes of parameter g of the interatomic interaction. The results are compared to those computed using the perturbation theory.