Antiferromagnetic Heisenberg Spin Chain of a Few Cold Atoms in a One-Dimensional Trap

We report on the deterministic preparation of antiferromagnetic Heisenberg spin chains consisting of up to four fermionic atoms in a one-dimensional trap. These chains are stabilized by strong repulsive interactions between the two spin components without the need for an external periodic potential. We independently characterize the spin configuration of the chains by measuring the spin orientation of the outermost particle in the trap and by projecting the spatial wave function of one spin component on single-particle trap levels. Our results are in good agreement with a spin-chain model for fermionized particles and with numerically exact diagonalizations of the full few-fermion system.

Figure 1

Heisenberg spin chain of three fermions. (a) Sketch of two spin-up and one spin-down atom with diverging 1D coupling constant ($g_{1\mathrm{D}}=\pm \infty$) in a harmonic trap. If the relative spatial wave function of two distinguishable fermions is symmetric, the strong interactions induce a cusp in the relative wave function of the two particles (left-hand side). This causes them to separate like identical fermions (right-hand side). In this fermionization limit the system forms a Wigner-crystal-like state with fixed ordering of the particles. (b) Single-particle contributions to the total (gray), the spin-up (green), and the spin-down density (blue) of two spin-up and one spin-down atom in the fermionization regime in a harmonic trap. Like in a Wigner crystal, the total densities of the ferromagnetic (left), the intermediate (middle), and the antiferromagnetic state (right) are identical, while their spin densities differ and are determined by a Heisenberg spin-chain Hamiltonian. (c) Densities of three particles before (left) and after (right) the tunneling of one atom with energy $E$ out of a tilted trap. At fermionization, only the rightmost particle can leave the trap in the tunneling process.

Figure 2

Energies in spin-chain regime. Eigenenergies of two (green) and three (red) strongly interacting spin-$1/2$ fermions in a 1D harmonic trap as a function of the interaction strength. In the Tonks regime, the antiferromagnetic states are the ground states of each multiplet, while the ferromagnetic states have the highest energies. In the super-Tonks regime, the ordering of the energy levels is inverted. Close to the confinement-induced resonance (CIR), the energy shifts are linear in $-1/g_{1\mathrm{D}}$ and can be determined by a Heisenberg spin-chain Hamiltonian. The system is initially prepared in the noninteracting ground state of the three-particle system at $-1/g_{1\mathrm{D}}=-\infty$, which evolves, for increasing $-1/g_{1\mathrm{D}}$, into the antiferromagnetic state around the CIR (red solid line). During a ramp across the CIR, the system stays in the antiferromagnetic state, since all eigenstates of the system are decoupled. The blue arrows indicate the predominant channels for the tunneling of one atom below (left) and above (right) the fermionization regime.

Figure 3

Probing the spin distribution. Tunneling probabilities of the spin-down atom in a (2, 1) system (a) and a (3, 1) system (b) and tunneling probability of both spin-down atoms in a (2, 2) system (c) as a function of the interaction strength. The red lines are the solutions of a tunneling model for the antiferromagnetic (solid), the ferromagnetic (dashed), and the intermediate state (dotted). The gray points in (a) indicate a narrow resonance between the antiferromagnetic and the intermediate state of the (2, 1) system close to $-1/g_{1\mathrm{D}}=0$. Error bars denote the $1\sigma$ statistical uncertainties.

Figure 4

Probing the spatial wave function. Occupation-number distribution of the spin-down atom on single-particle trap levels for an initial (a) (2, 1) and (b) (3, 1) system. The red and gray symbols show theoretical predictions for the antiferromagnetic state (red circles), the ferromagnetic state (gray squares), and intermediate states (gray diamonds). Both measurements (blue points) were made in the super-Tonks regime [$-1/g_{1\mathrm{D}}=0.586\pm 0.014$ for the (2, 1) system and $-1/g_{1\mathrm{D}}=0.536\pm 0.013$ for the (3, 1) system] to show that both systems stay in the respective antiferromagnetic states throughout the regime of fermionization. Error bars denote the $1\sigma$ statistical uncertainty.