The maximum speed with which information can propagate in a quantum many-body system directly affects how quickly disparate parts of the system can become correlated [1][2][3][4] and how difficult the system will be to describe numerically [5]. For systems with only short-range interactions, Lieb and Robinson derived a constant-velocity bound that limits correlations to within a linear effective light cone [6]. However, little is known about the propagation speed in systems with long-range interactions, since the best long-range bound [7] is too loose to give the correct light-cone shape for any known spin model and since analytic solutions rarely exist. In this work, we experimentally determine the spatial and time-dependent correlations of a far-from-equilibrium quantum many-body system evolving under a long-range Ising-or XY-model Hamiltonian. For several different interaction ranges, we extract the shape of the light cone and measure the velocity with which correlations propagate through the system. In many cases we find increasing propagation velocities, which violate the Lieb-Robinson prediction, and in one instance cannot be explained by any existing theory. Our results demonstrate that even modestly-sized quantum simulators are well-poised for studying complicated many-body systems that are intractable to classical computation.Lieb-Robinson bounds [6] have strongly influenced our understanding of locally-interacting quantum many-body systems. These bounds restrict the many-body dynamics to a well-defined causal region outside of which correlations are exponentially suppressed [8], analogous to causal light cones that arise in relativistic theories. Their existence has enabled proofs linking the decay of correlations in ground states to the presence of a spectral gap [7,9], as well as the area law for entanglement entropy [5,10,11], which can indicate the computational complexity of classically simulating a quantum system. Furthermore, Lieb-Robinson bounds constrain the timescales on which quantum systems might thermalize [12][13][14] and the maximum speed with which information can be sent through a quantum channel [15]. Recent experimental work has observed an effective Lieb-Robinson (i.e. linear) light cone in a 1D quantum gas [16].When interactions in a quantum system are longrange, the speed with which correlations build up between distant particles is no longer guaranteed to obey the Lieb-Robinson prediction. Indeed, for sufficiently long-ranged interactions, the notion of locality is expected to break down completely [17]. Violation of the Lieb-Robinson bound means that comparatively little can be predicted about the growth and propagation of correlations in long-range interacting systems, though there have been several recent theoretical and numerical advances [2,3,7,[17][18][19].Here we report an experiment that directly measures the shape of the causal region and the speed at which correlations propagate within Ising and XY spin chains. To induce the spread of correlations, we perform a global q...

Quantum simulation of spin models can provide insight into complex 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.Here we study non-equilibrium, quantum spin dynamics arising from an engineered, homogeneous Ising interaction in a two-dimensional array of 9 Be + ions in a Penning trap. We verify entanglement in the form of spin-squeezed states for up to 219 ions, directly observing 4.0±0.9 dB of spectroscopic enhancement. We also observe evidence of non-Gaussian, over-squeezed states in the full counting statistics. We find good agreement with ab-initio theory that includes competition between entanglement and decoherence, laying the Main Text: Quantum simulation, where one well-controlled quantum system emulates another system to be studied, anticipates solutions to intractable problems in fields including condensed-matter and high-energy physics, cosmology, and chemistry, before the development of a general purpose quantum computer (1-3). Of particular interest are simulations of the transverse-field Ising spin model (4), described by the Hamiltonianwhere N is the number of spins, J i,j parameterizes the spin-spin interaction, B x parameterizes a transverse magnetic field, andσ z ,σ x are Pauli spin matrices. A quantum simulation ofĤ T could illuminate complex phenomena in quantum magnetism, including quantum phase transitions, many-body localization, and glassiness in spin systems (5-8), and clarify whether quantum annealing can provide a speed-up for solving hard optimization problems (9, 10).Ensembles of photons, ions, neutral atoms, molecules, and superconducting circuits are all developing as quantum simulation platforms (3). For example, a variety of quantum spin models have been realized with large ensembles of neutral atoms (11-14) and molecules (15), using contact or dipolar interactions in optical lattices and using infinite-range interactions mediated by photons in optical cavities (16 Our experimental system consists of between 20 and 300 9 Be + ions confined to a singleplane Coulomb crystal in a Penning trap, described in Resonant microwave radiation for coupling ground states |↑ and |↓ is delivered through a waveguide. State-dependent fluorescence is collected through the pair of imaging objectives, where the bright state corresponds to |↑ . (B) Coulomb crystal images in a frame rotating at ω r with 9 Be + ions in |↑ . (C) The typical experiment pulse sequence, composed of cooling laser pulses (blue), microwave pulses (grey), and ODF laser pulses (green). Cooling and repumping initialize each ion in |↑ , then a microwave π/2 pulse prepares the spins along the x-axis. Suddenly switching onĤ I initiates the non-equilibrium spin dynamics. The microwave π pulse implements a spin-echo, reducing dephasing from magnetic field fluctuations and ODF laser light shifts. State readout consists of a final global rotation and flu...

In non-relativistic quantum theories with short-range Hamiltonians, a velocity v can be chosen such that the influence of any local perturbation is approximately confined to within a distance r until a time t ∼ r/v, thereby defining a linear light cone and giving rise to an emergent notion of locality. In systems with power-law (1/r α ) interactions, when α exceeds the dimension D, an analogous bound confines influences to within a distance r only until a time t ∼ (α/v) log r, suggesting that the velocity, as calculated from the slope of the light cone, may grow exponentially in time. We rule out this possibility; light cones of power-law interacting systems are algebraic for α > 2D, becoming linear as α → ∞. Our results impose strong new constraints on the growth of correlations and the production of entangled states in a variety of rapidly emerging, long-range interacting atomic, molecular, and optical systems. Though non-relativistic quantum theories are not explicitly causal, Lieb and Robinson [1] proved that an effective speed limit emerges dynamically in systems with short-ranged interactions, thereby extending the notion of causality into the fields of condensed matter physics, quantum chemistry, and quantum information science. Specifically, they proved that when interactions have a finite range or decay exponentially in space, the influence of a local perturbation decays exponentially outside of a space-time region bounded by the line t = r/v, which therefore plays the role of a light cone [ Fig. 1(a)]. However, many of the systems to which non-relativistic quantum theory is routinely applied-ranging from frustrated magnets and spin glasses [2,3] The results of Lieb and Robinson were first generalized to power-law (1/r α ) interacting systems by Hastings and Koma [17], with the following picture emerging. For α > D, the influence of a local perturbation is bounded by a function ∝ e vt /r α , and while a light cone can still be defined as the boundary outside of which this function falls below some threshold value, yielding t ∼ log r, that boundary is logarithmic rather than linear [ Fig. 1(b)]. Improvements upon these results exist, revealing, e.g., that the light-cone remains linear at intermediate distance scales [12], but all existing bounds consistently predict an asymptotically logarithmic light cone. An immediate and striking consequence is that the maximum group velocity, defined by the slope of the light cone, grows exponentially with time, thus suggesting that the aforementioned processes -thermalization, entanglement growth after a quench, etc. -may in principle be sped up exponentially by the presence of long-range interactions. In this manuscript, we show that this scenario is not possible. While light cones can potentially be sub-linear for any finite α, thus allowing a velocity that grows with time, for α > 2D they remain bounded by a polynomial t ∼ r ζ , and ζ ≤ 1 approaches unity for increasing α [ Fig. 3(c)].Model and formalism.-We assume a generic spin model with time-independent Ham...