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arXiv:2403.02349 · Physical Review A

Two ultracold atoms in a quasi-two-dimensional box confinement

Fan Yang, Ruijie Du, Ran Qi, Peng Zhang

Published Sep 16, 2024• 2 citations
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Abstract

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 ($z$) 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.

DOI
10.1103/physreva.110.033318

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We investigate the low-energy scattering and bound states of two two-component fermionic atoms in pure two-dimensional (2D) and quasi-2D confinements with Rashba spin-orbit coupling (SOC). We find that the SOC qualitatively changes the behavior of the 2D scattering amplitude in the low-energy limit. For quasi-2D systems we obtain the analytic expression for the effective-2D scattering amplitude and the algebraic equations for the two-atom bound state energy. Based on these results, we further derive the effective 2D interaction potential between two ultracold atoms in the quasi-2D confinement with Rashba SOC. These results are crucial for the control of the 2D effective physics in quasi-2D geometry via the confinement intensity and the atomic three-dimensional scattering length.
In this undergraduate thesis, we discuss the two-body quantum mechanical problem of two ultra- cold bosonic atoms at low energy limit, interacting through a square potential with and without an external confinement. The two-body s-wave scattering length is introduced in terms of the radial wavefunction. In free space with no confinement, the interaction-scattering length relation is calculated and compared in 3- and 1-dimensions; we show the recurring resonance regions where the scattering length diverges and the existence of dimer bound state with positively large scattering length. Lastly, in the case of a 3-dimensional isotropic harmonic confinement, again under the ultracold limit, we obtain the universal (s-state) energy spectrum of the two atoms in terms of the scattering length.
We solve the three-body problem of a quasi-one-dimensional ultracold Fermi gas with parabolic confinement length a (perpendicular) and 3D scattering length a. On the two-body level, there is a Feshbach-type resonance at a (perpendicular)/a approximately 1.46, and a dimer state for arbitrary a (perpendicular)/a. The three-body problem is shown to be universal, and described by the atom-dimer scattering length a(ad) and a range parameter b(ad). In the dimer limit a (perpendicular)/a>>1, we find a repulsive zero-range atom-dimer interaction. For a (perpendicular)/a<<-1, however, the potential has long range, with a(ad)>0 and b(ad)>>a(ad). There is no trimer state, and despite a(ad)=0 at a( perpendicular)/a approximately 2.6, there is no resonance enhancement of the interaction.
We study the three-body problem for both fermionic and bosonic cold-atom gases in a parabolic transverse trap of length scale ${a}_{\ensuremath{\perp}}$. For this quasi-one-dimensional (quasi-1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length aaa and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths aad{a}_{\mathit{ad}}aad​ and bad{b}_{\mathit{ad}}bad​. In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find bad=0{b}_{\mathit{ad}}=0bad​=0 and aad{a}_{\mathit{ad}}aad​ is linked to the 3D atom-dimer scattering length. In the weakly bound ``BCS limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal: aad{a}_{\mathit{ad}}aad​ and bad{b}_{\mathit{ad}}bad​ depend both on ${a}_{\ensuremath{\perp}}∕a$ and on a parameter R∗{R}^{*}R∗ related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.