Controllable, coherent many-body systems can provide insights into the fundamental properties of quantum matter, enable the realization of new quantum phases and could ultimately lead to computational systems that outperform existing computers based on classical approaches. Here we demonstrate a method for creating controlled many-body quantum matter that combines deterministically prepared, reconfigurable arrays of individually trapped cold atoms with strong, coherent interactions enabled by excitation to Rydberg states. We realize a programmable Ising-type quantum spin model with tunable interactions and system sizes of up to 51 qubits. Within this model, we observe phase transitions into spatially ordered states that break various discrete symmetries, verify the high-fidelity preparation of these states and investigate the dynamics across the phase transition in large arrays of atoms. In particular, we observe robust many-body dynamics corresponding to persistent oscillations of the order after a rapid quantum quench that results from a sudden transition across the phase boundary. Our method provides a way of exploring many-body phenomena on a programmable quantum simulator and could enable realizations of new quantum algorithms.

Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations [1]. These fluctuations play a dominant role in the quantum critical region surrounding the transition point, where the dynamics are governed by the universal properties associated with the QPT. While time-dependent phenomena associated with classical, thermally driven phase transitions have been extensively studied in systems ranging from the early universe to Bose Einstein Condensates [2-5], understanding critical real-time dynamics in isolated, non-equilibrium quantum systems is an outstanding challenge [6]. Here, we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations while crossing the QPT, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM) [7-9] for an Ising-type QPT, explore scaling universality, and observe corrections beyond QKZM predictions. This approach is subsequently used to measure the critical exponents associated with chiral clock models [10,11], providing new insights into exotic systems that have not been understood previously, and opening the door for precision studies of critical phenomena, simulations of lattice gauge theories [12,13] and applications to quantum optimization [14,15].The celebrated Kibble-Zurek mechanism [2, 3] describes nonequilibrium dynamics and the formation of topological defects in a second-order phase transition driven by thermal fluctuations, and has been experimentally verified in a wide variety of physical systems [4,5]. Recently, the concepts underlying the Kibble-Zurek description have been extended to the quantum regime [7][8][9]. Here, the typical size of the correlated regions, ξ, after a dynamical sweep across the QPT scales as a power-law Critical point gc Control parameter g Correlation length » Phase 1 Phase 2 » » jg ¡ gcj ¡º Time ¿ » jg ¡ gcj ¡zº Fast ramp Slow ramp 0 1 2 3 4 5 Detuning ¢= 1 2 3 4 Interaction range RB=a ²±²±²±²±²±²±² ²±±²±±²±±²±±² ²±±±²±±±²±±±² ℤ2 ℤ3 ℤ4 t ¢ -0.05 0 0.05 -0.1 0 0.1 Density-density correlator G(r) 0 4 8 12 16 20 Distance r (sites) -0.2 -0.1 0 0.1 a b c d Ω Ω FIG. 1: Quantum Kibble-Zurek mechanism (QKZM) and phase diagram. a, Illustration of the QKZM. As the control parameter approaches its critical value, the response time, τ , given by the inverse energy gap of the system, diverges. When the temporal distance to the critical point becomes equal to the response time, as marked by red crosses, the correlation length, b, stops growing due to nonadiabatic excitations. c, Numerically calculated ground-state phase diagram. Circles (diamonds) denote numerically obtained points along the phase boundaries calculated using (infinite-size) Density-Matrix Renormalization Group techniques (Methods). The shaded regions are a guide to the eye. Dashed lines show the experimental trajectories across the phase transitions determined by the pulse diagram shown...