Coupled-channel pseudopotential description of the Feshbach resonance in two dimensions

We derive pseudopotentials that describe the scattering between two particles in two dimensions for any partial wave $m$, whose scattering strength is parametrized in terms of the phase shift ${\delta }_m$. Using our $m=0$ pseudopotential, we develop a coupled-channel model with two-dimensional (2D) zero-range interactions, which describes the two-body physics across a Feshbach resonance. Our model predicts the scattering length, the binding energy, and the “closed channel molecular fraction” of two particles; these observables can be measured in experiments on ultracold quasi-2D atomic Bose and Fermi gases with present-day technology.

Figure 1

(Color online) $E_{1\chi }$ as a function of $a_1(k=0)$ $(\omega =2\pi \nu )$. The asterisks show $E_{1\chi }$, Eq. (13), for the energy-dependent PP $V_1^{ps}\left(\rho \right)$ and the solid lines show $E_{1\chi }$ for $V_{SW}$ with ${\rho }_0=0.01a_{ho}$. For comparison, horizontal dotted lines indicate the energy levels for $a_1=0$.

Figure 2

(Color online) Molecular fractions $P_0$ (solid line) and $P_1$ (dotted lines), Eq. (19), for $b_1=0.5a_{ho}$, $b_2=0.05a_{ho}$, and different coupling constants $\beta$ as a function of $ϵ$. Inset: 2D scattering length $a_0^{CC}(E,ϵ)$ as a function of $ϵ$ for $E=0$, $b_1=0.5a_{ho}$, $b_2=0.05a_{ho}$, and $\beta =0.1$.

Figure 3

(Color online) Eigenenergies $E_{0n}$, Eq. (18), for $b_1=0.5a_{ho}$, $b_2=0.05a_{ho}$, and three different values of $\beta$, i.e., $\beta =0.03$ (solid lines), $0.1$ (dotted lines), and $0.3$ (dashed lines), as a function of $ϵ$.