Engineered spin-spin interactions on a 2D array of trapped ions

We work with laser cooled $^9\mbox{Be}^+$ ions confined in a Penning trap to simulate quantum magnetic interactions. The valence electron of each ion behaves as an ideal spin-$1/2$ particle. We recently demonstrated a uniform anti-ferromagnetic Ising interaction on a naturally occurring two-dimensional (2D) triangular crystal of $100 < N < 350$ ions. The Ising interaction is generated by a spin-dependent optical dipole force (ODF). For spins separated by distance $d$, we show that the range can be tuned according to $(d/d0)^{-a}$, for $0