We study the coherent atomic tunneling between two zero-temperature BoseEinstein condensates (BEC) confined in a double-well magnetic trap. TwoGross-Pitaevskii equations for the self-interacting BEC amplitudes, coupled by a transfer matrix element, describe the dynamics in terms of the inter-well phase-difference and population imbalance. In addition to the anharmonic generalization of the familiar ac Josephson effect and plasma oscillations occurring in superconductor junctions, the non-linear BEC tunneling dynamics sustains a self-maintained population imbalance: a novel "macroscopic quantum self-trapping effect".

Quantum technologies exploit entanglement to revolutionize computing, measurements, and communications. This has stimulated the research in different areas of physics to engineer and manipulate fragile many-particle entangled states. Progress has been particularly rapid for atoms. Thanks to the large and tunable nonlinearities and the well developed techniques for trapping, controlling and counting, many groundbreaking experiments have demonstrated the generation of entangled states of trapped ions, cold and ultracold gases of neutral atoms. Moreover, atoms can couple strongly to external forces and light fields, which makes them ideal for ultra-precise sensing and time keeping. All these factors call for generating non-classical atomic states designed for phase estimation in atomic clocks and atom interferometers, exploiting many-body entanglement to increase the sensitivity of precision measurements. The goal of this article is to review and illustrate the theory and the experiments with atomic ensembles that have demonstrated many-particle entanglement and quantum-enhanced metrology.References 60

We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The Boson Josephson Junction (BJJ) dynamics is described by the two-mode non-linear Gross-Pitaevskii equation, that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of new dynamical regimes for the phase difference across the junction and for the population imbalance, that are not accessible with Superconductor Josephson Junctions (SJJ). These include oscillations with either, or both of the following properties: 1) the time-averaged value of the phase is equal to π (π − phase oscillations); 2) the average population imbalance is nonzero, in states with "macroscopic quantum self-trapping" (MQST). The (non-sinusoidal) gener-alization of the SJJ 'ac' and 'plasma' oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and total number of condensate atoms) onto a single universal curve, for the inverse period of oscillations.Analogies with Josephson oscillations between two weakly coupled reservoirs of 3 He-B and the internal Josephson effect in 3 He-A are also discussed.

We show that the quantum Fisher information provides a sufficient condition to recognize multiparticle entanglement in a N qubit state. The same criterion gives a necessary and sufficient condition for sub shot-noise phase sensitivity in the estimation of a collective rotation angle θ. The analysis therefore singles out the class of entangled states which are useful to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the non-linear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions. Introduction. The ability to create and manipulate entangled states of many-particle systems is a far-reaching possibility of quantum mechanics. Several efforts have been devoted, in the last few years, to exploit entanglement to design new technologies for secure communication, metrology and fast computation or to unveil foundational problems of quantum mechanics. From the experimental point of view, trapped Bose-Einstein condensates (BECs) [1,2], cold/thermal atoms [3] and trapped ions [4] are important candidates for the creation of large scale quantum entanglement. It is important to emphasize, however, that not all entangled states are equally useful for developing protocols that outperform classical operations. Generally speaking, current measures of entanglement mostly focus on the algebraic separability properties of quantum states. This notion should be extended for quantum technological applications, where it is essential to classify entanglement on the basis of some additional physical/algebraic properties required by the specific task. These attributes are crucially related with non-separability, but are not necessarily possessed by all entangled states.

We report on the direct observation of an oscillating atomic current in a one-dimensional array of Josephson junctions realized with an atomic Bose-Einstein condensate. The array is created by a laser standing wave, with the condensates trapped in the valleys of the periodic potential and weakly coupled by the interwell barriers. The coherence of multiple tunneling between adjacent wells is continuously probed by atomic interference. The square of the small-amplitude oscillation frequency is proportional to the microscopic tunneling rate of each condensate through the barriers and provides a direct measurement of the Josephson critical current as a function of the intermediate barrier heights. Our superfluid array may allow investigation of phenomena so far inaccessible to superconducting Josephson junctions and lays a bridge between the condensate dynamics and the physics of discrete nonlinear media.

We study the dynamical phase diagram of a dilute Bose-Einstein conden-

The Fisher information $F$ gives a limit to the ultimate precision achievable in a phase estimation protocol. It has been shown recently that the Fisher information for a linear two-mode interferometer cannot exceed the number of particles if the input state is separable. As a direct consequence, with such input states the shot-noise limit is the ultimate limit of precision. In this work, we go a step further by deducing bounds on $F$ for several multiparticle entanglement classes. These bounds imply that genuine multiparticle entanglement is needed for reaching the highest sensitivities in quantum interferometry. We further compute similar bounds on the average Fisher information $\bar F$ for collective spin operators, where the average is performed over all possible spin directions. We show that these criteria detect different sets of states and illustrate their strengths by considering several examples, also using experimental data. In particular, the criterion based on $\bar F$ is able to detect certain bound entangled states.Comment: Published version. Notice also the following article [Phys. Rev. A 85, 022322 (2012), DOI: 10.1103/PhysRevA.85.022322] by Geza T\'oth on the same subjec

Interferometers with atomic ensembles are an integral part of modern precision metrology. However, these interferometers are fundamentally restricted by the shot noise limit, which can only be overcome by creating quantum entanglement among the atoms. We used spin dynamics in Bose-Einstein condensates to create large ensembles of up to 10(4) pair-correlated atoms with an interferometric sensitivity -1.61(-1.1)(+0.98) decibels beyond the shot noise limit. Our proof-of-principle results point the way toward a new generation of atom interferometers.