One-dimensional tunneling of the two-body bound state

TL;DR

This work analyzes the one-dimensional tunneling of a two-body bound state (a dimer) formed by non-identical atoms, where one atom also interacts with a central delta-like barrier. Using a momentum-space formulation of the Schrödinger equation with a delta-interaction between the atoms and a delta barrier at the origin, the authors derive coupled integral equations for the dimer amplitudes and characterize bound and scattering states via $t$-matrices and on-shell amplitudes. They compute symmetric and antisymmetric scattering amplitudes $f^{\pm}_{P}(\\mathcal{E})$, from which the reflection and transmission coefficients $R_P$ and $T_P$ are obtained, revealing a general enhancement of dimer reflection compared to a single atom and a narrow parameter region with enhanced dimer transmission for attractive barriers. The results have implications for resonant tunneling and escape dynamics in quasi-one-dimensional ultracold-atom systems and motivate future work on extensions to higher dimensions and more complex delta-potentials.

Abstract

We consider bound and scattering states of the one-dimensional dimer formed by two coupled non-identical atoms when one of them also interacts with the zero-range potential located at the origin. By calculating the dimer localized and scattering wave functions, we identify properties of the system without the two-body bound-state collapse. In general, we predict an enhancement of the dimer reflection compared to a single atom, except for a narrow region on the attractive side of the external potential.

Figures (4)

• Figure 1: Symmetric non-normalized wave functions $c^{+}_p$ of the dimer bound states in external potential at different couplings. The insert shows (in logarithmic scale) the appropriate energy $\epsilon_+$ in units of $\epsilon=-\frac{1}{2ma^2}$.
• Figure 2: Symmetric (left) and antisymmetric (right) on-shell scattering amplitudes for negative $a_1$s (repulsive potential) from weak (top) to strong (bottom) couplings $a_1/a=-10; \ -1; \ -0.1$.
• Figure 3: Even (left) and odd (right) on-shell scattering amplitudes for attractive interaction (positive $a_1$s) $a_1/a=1$ (top) and $a_1/a=10$ (bottom). Shaded areas represent the regions ($Pa\ge \sqrt{2}$ for $a_1/a=1$, and $Pa\ge 1.995$ not shown for $a_1/a=10$) of inapplicability of the considered scattering solution (\ref{['c_scatt']}).
• Figure 4: The dimer reflection probabilities as a function of the potential barrier (well) height (depth) for two center-of-mass momenta $Pa=0.1$ (top) and $Pa=1.0$ (bottom). Small $a_1/a$s correspond to strong repulsion (negative ratios) and strong attraction (negative ratios). In the shaded areas, the dimer can dissociate into two particles, which is not accounted for in the present analysis.