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arXiv:2002.02168 · Physical Review A

Analytical solution for the spectrum of two ultracold atoms in a completely anisotropic confinement

Yue Chen, Da-Wu Xiao, Ren Zhang, Peng Zhang

Published May 14, 2020• 8 citations
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Abstract

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.

DOI
10.1103/physreva.101.053624

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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 J3D(E)=1/a3D{J}_{3\mathrm{D}}(E)=1/{a}_{3\mathrm{D}}J3D​(E)=1/a3D​ (J2D(E)=lna2D{J}_{2\mathrm{D}}(E)=ln{a}_{2\mathrm{D}}J2D​(E)=lna2D​) for the eigenenergy EEE of this system in the 3D (2D) case, with a3D{a}_{3\mathrm{D}}a3D​ and a2D{a}_{2\mathrm{D}}a2D​ being the corresponding sss-wave scattering lengths, and provide the analytical expressions of the functions J3D(E){J}_{3\mathrm{D}}(E)J3D​(E) and J2D(E){J}_{2\mathrm{D}}(E)J2D​(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.
Abstract We calculate the energy spectrum of three identical fermionic ultracold atoms in two different internal states confined in a two-dimensional anisotropic harmonic trap. Using the solutions of the corresponding two-body problems obtained in our previous work (Chen et al 2020 Phys. Rev. A 101, 053624), we derive the explicit transcendental equation for the eigen-energies, from which the energy spectrum is derived. Our results can be used for the calculation of the 3rd Virial coefficients or the studies of few-body dynamics.
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 present highly accurate solutions of the Schr\"odinger equation for three fermions in two different spin states with zero-range sss-wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all sss-wave scattering lengths.
We solve the two-body problem of ultracold heteronuclear atoms in a quasi-two-dimensional (quasi-2D) geometry. The quasi-2D confinement is realized by a harmonic trap along the longitudinal (zzz-) direction, with different trap frequencies for the two atoms, as in many current experiments on ultracold heteronuclear gases. As a consequence, the longitudinal center-of-mass (CoM) motion is coupled to the relative motion, which significantly complicates the two-body problem. We solve this problem exactly and derive the 2D scattering length a2Da_{\rm 2D}a2D​, the 2D effective range parameter R2DR_{\rm 2D}R2D​, and the bound-state energies, as functions of the sss-wave scattering length and effective range of the two atoms in free three-dimensional (3D) space. We show that multiple 2D scattering resonances can be induced by the coupling between the longitudinal CoM and relative motion. Around these resonances, a2Da_{\rm 2D}a2D​ varies rapidly with the 3D scattering parameters, while R2DR_{\rm 2D}R2D​ is strongly enhanced. Since the effective pairwise interaction in quasi-2D ultracold gases is determined by i.e., the two-body scattering amplitudes and bound-state energies, our results can be used for manipulating the effective 2D interatomic interaction in quasi-2D ultracold heteronuclear gases by tuning the confinement frequencies and the 3D scattering parameters.
We investigate the scattering and two-body bound states of two ultracold atoms in a quasi-two-dimensional (quasi-2D) confinement, with the confinement potential being an infinitely deep square well (box potential) in the transverse (zzz) direction, and the motion of the atoms in the $x\text{\ensuremath{-}}y$ plane being free. Specifically, we calculate the effective 2D scattering length and 2D effective range parameter of the low-energy scattering, as well as the energy and the transverse-excited-mode probability of the bound states. Comparing these results with those obtained under a harmonic transverse confinement potential, which has the same characteristic length as the box potential, we find that in most of the cases the 2D effective range parameter for the box confinement is approximately 0.28 of the one for the harmonic confinement. Moreover, the transverse-excited-mode probability of the bound states for the box confinement is also much lower than the one for the harmonic confinement. These results suggest that the transverse excitation in the box confinement is notably weaker than the one in the harmonic confinement with the same characteristic length. Therefore, achieving quasi-2D ultracold gases well described by pure-2D effective models, particularly those with 2D contact interaction, is more feasible through box confinement. Our results are helpful for the quantum simulation of 2D many-body physics with ultracold atoms, e.g., the suppression of 2D effective range parameter may lead to an enhancement of quantum anomaly in two-dimensional Fermi gases. Additionally, our calculation method is applicable to the two-body problems of ultracold atoms in other types of quasi-2D confinements.