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Weinvestigate an extension of the quantum Ising (QI) model in one spatial dimension including long-range 1/rα interactions in its statics and dynamicswith possible applications fromheteronuclear polar molecules in optical lattices to trapped ions described by two-state spin systems. We introduce the statics of the system via both numerical techniques with finite size and infinite size matrix product states (MPSs) and a theoretical approaches using a truncated Jordan-Wigner transformation for the ferromagnetic and antiferromagnetic case and show that finite size effects have a crucial role shifting the quantum critical point of the external field by fifteen percent between thirty-two and around fivehundred spins. We numerically study the Kibble-Zurek hypothesis in the long-range QI model with MPSs. A linear quench of the external field through the quantum critical point yields a power-law scaling of the defect density as a function of the total quench time. For example, the increase of the defect density is slower for longer-range models and the critical exponent changes by twenty-five percent. Our study emphasizes the importance of such long-range interactions in statics and dynamics that could point to similar phenomena in a different setup of dynamical systems or for other models. [ABSTRACT FROM AUTHOR]

Quantum simulation of spin models can provide insight into problems that are difficult or impossible to study with classical computers.Trapped ions are an established platform for quantum simulation, but only systems with fewer than 20 ions have demonstrated quantum correlations.We studied quantum spin dynamics arising from an engineered, homogeneous Ising interaction in a two-dimensional array of 9Be+ ions in a Penning trap.We verified entanglement in spin-squeezed states of up to 219 ions, directly observing 4.0 ± 0.9 decibels of spectroscopic enhancement, and observed states with non-Gaussian statistics consistent with oversqueezed states.The good agreement with ab initio theory that includes interactions and decoherence lays the groundwork for simulations of the transverse-field Ising model with variable-range interactions, which are generally intractable with classical methods. [ABSTRACT FROM AUTHOR]