Laser-cooled and trapped atomic ions form an ideal standard for the simulation of interacting quantum spin models. Effective spins are represented by appropriate internal energy levels within each ion, and the spins can be measured with near-perfect efficiency using state-dependent fluorescence techniques. By applying optical fields that exert optical dipole forces on the ions, their Coulomb interaction can be modulated in ways that give rise to long-range and tunable spin-spin interactions that can be reconfigured by shaping the spectrum and pattern of the laser fields. Here we review the theoretical mapping of atomic ions to interacting spin systems, the experimental preparation of complex equilibrium states, and the study of dynamical processes of this many-body interacting quantum system. The use of such quantum simulators for studying spin models may inform our understanding of exotic quantum materials and shed light on interacting quantum systems that cannot be modeled with conventional computers.

1 aMonroe, C.1 aCampbell, W., C.1 aDuan, L., -M.1 aGong, Z., -X.1 aGorshkov, Alexey, V.1 aHess, P.1 aIslam, R.1 aKim, K.1 aPagano, G.1 aRicherme, P.1 aSenko, C.1 aYao, N., Y. uhttps://arxiv.org/abs/1912.0784501207nas a2200145 4500008004100000245008800041210006900129260001500198300001100213490000700224520075600231100001900987700001801006856003701024 2013 eng d00aPrethermalization and dynamical transition in an isolated trapped ion spin chain 0 aPrethermalization and dynamical transition in an isolated trappe c2013/11/26 a1130510 v153 a We propose an experimental scheme to observe prethermalization and dynamical transition in one-dimensional XY spin chain with long range interaction and inhomogeneous lattice spacing, which can be readily implemented with the recently developed trapped-ion quantum simulator. Local physical observables are found to relax to prethermal values at intermediate time scale, followed by complete relaxation to thermal values at much longer time. The physical origin of prethermalization is explained by spotting a non-trivial structure in lower half of the energy spectrum. The dynamical behavior of the system is shown to cross difference phases when the interaction range is continuously tuned, indicating the existence of dynamical phase transition. 1 aGong, Zhe-Xuan1 aDuan, L., -M. uhttp://arxiv.org/abs/1305.0985v101995nas a2200205 4500008004100000245005100041210005100092260001300143490000700156520142000163100002001583700001901603700002301622700002401645700001801669700002301687700001701710700002501727856003701752 2013 eng d00aQuantum Logic between Remote Quantum Registers0 aQuantum Logic between Remote Quantum Registers c2013/2/60 v873 a We analyze two approaches to quantum state transfer in solid-state spin systems. First, we consider unpolarized spin-chains and extend previous analysis to various experimentally relevant imperfections, including quenched disorder, dynamical decoherence, and uncompensated long range coupling. In finite-length chains, the interplay between disorder-induced localization and decoherence yields a natural optimal channel fidelity, which we calculate. Long-range dipolar couplings induce a finite intrinsic lifetime for the mediating eigenmode; extensive numerical simulations of dipolar chains of lengths up to L=12 show remarkably high fidelity despite these decay processes. We further consider the extension of the protocol to bosonic systems of coupled oscillators. Second, we introduce a quantum mirror based architecture for universal quantum computing which exploits all of the spins in the system as potential qubits. While this dramatically increases the number of qubits available, the composite operations required to manipulate "dark" spin qubits significantly raise the error threshold for robust operation. Finally, as an example, we demonstrate that eigenmode-mediated state transfer can enable robust long-range logic between spatially separated Nitrogen-Vacancy registers in diamond; numerical simulations confirm that high fidelity gates are achievable even in the presence of moderate disorder. 1 aYao, Norman, Y.1 aGong, Zhe-Xuan1 aLaumann, Chris, R.1 aBennett, Steven, D.1 aDuan, L., -M.1 aLukin, Mikhail, D.1 aJiang, Liang1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1206.0014v100674nas a2200193 4500008004100000245006100041210005900102260001500161520012900176100001700305700001900322700001800341700001700359700001600376700001600392700001800408700001700426856003700443 2012 eng d00aReply to Comment on "Space-Time Crystals of Trapped Ions0 aReply to Comment on SpaceTime Crystals of Trapped Ions c2012/10/153 a This is a reply to the comment from Patrick Bruno (arXiv:1211.4792) on our paper (Phys. Rev. Lett. 109, 163001 (2012)). 1 aLi, Tongcang1 aGong, Zhe-Xuan1 aYin, Zhang-qi1 aQuan, H., T.1 aYin, Xiaobo1 aZhang, Peng1 aDuan, L., -M.1 aZhang, Xiang uhttp://arxiv.org/abs/1212.6959v200744nas a2200121 4500008004100000245009800041210006900139260001500208520032500223100001900548700001800567856003700585 2011 eng d00aComment on "Foundation of Statistical Mechanics under Experimentally Realistic Conditions" 0 aComment on Foundation of Statistical Mechanics under Experimenta c2011/09/223 a Reimann [Phys. Rev. Lett. 101, 190403 (2008)] claimed that generic isolated macroscopic quantum system will equilibrate under experimentally realistic conditions by proving a theorem. We here show that the proof is invalid for most many-body systems and is unable to demonstrate equilibration in realistic experiment. 1 aGong, Zhe-Xuan1 aDuan, L., -M. uhttp://arxiv.org/abs/1109.4696v101276nas a2200157 4500008004100000245008200041210006900123260001500192300001100207490000700218520080100225100001901026700001801045700001801063856003701081 2011 eng d00aDynamics of Overhauser Field under nuclear spin diffusion in a quantum dot 0 aDynamics of Overhauser Field under nuclear spin diffusion in a q c2011/03/25 a0330360 v133 a The coherence of electron spin can be significantly enhanced by locking the Overhauser field from nuclear spins using the nuclear spin preparation. We propose a theoretical model to calculate the long time dynamics of the Overhauser field under intrinsic nuclear spin diffusion in a quantum dot. We obtain a simplified diffusion equation that can be numerically solved and show quantitatively how the Knight shift and the electron-mediated nuclear spin flip-flop affect the nuclear spin diffusion. The results explain several recent experimental observations, where the decay time of Overhauser field is measured under different configurations, including variation of the external magnetic field, the electron spin configuration in a double dot, and the initial nuclear spin polarization rate. 1 aGong, Zhe-Xuan1 aYin, Zhang-qi1 aDuan, L., -M. uhttp://arxiv.org/abs/0912.4322v101404nas a2200193 4500008004100000245006800041210006800109260001400177490000800191520083700199100002001036700001701056700002501073700001901098700001501117700001801132700002301150856003701173 2011 eng d00aRobust Quantum State Transfer in Random Unpolarized Spin Chains0 aRobust Quantum State Transfer in Random Unpolarized Spin Chains c2011/1/270 v1063 a We propose and analyze a new approach for quantum state transfer between remote spin qubits. Specifically, we demonstrate that coherent quantum coupling between remote qubits can be achieved via certain classes of random, unpolarized (infinite temperature) spin chains. Our method is robust to coupling strength disorder and does not require manipulation or control over individual spins. In principle, it can be used to attain perfect state transfer over arbitrarily long range via purely Hamiltonian evolution and may be particularly applicable in a solid-state quantum information processor. As an example, we demonstrate that it can be used to attain strong coherent coupling between Nitrogen-Vacancy centers separated by micrometer distances at room temperature. Realistic imperfections and decoherence effects are analyzed. 1 aYao, Norman, Y.1 aJiang, Liang1 aGorshkov, Alexey, V.1 aGong, Zhe-Xuan1 aZhai, Alex1 aDuan, L., -M.1 aLukin, Mikhail, D. uhttp://arxiv.org/abs/1011.2762v201144nas a2200145 4500008004100000245010200041210006900143260001500212490000800227520067200235100001900907700001700926700001800943856003700961 2010 eng d00aTemperature driven structural phase transition for trapped ions and its experimental detection 0 aTemperature driven structural phase transition for trapped ions c2010/12/290 v1053 a A Wigner crystal formed with trapped ion can undergo structural phase transition, which is determined only by the mechanical conditions on a classical level. Instead of this classical result, we show that through consideration of quantum and thermal fluctuation, a structural phase transition can be solely driven by change of the system's temperature. We determine a finite-temperature phase diagram for trapped ions using the renormalization group method and the path integral formalism, and propose an experimental scheme to observe the predicted temperature-driven structural phase transition, which is well within the reach of the current ion trap technology. 1 aGong, Zhe-Xuan1 aLin, G., -D.1 aDuan, L., -M. uhttp://arxiv.org/abs/1009.0089v1