Entanglement is the key quantum resource for improving measurement sensitivity beyond classical limits. However, the production of entanglement in mesoscopic atomic systems has been limited to squeezed states, described by Gaussian statistics. Here we report on the creation and characterization of non-Gaussian many-body entangled states. We develop a general method to extract the Fisher information, which reveals that the quantum dynamics of a classically unstable system creates quantum states that are not spin squeezed but nevertheless entangled. The extracted Fisher information quantifies metrologically useful entanglement which we confirm by Bayesian phase estimation with sub shot-noise sensitivity. These methods are scalable to large particle numbers and applicable directly to other quantum systems. [14,15]. The production of these fragile states in large systems remains a challenge and efficient methods for characterization are necessary because full state reconstruction becomes intractable. Here, we generate a class of non-Gaussian many-particle entangled states and reveal their quantum properties by studying the distinguishability of experimental probability distributions.A measure of the distinguishability with respect to small phase changes of the state is provided by the Fisher information F [16]. It is related to the highest attainable interferometric phase sensitivity by the Cramer-Rao bound ∆θ CR = 1/ √ F [17]. This limit follows from general statistical arguments for a measurement device with fluctuating output [18]. The Fisher information is limited by quantum fluctuations of the input state as well as the performance of the device. Even in the absence of technical noise, the Fisher information of a classical input state is F ≤ N because of the intrinsic granularity of N independent particles which translates into the shot-noise limit ∆θ ≥ 1/ √ N for phase estimation. This classical bound can be surpassed with a reduction of the input fluctuations by introducing entanglement between the N particles [5]. These states, known as squeezed states, are fully characterized by mean and variance of the observable and already employed in precision measurements [19][20][21]. In contrast, non-Gaussian quantum states can have increased fluctuations of the observable but nevertheless allow surpassing shot-noise limited performance. A textbook example is the Schrödinger cat state characterized by macroscopic fluctuations but achieving the best interferometric performance allowed by quantum mechanics, i.e. at the fundamental Heisenberg limit F = N . The initial coherent spin state (green) ideally evolves into a squeezed state (orange) followed by non-Gaussian states at later evolution times (violet). Edges of shaded areas are contours of the Husimi distribution for N = 380 at 1/e 2 of its maximum. (C) Experimental absorption picture, showing the site-and state-resolved optical lattice after a Stern-Gerlach separation. Shaded boxes indicate the sites with a total atom number in the range 380 ± 15, which are sele...

We experimentally demonstrate a nonlinear detection scheme exploiting time-reversal dynamics that disentangles continuous variable entangled states for feasible readout. Spin-exchange dynamics of Bose-Einstein condensates is used as the nonlinear mechanism which not only generates entangled states but can also be time reversed by controlled phase imprinting. For demonstration of a quantumenhanced measurement we construct an active atom SU(1,1) interferometer, where entangled state preparation and nonlinear readout both consist of parametric amplification. This scheme is capable of exhausting the quantum resource by detecting solely mean atom numbers. Controlled nonlinear transformations widen the spectrum of useful entangled states for applied quantum technologies.Nonlinear dynamics is the basis of generating nonclassical states of many particles. These entangled states are capable of improving a large variety of operations, e.g., computational tasks [1], communication and measurements [2]. Unlocking their full potential for quantum technologies requires both the generation and detection at the fundamental quantum limit. The generation of such highly entangled states with many particles has witnessed tremendous advances [3,4]. However, to fully exploit this quantum resource, the complete correlations on the single particle level need to be accessed, which still limits current experiments.To address this challenge, nonlinear readout schemes have been proposed [5][6][7][8]. Most of these employ a time inversion sequence. For this the nonlinear evolution that is used to produce the entangled state is inverted and reapplied for readout. If the state remains unperturbed, the second period of nonlinear evolution counteracts the first. This time-reversed readout disentangles the probe state such that the known separable initial state is recovered. This reversibility is nonperfect if the state is changed in between, similar to an incomplete Loschmidt-Echo [9]. By this sensitive mechanism, minute state perturbations are mapped onto readily discernable quantities.Experimentally, we use spin-changing collisions [10] in a mesoscopic spinor Bose-Einstein condensate. This nonlinear mechanism is the atomic analogue of parametric amplification, which is the textbook example of entangled state generation in quantum optics. At the same time, both the sign and the strength of the nonlinear coupling are experimentally adjustable, which makes this system ideally suited for realizing time reversal readout schemes.Spin exchange is performed in an effective three-level system within the spin F = 2 manifold of 87 Rb. For this the external degrees of freedom are frozen out such that dynamics is restricted to the spin degree of freedom. We start with a pure |F = 2, m F = 0 condensate (pump mode). Population in any m F = 0 state is carefully cleaned. During spin mixing atoms of the pump mode are coherently and pairwise scattered into the signal |↑ ≡ |2, 1 and idler |↓ ≡ |2, −1 mode, which we refer to as side modes (see Fig. 1). For small p...

The dynamics of quantum systems far from equilibrium represents one of the most challenging problems in theoretical many-body physics [1,2]. While the evolution is in general intractable in all its details, relevant observables can become insensitive to microscopic system parameters and initial conditions. This is the basis of the phenomenon of universality. Far from equilibrium, universality is identified through the scaling of the spatio-temporal evolution of the system, captured by universal exponents and functions. Theoretically, this has been studied in examples as different as the reheating process in inflationary universe cosmology [3,4], the dynamics of nuclear collision experiments described by quantum chromodynamics [5,6], or the postquench dynamics in dilute quantum gases in nonrelativistic quantum field theory [7][8][9][10][11]. However, an experimental demonstration of such scaling evolution in space and time in a quantum many-body system is lacking so far. Here we observe the emergence of universal dynamics by evaluating spatially resolved spin correlations in a quasi one-dimensional spinor Bose-Einstein condensate [12][13][14][15][16]. For long evolution times we extract the scaling properties from the spatial correlations of the spin excitations. From this we find the dynamics to be governed by transport of an emergent conserved quantity towards low momentum scales. Our results establish an important class of non-stationary systems whose dynamics is encoded in time-independent scaling exponents and functions signaling the existence of non-thermal fixed points [10,17,18]. We confirm that the non-thermal scaling phenomenon involves no fine-tuning, by preparing different initial conditions and observing the same scaling behaviour. Our analog quantum simulation approach provides the basis to reveal the underlying mechanisms and characteristics of non-thermal universality classes. One may use this universality to learn, from experiments with ultra-cold gases, about fundamental aspects of dynamics studied in cosmology and quantum chromodynamics.Isolated quantum many-body systems offer particularly clean settings for studying fundamental properties of the underlying unitary time evolution [19]. For sys-tems initialised far from equilibrium different scenarios have been identified, including the occurence of manybody oscillations [20] and revivals [21], the manifestation of many-body localisation [22], and quasi-stationary behaviour in a prethermalised stage of the evolution [23].Here we observe a new scenario associated to the notion of non-thermal fixed points. This is illustrated schematically in Fig. 1a: Starting from a class of farfrom-equilibrium initial conditions, the system develops a universal scaling behaviour in time and space. This is a consequence of the effective loss of details about initial conditions and system parameters long before a quasistationary or equilibrium situation may be reached. The transient scaling behaviour is found to be governed by the transport of an emergent collectiv...

A major challenge in quantum metrology is the generation of entangled states with macroscopic atom number. Here, we demonstrate experimentally that atomic squeezing generated via non-linear dynamics in Bose Einstein condensates, combined with suitable trap geometries, allows scaling to large ensemble sizes. We achieve a suppression of fluctuations by 5.3(5) dB for 12300 particles, which implies that similar squeezing can be achieved for more than 10 7 atoms. With this resource, we demonstrate quantum-enhanced magnetometry by swapping the squeezed state to magnetically sensitive hyperfine levels that have negligible nonlinearity. We find a quantum-enhanced single-shot sensitivity of 310 (47) [2][3][4][5]. Since state-of-the-art atom interferometers already operate at the classical limit for phase precision, given by the projection noise, quantum entangled input states are a viable route for further improving the sensitivity of these devices. One class of such states are spin squeezed states, which outperform the classical limit at a level given by the metrological spin squeezing parameter ξ R with ∆θ sq = ξ R · ∆θ cl [7, 8]. In the photonic case, quantumenhanced interferometry with squeezed states is routinely employed in optical gravitational wave detectors [9]. For atoms, proof-of-principle experiments have shown that spin squeezed states can be generated in systems ranging from high-temperature vapors to ultracold Bose-Einstein condensates (BEC) [10][11][12][13][14][15][16][17], surpassing the classical limit in atom interferometry [12, 16,18] and atomic clocks [19,20]. BECs are particularly well-suited for applications that require long interrogation times, high spatial resolution or control of motional degrees of freedom, such as in measurements of acceleration and rotation, due to their high phase-space density. However, in these systems scaling of the squeezed states to large particle numbers is intrinsically limited by density dependent losses, e.g. due to molecule formation. Keeping these processes negligible implies that for larger numbers the volume has to be increased, which in turn limits the generation of squeezed states due to uncontrolled nonlinear multimode dynamics. Here, we show how this limitation can be overcome by realizing an array of many individual condensates with an optical lattice, increasing the local trap frequencies but keeping the density small. In our experiment, we simultaneously prepare up to 30 independent BECs by superimposing a deep 1D optical lattice (period 5.5 µm) on a harmonic trap with large aspect ratio, which provides transverse confinement (see Fig. 1(a)). Each lattice site contains a condensate with N = 300 to 600 atoms in a localized spatial mode and the internal state |a = |F, m F = |1, 1 of the lowest hyperfine manifold. Using a two-photon radio frequency and microwave transition, we apply phase