Exploring exchange mechanisms with a cold-atom gas

Fermionic atoms trapped in a double-well potential are an ideal setting to study fundamental exchange mechanisms. We use exact diagonalization and complementary analytic calculations to demonstrate that two trapped fermions deliver a minimal model of the direct-exchange mechanism. This is an ideal quantum simulator of the Heisenberg antiferromagnet, exposes the competition between covalent and ionic bonding, and can create, manipulate, and detect quantum entanglement. Three trapped atoms form a faithful simulator of the double-exchange mechanism that is the fundamental building block behind many Heisenberg ferromagnets.

Figure 1

(a) Exchange energy from exact diagonalization (black lines), with the open channel (red line), the lowest $S=1$ state (blue line), and perturbation theory (green dashed line). (b) Bottom: Normalized exchange energy of the $S=0$ level as a function of relative well depth $\epsilon$ from exact diagonalization (blue crosses) and from the Hubbard Hamiltonian (red line). Top: The density profile (green) in the trap (magenta) at three values of the tilt and interaction strength. (c) Schematic of the experimental protocol to study quantum entanglement.

Figure 2

The phase diagrams predicted by exact diagonalization (left) and perturbation theory (right) at three different trap ratios. The phase corresponding to the each color is identified in each diagram, with “L” indicating a longitudinal phase and “T” a transverse phase. The arrows show the critical interaction strength predicted by the Hubbard model.