Interatomic collisions in two-dimensional and quasi-two-dimensional confinements with spin-orbit coupling

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. We further derive the effective 2D contact potential for the interaction between ultracold atoms in the quasi-2D confinement. This effective potential can be used in the research on many-body physics of quasi-2D ultracold Fermi gases with Rashba SOC.

Figure 1

Variation of function ${\left|F\right|}^2$ with 2D scattering energy $\varepsilon \left(c\right)$. Here, $F$ is defined in Eq. (26) and $d$ is the parameter in Eq. (13). We show results in cases with zero center-of-mass momentum $q=0$ and SOC intensity $\xi =0$ [solid (red) line], $1/d$ [dashed-dotted (blue) line], $2/d$ [dashed (black) line], and $5/d$ [open (green) circles].

Figure 2

The variation of $|F_{\mathrm{eff}}{|}^2$ as a function of the quasi-2D scattering energy $\varepsilon \left(c\right)$. Here, $F_{\mathrm{eff}}$ is defined in Eq. (39) and $l_0=1/\sqrt{\omega }$. We show the results with zero center-of-mass momentum $q=0$, SOC intensities (a) $\xi =0$, (b) $\xi =0.3/l_0$, (c) $\xi =1/l_0$, and (d) $\xi =3/l_0$, and 3D scattering lengths $a=\infty$ [solid (red) line], $l_0$ [dashed (black) line], and $-l_0$ [dashed-dotted (blue) line].

Figure 3

Variation of $|F_{\mathrm{eff}}{|}^2$ with 3D scattering length. The function $F_{\mathrm{eff}}$ is defined in Eq. (39) and $l_0=1/\sqrt{\omega }$. We show the results with zero center-of-mass momentum $q=0$, SOC intensities (a) $\xi =0$, (b) $\xi =0.3/l_0$, (c) $\xi =1/l_0$, and (d) $\xi =3/l_0$, and the quasi-2D scattering energy $\varepsilon \left(c\right)=-{\xi }^2/4+0.006\omega$ [solid (red) line], $-{\xi }^2/4+0.02\omega$ [dashed (black) line], and $-{\xi }^2/4+0.2\omega$ [dashed-dotted (blue) line].

Figure 4

The binding energy $E_{\mathrm{binding}}={\varepsilon }_{\mathrm{thre}}\left(q\right)-E_b$ of the quasi-2D two-atom bound state as a function of the 3D scattering length. In this plot, we set the center-of-mass momentum $q=0$ and the SOC intensities $\xi =0$ [(black) dots], $0.5/l_0$ [dashed (blue) line], $1/l_0$ [dashed-dotted (green) line], and $3/l_0$ [solid (red) line].

Figure 5

(a) Variation of the effective mass $m_{\mathrm{eff}}$ of the two-atom bound state in the quasi-2D system as a function of the 3D scattering length. The SOC intensities used in this plot are $\xi =1/l_0$ [dashed (black) line], $2/l_0$ [dashed-dotted (blue) line], and $5/l_0$ [solid (red) line]. (b) Variation of $m_{\mathrm{eff}}$ as a function of the SOC intensity $\xi$. The 3D scattering lengths used in this plot are $a=-l_0$ [dashed (black) line], $a=\infty$ [dashed-dotted (blue) line], and $a=l_0$ [solid (red) line].