A quantum simulator is a restricted class of quantum computer that controls the interactions between quantum bits in a way that can be mapped to certain difficult quantum many-body problems. As more control is exerted over larger numbers of qubits, the simulator can tackle a wider range of problems, with the ultimate limit being a universal quantum computer that can solve general classes of hard problems. We use a quantum simulator composed of up to 53 qubits to study a non-equilibrium phase transition in the transverse field Ising model of magnetism, in a regime where conventional statistical mechanics does not apply. The qubits are represented by trapped ion spins that can be prepared in a variety of initial pure states. We apply a global long-range Ising interaction with controllable strength and range, and measure each individual qubit with near 99% efficiency. This allows the single-shot measurement of arbitrary many-body correlations for the direct probing of the dynamical phase transition and the uncovering of computationally intractable features that rely on the long-range interactions and high connectivity between the qubits.There have been many recent demonstrations of quantum simulators with varying numbers of qubits and degrees of individual qubit control [1]. For instance, small numbers of qubits stored in trapped atomic ions [2,3] and superconducting circuits [4] have been used to simulate various magnetic spin or Hubbard models with individual qubit state preparation and measurement. Large numbers of atoms have simulated similar models, but with global control and measurements [5] or with correlations that only appear over a few atom sites [6]. An outstanding challenge is to increase qubit number while maintaining individual qubit control and measurement, with the goal of performing simulations or algorithms that cannot be efficiently solved classically. Atomic systems are excellent candidates for this scaling, because their qubits can be made virtually identical, with flexible and reconfigurable control through external optical fields and high initialization and detection efficiency for individual qubits. Recent work with neutral atoms [7,8] has demonstrated many-body quantum dynamics with up to 51 atoms coupled through van der Waals Rydberg interactions, and the current work presents the optical control and measurement of a similar number of atomic ions interacting through their long-range Coulomb-coupled motion.We perform a quantum simulation of a dynamical phase transition (DPT) with up to 53 trapped ion qubits. The understanding of such nonequilibrium behavior is of great interest to a wide range of subjects, from social science [9] and cellular biology [10] to astrophysics [11] and quantum condensed matter physics [12]. Recent theoretical studies of DPT [13][14][15][16][17][18][19][20] involve the transverse field Ising model (TFIM), the quintessential model of quantum phase transitions [21]. A recent experiment investigated a DPT with up to 10 trapped ion qubits, where the transverse field ...

Fermionic alkaline-earth atoms have unique properties that make them attractive candidates for the realization of novel atomic clocks and degenerate quantum gases. At the same time, they are attracting considerable theoretical attention in the context of quantum information processing. Here we demonstrate that when such atoms are loaded in optical lattices, they can be used as quantum simulators of unique many-body phenomena. In particular, we show that the decoupling of the nuclear spin from the electronic angular momentum can be used to implement many-body systems with an unprecedented degree of symmetry, characterized by the SU(N) group with N as large as 10. Moreover, the interplay of the nuclear spin with the electronic degree of freedom provided by a stable optically excited state allows for the study of spin-orbital physics. Such systems may provide valuable insights into strongly correlated physics of transition metal oxides, heavy fermion materials, and spin liquid phases.The interest in fermionic alkaline-earth atoms [1, 2, 3, 4, 5, 6, 7, 8] stems from their two key features: (1) the presence of a metastable excited state 3 P 0 coupled to the ground 1 S 0 state via an ultranarrow doubly-forbidden transition [1] and (2) the almost perfect decoupling [1] of the nuclear spin I from the electronic angular momentum J in these two states, since they both have J = 0. This decoupling implies that s-wave scattering lengths involving states 1 S 0 and 3 P 0 are independent of the nuclear spin, aside from the restrictions imposed by fermionic antisymmetry. We show that the resulting SU(N) spin symmetry (where N = 2I + 1 can be as large as 10) together with the possibility of combining (nuclear) spin physics with (electronic) orbital physics open up a wide field of extremely rich many-body systems with alkaline-earth atoms.In what follows, we derive the two-orbital SU(N)-symmetric Hubbard model describing alkaline-earth atoms in 1 S 0 and 3 P 0 states trapped in an optical lattice. We focus on specific parameter regimes characterized by full or partial atom localization due to strong atomic interactions, where simpler effective spin Hamiltonians can be derived. The interplay between orbital and spin degrees of freedom in such effective models is a central topic in quantum magnetism and has attracted tremendous interest in the condensed matter community. Alkaline earth atoms thus provide, on the one hand, a unique opportunity for the implementation of some of these models for the first time in a defect-free and fully controllable environment. On the other hand, they open a new arena to study a wide range of models, many of which have not been discussed previously, even theoretically. We demonstrate, in particular, how to implement the KugelKhomskii model studied in the context of transition metal oxides [9, 10,11,12,13], the Kondo lattice model [14,15,16,17,18,19,20,21,22,23,24,25,26] [27,28,29,30,31,32,33,34]. For example, we discuss how, by appropriately choosing the initial state, a single alkaline-earth atom s...

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...

The realization of strong nonlinear interactions between individual light quanta (photons) is a long-standing goal in optical science and engineering 1, 2 that is both of fundamental and technological significance. In conventional optical materials, the nonlinearity at light powers corresponding to single photons is negligibly weak. Here we demonstrate a medium that is nonlinear at the level of individual quanta, exhibiting strong absorption of photon pairs while remaining transparent to single photons. The quantum nonlinearity is obtained by coherently coupling slowly propagating photons [3][4][5] to strongly interacting atomic Rydberg states [6][7][8][9][10][11][12] in a cold, dense atomic gas 13 . Our approach opens the door for quantum-byquantum control of light fields, including single-photon switching 14 , all-optical deterministic 1 quantum logic 15 , and the realization of strongly correlated many-body states of light 16 .Recently, remarkable advances have been made towards optical systems that are nonlinear at the level of individual photons. The most promising approaches have used high-finesse optical cavities to enhance the atom-photon interaction probability 2,[17][18][19][20][21] . In contrast, our present method is cavity-free and is based on mapping photons onto atomic states with strong interactions in an extended atomic ensemble 13,14,22,23 . The central idea is illustrated in Fig. 1, where a quantum probe field incident onto a cold atomic gas is coupled to high-lying atomic states (Rydberg levels 24 ) by means of a second, stronger laser field (control field). For a single incident probe photon, the control field induces a transparency window in the otherwise opaque medium via Electromagnetically Induced Transparency (EIT), and the probe photon travels at much reduced speed in the form of a coupled excitation of light and matter (Rydberg polariton). However, in stark contrast to conventional EIT 5 , if two probe photons are incident onto the Rydberg EIT medium, the strong interaction between two Rydberg atoms tunes the EIT transition out of resonance, thereby destroying the EIT and leading to absorption 14,22,23, 25, 26 . The experimental demonstration of an extraordinary optical material exhibiting strong two-photon attenuation in combination with single-photon transmission is the central result of this work.The quantum nonlinearity can be viewed as a photon-photon blockade mechanism that prevents the transmission of any multi-photon state. It arises from the Rydberg excitation blockade 27 , which precludes the simultaneous excitation of two Rydberg atoms that are separated by less than a blockade radius r b (see Figure 1). During the optical excitation, an incident single photon is 2 converted, under the EIT conditions, into a Rydberg polariton inside the medium. However, due to the Rydberg blockade, a second polariton cannot travel within a blockade radius from the first one, and EIT is destroyed. Accordingly if the second photon approaches the single Rydberg polariton, it will be signific...

Using cold atoms to simulate strongly interacting quantum systems is an exciting frontier of physics. However, because atoms are nominally neutral point particles, this limits the types of interaction that can be produced. We propose to use the powerful new platform of cold atoms trapped near nanophotonic systems to extend these limits, enabling a novel quantum material in which atomic spin degrees of freedom, motion and photons strongly couple over long distances. In this system, an atom trapped near a photonic crystal seeds a localized, tunable cavity mode around the atomic position. We find that this effective cavity facilitates interactions with other atoms within the cavity length, in a way that can be made robust against realistic imperfections. Finally, we show that such phenomena should be accessible using onedimensional photonic crystal waveguides in which coupling to atoms has already been experimentally demonstrated.T rapped ultracold atoms are a rich resource for physicists. Isolated from the environment and routinely manipulated, they can act as a quantum simulator for a wide variety of physical models 1 . However, although short-range interactions between atoms can be adjusted by Feshbach resonance, these systems typically lack the long-range interactions required to produce some of the most interesting condensed-matter phenomena. For example, exotic phases such as supersolids are predicted in systems with long-range interactions 2 , as well as Wigner crystallization 3 and topological states 4 . Long-range interactions can also lead to the breakdown of concepts such as additivity in statistical mechanics 5,6 and the violation of speed limits (Lieb-Robinson bounds) for the propagation of information [7][8][9] . As a result, there are active efforts to achieve long-range interactions using specific properties of the atoms 10 , such as their magnetic moment 11,12 , Rydberg excitation 13 or by using polar molecules 14 .In this Article we investigate another paradigm, where, instead of relying on atomic properties, we design the medium through which the atoms interact-specifically, by coupling the atoms via the photon modes of a photonic crystal. Our proposal is inspired by demonstrations of strong coupling of photons in nanophotonic systems with individual solid-state emitters 15 and, more recently, with cold atoms [16][17][18][19] . For example, systems of ∼10 3 atoms have been trapped by and coupled to the evanescent guided modes of nanofibres 16,17 , and single atoms have been coupled to photonic crystal cavities 18 and waveguides 19. One aim of these efforts is to utilize strong, controlled light-matter interactions for quantum information processing and networks 20. Here, we show that atoms interfaced with photonic crystals can also have remarkable consequences for the exploration of quantum many-body physics [21][22][23] .A photonic crystal is a periodic dielectric structure that controls the propagation of light 24 . By introducing a defect into this regular structure, it is possible to induce c...

In a recent paper [Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007)], we used a universal physical picture to optimize and demonstrate equivalence between a wide range of techniques for storage and retrieval of photon wave packets in Λ-type atomic media in free space, including the adiabatic reduction of the photon group velocity, pulse-propagation control via off-resonant Raman techniques, and photon-echo-based techniques. In the present paper, we perform the same analysis for the cavity model. In particular, we show that the retrieval efficiency is equal to C/(1 + C) independent of the retrieval technique, where C is the cooperativity parameter. We also derive the optimal strategy for storage and, in particular, demonstrate that at any detuning one can store, with the optimal efficiency of C/(1 + C), any smooth input mode satisfying T Cγ ≫ 1 and a certain class of resonant input modes satisfying T Cγ ∼ 1, where T is the duration of the input mode and 2γ is the transition linewidth. In the two subsequent papers of the series, we present the full analysis of the free-space model and discuss the effects of inhomogeneous broadening on photon storage.

We develop the theory of light propagation under the conditions of electromagnetically induced transparency (EIT) in systems involving strongly interacting Rydberg states. Taking into account the quantum nature and the spatial propagation of light, we analyze interactions involving fewphoton pulses. We demonstrate that this system can be used for the generation of nonclassical states of light including trains of single photons with an avoided volume between them, for implementing photon-photon quantum gates, as well as for studying many-body phenomena with strongly correlated photons.PACS numbers: 42.50. Nn, 32.80.Ee, 42.50.Gy, 34.20.Cf The phenomenon of electromagnetically induced transparency (EIT) [1] in systems involving Rydberg states [2] has recently attracted significant experimental [3][4][5][6][7][8][9][10] and theoretical [11][12][13][14][15][16][17][18][19][20][21] attention. While EIT allows for strong atom-light interactions without absorption, Rydberg states provide strong long-range atom-atom interactions. Therefore, the resulting combination of EIT with Rydberg atoms is ideal for implementing mesoscopic quantum gates [2,16] and for inducing strong photonphoton interactions, with applications to photonic quantum information processing [2,[11][12][13][14][19][20][21][22] and to the realization of many-body phenomena with strongly interacting photons [23]. At the same time, the many-body theoretical description of EIT with arbitrarily strongly interacting Rydberg atoms, taking into account the full quantum dynamics and the spatial propagation of light, has not been reported previously.In this Letter, we develop such a theory by analyzing the problem for at most two incident photons, which, in turn, provides intuition for understanding the full multiphoton problem. We show that Rydberg atom interactions give rise to photon-photon interactions, which, below a critical inter-photonic distance, turn the EIT medium into an effective two-level medium. This can be used to implement photon-atom and photon-photon phase gates and to enable deterministic single-photon sources.The basic physics is illustrated by considering a simple case [ Fig. 1(b)], in which a single-photon wavepacket E propagates in an EIT medium [level scheme in Fig. 1(a)] with a central control atom at z = 0 prepared in a Rydberg state |r . Atoms in another Rydberg state |r , coupled by the EIT control laser [ Fig. 1(a)], experience a van der Waals potential V (z) = C 6 /z 6 due to the interaction with the control atom, which is decoupled from the applied fields. Alternatively, one could apply an electric field to induce dipole moments in states |r and |r ,Sufficiently far away from z = 0, the incident photon propagates in a standard EIT medium featuring a twophoton-resonant control field with Rabi frequency Ω. In the vicinity of the control atom, however, the state |r is shifted so strongly out of resonance that the photon sees only a two-level (|g , |e ) medium with transition linewidth 2γ. The critical z, at which the interaction is ...

* These authors contributed equally to this workThe fundamental properties of light derive from its constituent particles (photons) that are massless and do not interact with one another 1 . At the same time, it has been long known that the realization of coherent interactions between individual photons, akin to those associated with conventional massive particles, could enable a wide variety of unique scientific and engineering applications 2,3 . Here, by coupling light to strongly interacting atomic Rydberg states in a dispersive regime, we demonstrate a quantum nonlinear medium inside which individual photons travel as massive particles with strong mutual attraction, such that the propagation of photon pairs is dominated by a two-photon bound state 4-7 . We measure the dynamical evolution of the two-photon wavefunction using time-resolved quantum state tomography, and demonstrate a conditional phase shift 8 exceeding one radian, resulting in polarization-entangled photon pairs. Unique applications include all-optical switching, deterministic photonic quantum logic, and the generation of strongly correlated states of light 9 .Interactions between individual photons are being explored in cavity quantum electrodynamics, where a single, confined electromagnetic mode is coupled to an atomic system 10-12 . Our approach is to couple a light field propagating in a dispersive medium to highly excited atomic states with strong mutual interactions (Rydberg states) 13,14 . Similar to previous studies of quantum nonlinearities via Rydberg states that were based on dissipation 15-19 rather than dispersion 20 , we make use of electromagnetically induced transparency (EIT) to slow down the propagation of light 21 in a cold atomic gas. By operating in a dispersive regime away from the intermediate atomic resonance (Fig. 1b), where atomic absorption is low and only weakly nonlinear 22 , we realize a situation where Rydberg-atom-mediated coherent interactions between individual photons dominate the propagation dynamics of weak light pulses. Previous theoretical studies have proposed various scenarios for inducing strong interactions between individual photons 2,3,23 and for creating bound states of a few quanta 4,5,7,24 , a feature generic to strongly interacting quantum field theories. The first experimental realization of a photonic system with strong attractive interactions, including evidence for a predicted two-photon bound state, represents the main result of this work.Our experiment (outlined in Fig. 1a) makes use of an ultracold rubidium gas loaded into a dipole trap, as described previously 19 . The probe light of interest is σ + polarized, coupling the ground state |g to the Rydberg state |r via an intermediate state |e of linewidth Γ/(2π) = 6.1 MHz by means of a control field that is detuned by ∆ below the resonance frequency of the upper transition |e → |r (Fig. 1b). Under these conditions, for a very weak probe field with mean incident photon rate R i = 0.5 µs −1 , EIT is established when the probe detuning matches...