Entanglement is one of the most intriguing features of quantum mechanics. It describes non-local correlations between quantum objects, and is at the heart of quantum information sciences. Entanglement is now being studied in diverse fields ranging from condensed matter to quantum gravity. However, measuring entanglement remains a challenge. This is especially so in systems of interacting delocalized particles, for which a direct experimental measurement of spatial entanglement has been elusive. Here, we measure entanglement in such a system of itinerant particles using quantum interference of many-body twins. Making use of our single-site-resolved control of ultracold bosonic atoms in optical lattices, we prepare two identical copies of a many-body state and interfere them. This enables us to directly measure quantum purity, Rényi entanglement entropy, and mutual information. These experiments pave the way for using entanglement to characterize quantum phases and dynamics of strongly correlated many-body systems.

Statistical mechanics relies on the maximization of entropy in a system at thermal equilibrium. However, an isolated quantum many-body system initialized in a pure state remains pure during Schrödinger evolution, and in this sense it has static, zero entropy. We experimentally studied the emergence of statistical mechanics in a quantum state and observed the fundamental role of quantum entanglement in facilitating this emergence. Microscopy of an evolving quantum system indicates that the full quantum state remains pure, whereas thermalization occurs on a local scale. We directly measured entanglement entropy, which assumes the role of the thermal entropy in thermalization. The entanglement creates local entropy that validates the use of statistical physics for local observables. Our measurements are consistent with the eigenstate thermalization hypothesis.

A logarithmic signature Some one-dimensional disordered interacting quantum systems have been theoretically predicted to display a property termed many-body localization (MBL), where the system retains the memory of its initial state and fails to thermalize. However, proving experimentally that something does not occur is tricky. Instead, physicists have proposed monitoring the entanglement entropy of the system, which should grow logarithmically with evolution time in an MBL system. Lukin et al. observed this characteristic logarithmic trend in a disordered chain of interacting atoms of rubidium-87. This method should be generalizable to other experimental platforms and higher dimensions. Science , this issue p. 256

Full control over the dynamics of interacting, indistinguishable quantum particles is an important prerequisite for the experimental study of strongly correlated quantum matter and the implementation of high-fidelity quantum information processing. Here we demonstrate such control over the quantum walk -the quantum mechanical analogue of the classical random walk -in the strong interaction regime. Using interacting bosonic atoms in an optical lattice, we directly observe fundamental effects such as the emergence of correlations in two-particle quantum walks, as well as strongly correlated Bloch oscillations in tilted optical lattices. Our approach can be scaled to larger systems, greatly extending the class of problems accessible via quantum walks.Quantum walks are the quantum-mechanical analogues of the classical random walk process, describing the propagation of quantum particles on periodic potentials [1,2]. Unlike classical objects, particles performing a quantum walk can be in a superposition state and take all possible paths through their environment simultaneously, leading to faster propagation and enhanced sensitivity to initial conditions. These properties have generated considerable interest in using quantum walks for the study of position-space quantum dynamics and for quantum information processing [3]. Two distinct models of quantum walk with similar physical behavior were devised: The discrete time quantum walk [1], in which the particle propagates in discrete steps determined by a dynamic internal degree of freedom, and the continuous time quantum walk [2], in which the dynamics is described by a time-independent lattice Hamiltonian.Experimentally, quantum walks have been implemented for photons [4], trapped ions [5,6], and neutral atoms [7][8][9], among other platforms [4]. Until recently, most experiments were aimed at observing the quantum walks of a single quantum particle, which are described by classical wave equations.An enhancement of quantum effects emerges when more than one indistinguishable particle participates in the quantum walk simultaneously. In such cases, quantum correlations can develop as a consequence of Hanbury Brown-Twiss (HBT) interference and quantum statistics, as was investigated theoretically [10,11] and experimentally [12][13][14][15][16][17]. In the absence of interactions or auxiliary feed-forward measurements of the KnillLaflamme-Milburn type [18] this problem is believed to lack full quantum complexity, although it can still become intractable by classical computing [11].The inclusion of interaction between indistinguishable quantum walkers [19,20] may grant access to a much wider class of computationally hard problems, such as many-body localization and the dynamics of interacting quantum disordered systems [21]. Similarly, in the presence of interactions the quantum walk can yield universal Starting from a localized initial state (I), individual atoms perform independent quantum walks in an optical lattice (II). Right: The single-particle density distribution ex...

The interplay between magnetic fields and interacting particles can lead to exotic phases of matter that exhibit topological order and high degrees of spatial entanglement. Although these phases were discovered in a solid-state setting, recent innovations in systems of ultracold neutral atoms-uncharged atoms that do not naturally experience a Lorentz force-allow the synthesis of artificial magnetic, or gauge, fields. This experimental platform holds promise for exploring exotic physics in fractional quantum Hall systems, owing to the microscopic control and precision that is achievable in cold-atom systems. However, so far these experiments have mostly explored the regime of weak interactions, which precludes access to correlated many-body states. Here, through microscopic atomic control and detection, we demonstrate the controlled incorporation of strong interactions into a two-body system with a chiral band structure. We observe and explain the way in which interparticle interactions induce chirality in the propagation dynamics of particles in a ladder-like, real-space lattice governed by the interacting Harper-Hofstadter model, which describes lattice-confined, coherently mobile particles in the presence of a magnetic field. We use a bottom-up strategy to prepare interacting chiral quantum states, thus circumventing the challenges of a top-down approach that begins with a many-body system, the size of which can hinder the preparation of controlled states. Our experimental platform combines all of the necessary components for investigating highly entangled topological states, and our observations provide a benchmark for future experiments in the fractional quantum Hall regime.