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 σ_x^i σ_x^j + J_ij^y σ_y^i σ_y^j) Hamiltonians on trapped ion spins with independent control over the J_ij^x and J_ij^y terms. The Ising-type interactions σ_x^i σ_x^j and σ_y^i σ_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 ^171Yb^+ 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.