Tunneling theory for tunable open quantum systems of ultracold atoms in one-dimensional traps

The creation of tunable open quantum systems is becoming feasible in current experiments with ultracold atoms in low-dimensional traps. In particular, the high degree of experimental control over these systems allows detailed studies of tunneling dynamics, e.g., as a function of the trapping geometry and the interparticle interaction strength. In order to address this exciting opportunity we present a theoretical framework for two-body tunneling based on the rigged Hilbert space formulation. In this approach, bound, resonant, and scattering states are included on an equal footing and we argue that the coupling of all these components is vital for a correct description of the relevant threshold phenomena. In particular, we study the tunneling mechanism for two-body systems in one-dimensional traps and different interaction regimes. We find a strong dominance of sequential tunneling of single particles for repulsive and weakly attractive systems, while there is a signature of correlated pair tunneling in the calculated many-particle flux for strongly attractive interparticle interaction.

Figure 1

(a) Trap potential, indicating the position of SP and two-body resonance states. (b) Complex-momentum contour and Berggren basis states, highlighting the position of the SP resonance pole.

Figure 2

Two-fermion resonance state as a function of the interaction strength $g$ for $c_B=1$. (a) Interaction energy (5) compared with the corresponding energy obtained using the pole approximation. (b) Tunneling rates obtained from the imaginary part of the resonance energy (from the full calculation and the pole approximation, respectively) compared with the rate obtained from the flux calculation (3).

Figure 3

(a), (c), and (e) Density contour plots and (b), (d), and (f) logarithm of the outward particle flux for repulsive ($g=+g_0$), slightly attractive ($g=-0.1g_0$), and strongly attractive ($g=-g_0$) interactions (with $g_0=100\text{nK}{k}_B\mu \text{m}$) from top to bottom, respectively.