Chinese Physics Letters

Exact Treatment for Two Ultracold Ions in a Two-Dimensional Penning Trap

Mang Feng, Xi-ming Fang, Lei Shi, Min Yan, Xiwen Zhu

Published Apr 1, 1998• 0 citations

Abstract

The Schrödinger equation of two ultracold ions confined in xy plane of the Penning trap is exactly solved and the analytical form of the wave function of the system is obtained. A numerical calculation for a continued fraction presents the relation among the eigenenergies, the Coulomb interaction and the azimuthal number.

DOI: 10.1088/0256-307x/15/4/009

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Chinese Physics Letters

Exact Treatment for Two Ultracold Ions in a Two-Dimensional Penning Trap

Mang Feng, Xi-ming Fang, Lei Shi, Min Yan, Xiwen Zhu

Published Apr 1, 1998• 0 citations

Publisher

Abstract

DOI: 10.1088/0256-307x/15/4/009

Related papers

Analytical solutions for the dynamics of two trapped interacting ultracold atoms

Zbigniew Idziaszek, Tommaso Calarco

We discuss exact solutions of the Schr\"odinger equation for the system of two ultracold atoms confined in an axially symmetric harmonic potential. We investigate different geometries of the trapping potential, in particular we study the properties of eigenenergies and eigenfunctions for quasi-one-dimensional and quasi-two-dimensional traps. We show that the quasi-one-dimensional and the quasi-two-dimensional regimes for two atoms can be already realized in the traps with moderately large (or small) ratios of the trapping frequencies in the axial and the transverse directions. Finally, we apply our theory to Feshbach resonances for trapped atoms. Introducing in our description an energy-dependent scattering length we calculate analytically the eigenenergies for two trapped atoms in the presence of a Feshbach resonance.We discuss exact solutions of the Schr\"odinger equation for the system of two ultracold atoms confined in an axially symmetric harmonic potential. We investigate different geometries of the trapping potential, in particular we study the properties of eigenenergies and eigenfunctions for quasi-one-dimensional and quasi-two-dimensional traps. We show that the quasi-one-dimensional and the quasi-two-dimensional regimes for two atoms can be already realized in the traps with moderately large (or small) ratios of the trapping frequencies in the axial and the transverse directions. Finally, we apply our theory to Feshbach resonances for trapped atoms. Introducing in our description an energy-dependent scattering length we calculate analytically the eigenenergies for two trapped atoms in the presence of a Feshbach resonance.

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We study the system of two ultracold atoms in a three-dimensional (3D) or two-dimensional (2D) completely anisotropic harmonic trap. We derive the transcendental equation ${J}_{3\mathrm{D}}(E)=1/{a}_{3\mathrm{D}}$ ( ${J}_{2\mathrm{D}}(E)=ln{a}_{2\mathrm{D}}$ ) for the eigenenergy $E$ of this system in the 3D (2D) case, with ${a}_{3\mathrm{D}}$ and ${a}_{2\mathrm{D}}$ being the corresponding $s$ -wave scattering lengths, and provide the analytical expressions of the functions ${J}_{3\mathrm{D}}(E)$ and ${J}_{2\mathrm{D}}(E)$ . In previous research, this type of equation was obtained for spherically or axially symmetric harmonic traps [T. Busch et al., Found. Phys. 28, 549 (1998); Z. Idziaszek and T. Calarco, Phys. Rev. A 74, 022712 (2006)]. However, for our cases with a completely anisotropic trap, only the equation for the ground-state energy of some cases has been derived [J.-J. Liang and C. Zhang, Phys. Scr. 77, 025302 (2008)]. Our results in this work are applicable for arbitrary eigenenergy of this system and can be used for the study of dynamics and thermal dynamics of interacting ultracold atoms in this trap, e.g., the calculation of the second virial coefficient or the evolution of two-body wave functions. In addition, our approach for the derivation of the above equations can also be used for other two-body problems of ultracold atoms.We study the system of two ultracold atoms in a three-dimensional (3D) or two-dimensional (2D) completely anisotropic harmonic trap. We derive the transcendental equation ${J}_{3\mathrm{D}}(E)=1/{a}_{3\mathrm{D}}$ (${J}_{2\mathrm{D}}(E)=ln{a}_{2\mathrm{D}}$) for the eigenenergy $E$ of this system in the 3D (2D) case, with ${a}_{3\mathrm{D}}$ and ${a}_{2\mathrm{D}}$ being the corresponding $s$-wave scattering lengths, and provide the analytical expressions of the functions ${J}_{3\mathrm{D}}(E)$ and ${J}_{2\mathrm{D}}(E)$. In previous research, this type of equation was obtained for spherically or axially symmetric harmonic traps [T. Busch et al., Found. Phys. 28, 549 (1998); Z. Idziaszek and T. Calarco, Phys. Rev. A 74, 022712 (2006)]. However, for our cases with a completely anisotropic trap, only the equation for the ground-state energy of some cases has been derived [J.-J. Liang and C. Zhang, Phys. Scr. 77, 025302 (2008)]. Our results in this work are applicable for arbitrary eigenenergy of this system and can be used for the study of dynamics and thermal dynamics of interacting ultracold atoms in this trap, e.g., the calculation of the second virial coefficient or the evolution of two-body wave functions. In addition, our approach for the derivation of the above equations can also be used for other two-body problems of ultracold atoms.

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We study the system of two ultracold atoms in a three-dimensional (3D) or two-dimensional (2D) completely anisotropic harmonic trap. We derive the transcendental equation

{J}_{3\mathrm{D}}(E)=1/{a}_{3\mathrm{D}}

(

{J}_{2\mathrm{D}}(E)=ln{a}_{2\mathrm{D}}

) for the eigenenergy

E

of this system in the 3D (2D) case, with

{a}_{3\mathrm{D}}

and

{a}_{2\mathrm{D}}

being the corresponding

s

-wave scattering lengths, and provide the analytical expressions of the functions

{J}_{3\mathrm{D}}(E)

and

{J}_{2\mathrm{D}}(E)

. In previous research, this type of equation was obtained for spherically or axially symmetric harmonic traps [T. Busch et al., Found. Phys. 28, 549 (1998); Z. Idziaszek and T. Calarco, Phys. Rev. A 74, 022712 (2006)]. However, for our cases with a completely anisotropic trap, only the equation for the ground-state energy of some cases has been derived [J.-J. Liang and C. Zhang, Phys. Scr. 77, 025302 (2008)]. Our results in this work are applicable for arbitrary eigenenergy of this system and can be used for the study of dynamics and thermal dynamics of interacting ultracold atoms in this trap, e.g., the calculation of the second virial coefficient or the evolution of two-body wave functions. In addition, our approach for the derivation of the above equations can also be used for other two-body problems of ultracold atoms.