In relativistic quantum field theory, information propagation is bounded by the speed of light. No such limit exists in the non-relativistic case, although in real physical systems, short-range interactions may be expected to restrict the propagation of information to finite velocities. The question of how fast correlations can spread in quantum many-body systems has been long studied. The existence of a maximal velocity, known as the Lieb-Robinson bound, has been shown theoretically to exist in several interacting many-body systems (for example, spins on a lattice)--such systems can be regarded as exhibiting an effective light cone that bounds the propagation speed of correlations. The existence of such a 'speed of light' has profound implications for condensed matter physics and quantum information, but has not been observed experimentally. Here we report the time-resolved detection of propagating correlations in an interacting quantum many-body system. By quenching a one-dimensional quantum gas in an optical lattice, we reveal how quasiparticle pairs transport correlations with a finite velocity across the system, resulting in an effective light cone for the quantum dynamics. Our results open perspectives for understanding the relaxation of closed quantum systems far from equilibrium, and for engineering the efficient quantum channels necessary for fast quantum computations.

Ultracold atoms in optical lattices are a versatile tool to investigate fundamental properties of quantum many body systems. In particular, the high degree of control of experimental parameters has allowed the study of many interesting phenomena such as quantum phase transitions and quantum spin dynamics. Here we demonstrate how such control can be extended down to the most fundamental level of a single spin at a specific site of an optical lattice. Using a tightly focussed laser beam together with a microwave field, we were able to flip the spin of individual atoms in a Mott insulator with sub-diffraction-limited resolution, well below the lattice spacing. The Mott insulator provided us with a large two-dimensional array of perfectly arranged atoms, in which we created arbitrary spin patterns by sequentially addressing selected lattice sites after freezing out the atom distribution. We directly monitored the tunnelling quantum dynamics of single atoms in the lattice prepared along a single line and observed that our addressing scheme leaves the atoms in the motional ground state. Our results open the path to a wide range of novel applications from quantum dynamics of spin impurities, entropy transport, implementation of novel cooling schemes, and engineering of quantum many-body phases to quantum information processing.The ability to observe and control the position of single atoms on a surface of a solid via scanning tunnelling and atomic force microscopy has revolutionised the field of condensed matter physics [1,2]. In few-atom systems, coherent control of single particles in e.g. an ion chain has proven crucial for the implementation of high-fidelity quantum gates and the readout of individual qubits in quantum information processing [3]. Bringing such levels of control to the regime of large scale many-body systems has been a longstanding goal in quantum physics. In the context of ultracold atoms in optical lattices, a major challenge has been to combine degenerate atomic samples with single-site addressing resolution and singleatom sensitivity. This full control is essential for many applications in condensed matter physics, such as the study of spin impurities [4] and quantum spin dynamics [5,6] within quantum magnetism, entropy transport, the implementation of novel cooling schemes [7,8] or digital quantum simulations based on Rydberg atoms [9]. For scalable quantum information processing, a Mott insulator with unity filling provides a natural quantum register with several hundreds of qubits. In order to exploit the full potential of such a large scale system for quantum computation, coherent manipulation of individual spins is indispensable, both within a circuit-based [10] or a one-way quantum computer architecture [11,12].The quest to address atoms on single sites of an optical lattice has a long history [7,[13][14][15][16][17][18][19][20][21][22]. In one dimension, single-site addressing was accomplished optically † present address: Department of Physics and Astronomy, University of Aarhus, DK-8000 ...

One of the elementary processes in quantum magnetism is the propagation of spin excitations. Here we study the quantum dynamics of a deterministically created spin-impurity atom, as it propagates in a one-dimensional lattice system. We probe the spatial probability distribution of the impurity at different times using single-site-resolved imaging of bosonic atoms in an optical lattice. In the Mott-insulating regime, the quantum-coherent propagation of a magnetic excitation in the Heisenberg model can be observed using a post-selection technique. Extending the study to the superfluid regime of the bath, we quantitatively determine how the bath affects the motion of the impurity, showing evidence of polaronic behaviour. The experimental data agree with theoretical predictions, allowing us to determine the effect of temperature on the impurity motion. Our results provide a new approach to studying quantum magnetism, mobile impurities in quantum fluids and polarons in lattice systems

The ability to control and tune interactions in ultracold atomic gases has paved the way for the realization of new phases of matter. So far, experiments have achieved a high degree of control over short-range interactions, but the realization of long-range interactions has become a central focus of research because it would open up a new realm of many-body physics. Rydberg atoms are highly suited to this goal because the van der Waals forces between them are many orders of magnitude larger than those between ground-state atoms. Consequently, mere laser excitation of ultracold gases can cause strongly correlated many-body states to emerge directly when atoms are transferred to Rydberg states. A key example is a quantum crystal composed of coherent superpositions of different, spatially ordered configurations of collective excitations. Here we use high-resolution, in situ Rydberg atom imaging to measure directly strong correlations in a laser-excited, two-dimensional atomic Mott insulator. The observations reveal the emergence of spatially ordered excitation patterns with random orientation, but well-defined geometry, in the high-density components of the prepared many-body state. Together with a time-resolved analysis, this supports the description of the system in terms of a correlated quantum state of collective excitations delocalized throughout the gas. Our experiment demonstrates the potential of Rydberg gases to realize exotic phases of matter, thereby laying the basis for quantum simulations of quantum magnets with long-range interactions.

More than eighty years ago, H. Bethe pointed out the existence of bound states of elementary spin waves in one-dimensional quantum magnets [1]. To date, identifying signatures of such magnon bound states has remained a subject of intense theoretical research [2][3][4][5] while their detection has proved challenging for experiments. Ultracold atoms offer an ideal setting to reveal such bound states by tracking the spin dynamics after a local quantum quench [6] with single-spin and singlesite resolution [7,8]. Here we report on the direct observation of two-magnon bound states using in-situ correlation measurements in a one-dimensional Heisenberg spin chain realized with ultracold bosonic atoms in an optical lattice. We observe the quantum walk of free and bound magnon states through time-resolved measurements of the two spin impurities. The increased effective mass of the compound magnon state results in slower spin dynamics as compared to single magnon excitations. In our measurements, we also determine the decay time of bound magnons, which is most likely limited by scattering on thermal fluctuations in the system. Our results open a new pathway for studying fundamental properties of quantum magnets and, more generally, properties of interacting impurities in quantum many-body systems.The study of non-equilibrium processes in quantum spin models can provide fundamental insight into elementary aspects of magnetism. Magnons are the basic quasiparticle exitations around the ground state of ferromagnets and govern their low temperature physics [9,10]. Due to the ferromagnetic interaction, two spin excitations can remain bound together, forming a so-called twomagnon bound state [1,9,11]. In one and two dimensions, bound states exist for all center of mass momenta, which prohibits the description of low energy properties in terms of free magnon states [9]. In the classical limit, magnon bound states can be regarded as the basic building blocks of magnetic solitons [12,13]. Next to these fundamental aspects, the study of non-equilibrium dynamics in quantum spin chains is also important for a variety of applications. The evolution of two localized spin excitations realizes an interacting quantum walk [14,15] in the spin domain, which can be a versatile tool for the study of complex many-body systems [16]. It is also of importance in the context of quantum information [17], where transport properties in a one-dimensional chain of qubits can be strongly influenced by magnon bound states [18].The spin-1/2 Heisenberg model is one of the foundational models for interacting quantum spins. This model could be solved analytically in one dimension in the early 1930's by H. Bethe using a systematic Ansatz for the form of the eigenvectors [1]. Later, the Bethe Ansatz proved to be far more general and allowed for solving many more one-dimensional models, such as the Lieb-Liniger or the fermionic Hubbard model [20], and recent, powerful extensions include the investigation of the dynamics * Electronic address: takeshi.fukuhara@mpq....

Spontaneous symmetry breaking plays a key role in our understanding of nature. In a relativistic field theory, a broken continuous symmetry leads to the emergence of two types of fundamental excitations: massless Nambu-Goldstone modes and a massive 'Higgs' amplitude mode. An excitation of Higgs type is of crucial importance in the standard model of elementary particles [1] and also appears as a fundamental collective mode in quantum many-body systems [2]. Whether such a mode exists in low-dimensional systems as a resonance-like feature or becomes over-damped through coupling to Nambu-Goldstone modes has been a subject of theoretical debate [2][3][4][5][6][7]. Here we experimentally reveal and study a Higgs mode in a two-dimensional neutral superfluid close to the transition to a Mott insulating phase. We unambiguously identify the mode by observing the expected softening of the onset of spectral response when approaching the quantum critical point. In this regime, our system is described by an effective relativistic field theory with a two-component quantum-field [2,8,9], constituting a minimal model for spontaneous breaking of a continuous symmetry. Additionally, all microscopic parameters of our system are known from first principles and the resolution of our measurement allows us to detect excited states of the many-body system at the level of individual quasiparticles. This allows for an in-depth study of Higgs excitations, which also addresses the consequences of reduced dimensionality and confinement of the system. Our work constitutes a first step in exploring emergent relativistic models with ultracold atomic gases.Higgs modes are amplitude oscillations of a quantum field and appear as collective excitations in quantum many-body systems as a consequence of spontaneous breaking of a continuous symmetry. Close to a quantum critical point, the low-energy physics of such systems is in many cases captured by an effective Lorentz invariant critical theory [2]. The minimal version of such a theory describes the dynamics of a complex order parameter Ψ = |Ψ|e iφ near a quantum phase transition between an ordered (|Ψ| > 0) and a disordered phase (|Ψ| = 0). Within the ordered phase, the classical energy density has the shape of a Mexican hat (Fig. 1a) and the order parameter takes on a non-zero value in the minimum of this potential. Hereby, its phase φ acquires a definite value through spontaneous breaking of the rotation symmetry (i.e., U (1) symmetry). Expanding the field around the symmetry broken ground state leads to two types of modes: a Nambu-Goldstone mode and a Higgs mode related to phase and amplitude variations of Ψ, respectively (Fig. 1a). In contrast to the phase mode, the amplitude mode has a finite excitation gap (i.e., a finite mass), which is expected to show a characteristic softening when approaching the disordered phase (Fig. 1a). The sketched minimal model of an order parameter with N = 2 components belongs to a class of O(N ) relativistic * Electronic address: manuel.endres@mpq.mpg.de field t...

Dominating finite-range interactions in many-body systems can lead to intriguing self-ordered phases of matter. Well known examples are crystalline solids or Coulomb crystals in ion traps. In those systems, crystallization proceeds via a classical transition, driven by thermal fluctuations. In contrast, ensembles of ultracold atoms laser-excited to Rydberg states provide a well-controlled quantum system [1], in which a crystalline phase transition governed by quantum fluctuations can be explored [2][3][4]. Here we report on the experimental preparation of the crystalline states in such a Rydberg many-body system. Fast coherent control on the many-body level is achieved via numerically optimized laser excitation pulses [2][3][4]. We observe an excitation-number staircase [2][3][4][5][6][7] as a function of the system size and show directly the emergence of incompressible ordered states on its steps. Our results demonstrate the applicability of quantum optical control techniques in strongly interacting systems, paving the way towards the investigation of novel quantum phases in long-range interacting quantum systems, as well as for detailed studies of their coherence and correlation properties [2][3][4][5][6][7][8].Rydberg atoms exhibit unique properties that are key to realize and explore novel quantum many-body Hamiltonians and their phases. The strong van der Waals interaction between them allows to create many-body systems with tailored long-range interactions in neutral ultra-cold atom samples [1,9,10]. Complete experimental control of these systems is possible using the well developed toolbox of quantum optics for the laserexcitation to the Rydberg states. The magnitude of the resulting interactions between the Rydberg atoms is determined by the choice of the excited state and it can exceed all other relevant energy scales on distances of several microns, thereby leading to an ensemble dominated by long-range interactions. In this regime, the ground state of the resulting many-body system is expected to show crystalline ordering of the Rydberg excitations, which can be understood in the limit of vanishing coupling as the classical closest packing of hard spheres [11]. The lattice constant of the crystal is set by the dipole blockade radius R b [12,13], defined as the inter-particle spacing at which the dipole interaction between two Rydberg atoms exceeds the spectral range of the optical coupling. To prepare the system in this crystalline phase, a dynamical approach has been suggested that adiabatically connects the ground state containing no Rydberg excitations with the targeted crystalline state. At the heart of this dynamical crystallization technique is the coherent control of the many-body system [2][3][4][14][15][16][17].Previous experiments showed direct or indirect evidence for correlations caused by the long-range interac- * Electronic address: peter.schauss@mpq.mpg.de tions in Rydberg many-body systems, such as a universal scaling of the Rydberg excitation number [18], subPoissonian counting statis...

Quantum phases of matter are characterized by the underlying correlations of the many-body system. Although this is typically captured by a local order parameter, it has been shown that a broad class of many-body systems possesses a hidden non-local order. In the case of bosonic Mott insulators, the ground state properties are governed by quantum fluctuations in the form of correlated particle-hole pairs that lead to the emergence of a non-local string order in one dimension. Using high-resolution imaging of low-dimensional quantum gases in an optical lattice, we directly detect these pairs with single-site and single-particle sensitivity and observe string order in the one-dimensional case.The realization of strongly correlated quantum manybody systems using ultracold atoms has enabled the direct observation and control of fundamental quantum effects [1][2][3]. A prominent example is the transition from a superfluid (SF) to a Mott insulator (MI), occurring when interactions between bosonic particles on a lattice dominate over their kinetic energy [4][5][6][7][8]. At zero temperature, and in the limit where the ratio of kinetic energy over interaction energy vanishes, particle fluctuations are completely suppressed and the lattice sites are occupied by an integer number of particles. However, at a finite tunnel coupling, but still in the Mott insulating regime, quantum fluctuations create correlated particlehole pairs on top of this fixed-density background, which can be understood as virtual excitations. These particlehole pairs fundamentally determine the properties of the Mott insulator such as its residual phase coherence [9] and lie at the heart of superexchange-mediated spin interactions that form the basis of quantum magnetism in multi-component quantum gas mixtures [10][11][12].In a one-dimensional system, the appearance of correlated particle-hole pairs at the transition point from a superfluid to a Mott insulator is intimately connected to the emergence of a hidden string-order parameter O P [13,14]:Here δn j =n j −n denotes the deviation in occupation of the jth lattice site from the average background density, and k is an arbitrary position along the chain. In the simplest case of a Mott insulator with unity filling * Electronic address: manuel.endres@mpq.mpg.de (n = 1), relevant to our experiments, each factor in the product of operators in Eq. 1 yields −1 instead of +1, when a single-particle fluctuation from the unit background density is encountered. In the superfluid, particle and hole fluctuations occur independently and are uncorrelated, such that O P = 0. However, in the Mott insulating phase, density fluctuations always occur as correlated particle-hole pairs, resulting in O P = 0. For a homogeneous system, O P is expected to follow a scaling of Berezinskii-Kosterlitz-Thouless (BKT) type [15]. Non-local correlation functions, like the string-order parameter defined above, have been introduced in the context of low-dimensional quantum systems. They classify many-body quantum phases that are n...