Programmable XY-type couplings through parallel spin-dependent forces on the same trapped ion motional modes

We propose and experimentally demonstrate an analog scheme for generating XY-type ($J_{ij}^x{\sigma }_x^i{\sigma }_x^j$ + $J_{ij}^y{\sigma }_y^i{\sigma }_y^j$) Hamiltonians on trapped ion spins with independent control over the $J_{ij}^x$ and $J_{ij}^y$ terms. The Ising-type interactions ${\sigma }_x^i{\sigma }_x^j$ and ${\sigma }_y^i{\sigma }_y^j$ are simultaneously generated by employing two spin-dependent forces operating in parallel on the same set of normal modes. We analytically calculate the region of validity of this scheme, and provide numerical and experimental validation with ${}^{171}{\mathrm{Y}\mathrm{b}}^{+}$ ions. This scheme inherits the programmability and scalability of the Ising-type interactions with trapped ions that have been explored in numerous quantum simulation experiments. Our approach extends the capabilities of existing trapped ion quantum simulators to access a large class of spin Hamiltonians relevant for exploring exotic quantum phases such as superfluidity and spin liquids.