We report on the first realization of a single bosonic Josephson junction, implemented by two weakly linked Bose-Einstein condensates in a double-well potential. In order to fully investigate the nonlinear tunneling dynamics we measure the density distribution in situ and deduce the evolution of the relative phase between the two condensates from interference fringes. Our results verify the predicted nonlinear generalization of tunneling oscillations in superconducting and superfluid Josephson junctions. Additionally, we confirm a novel nonlinear effect known as macroscopic quantum self-trapping, which leads to the inhibition of large amplitude tunneling oscillations.

We consider a two-component one-dimensional model of gap solitons (GSs), which is based on two nonlinear Schrödinger equations, coupled by repulsive XPM (cross-phase-modulation) terms, in the absence of the SPM (self-phase-modulation) nonlinearity. The equations include a periodic potential acting on both components, thus giving rise to GSs of the "symbiotic" type, which exist solely due to the repulsive interaction between the two components. The model may be implemented for "holo-graphic solitons" in optics, and in binary bosonic or fermionic gases trapped in the optical lattice. Fundamental symbiotic GSs are constructed, and their stability is investigated, in the first two finite bandgaps of the underlying spectrum. Symmetric solitons are destabilized, including their entire family in the second bandgap, by symmetry-breaking perturbations above a critical value of the total power. Asymmetric solitons of intra-gap and inter-gap types are studied too, with the propagation constants of the two components falling into the same or different bandgaps, respectively. The increase of the asymmetry between the components leads to shrinkage of the stability areas of the GSs. Inter-gap GSs are stable only in a strongly asymmetric form, in which the first-bandgap component is a dominating one. Intra-gap solitons are unstable in the second bandgap. Unstable two-component GSs are transformed into persistent breathers. In addition to systematic numerical considerations, analytical results are obtained by means of an extended ("tailed") Thomas-Fermi approximation (TFA).

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

Interference is fundamental to wave dynamics and quantum mechanics. The quantum wave properties of particles are exploited in metrology using atom interferometers, allowing for high-precision inertia measurements. Furthermore, the state-of-the-art time standard is based on an interferometric technique known as Ramsey spectroscopy. However, the precision of an interferometer is limited by classical statistics owing to the finite number of atoms used to deduce the quantity of interest. Here we show experimentally that the classical precision limit can be surpassed using nonlinear atom interferometry with a Bose-Einstein condensate. Controlled interactions between the atoms lead to non-classical entangled states within the interferometer; this represents an alternative approach to the use of non-classical input states. Extending quantum interferometry to the regime of large atom number, we find that phase sensitivity is enhanced by 15 per cent relative to that in an ideal classical measurement. Our nonlinear atomic beam splitter follows the 'one-axis-twisting' scheme and implements interaction control using a narrow Feshbach resonance. We perform noise tomography of the quantum state within the interferometer and detect coherent spin squeezing with a squeezing factor of -8.2 dB (refs 11-15). The results provide information on the many-particle quantum state, and imply the entanglement of 170 atoms.

Entanglement, a key feature of quantum mechanics, is a resource that allows the improvement of precision measurements beyond the conventional bound reachable by classical means [1]. This is known as the standard quantum limit, already defining the accuracy of the best available sensors for various quantities such as time [2] or position [3,4]. Many of these sensors are interferometers in which the standard quantum limit can be overcome by feeding their two input ports with quantum-entangled states, in particular spin squeezed states [5,6]. For atomic interferometers, Bose-Einstein condensates of ultracold atoms are considered good candidates to provide such states involving a large number of particles [7]. In this letter, we demonstrate their experimental realization by splitting a condensate in a few parts using a lattice potential. Site resolved detection of the atoms allows the measurement of the conjugated variables atom number difference and relative phase. The observed fluctuations imply entanglement between the particles [7,8,9], a resource that would allow a precision gain of 3.8 dB over the standard quantum limit for interferometric measurements.Spin squeezing was one of the first quantum strategies proposed to overcome the standard quantum limit in a precision measurement [5,6] that triggered many experiments [10,11,12,13,14,15,16,17]. It applies to measurements where the final readout is done by counting the occupancy difference between two quantum states, as in interferometry or in spectroscopy. The name "spin squeezing" originates from the fact that the N particles used in the measurement can be described by a fictitious spin J = N/2. In an interferometric sequence, the spin undergoes a series of rotations where one of the rotation angles is the phase shift to be measured. A sufficient criterion for the input state allowing for quantum enhanced metrology is given by ξ S < 1 where ξis the squeezing parameter introduced in ref. [6]. The fluctuations of the spin in one direction have to be reduced below shot-noise ∆J 2 z < J/2, and the spin polarization in the orthogonal plane J x 2 + J y 2 has to be large enough to maintain the sensitivity of the interferometer. A pictorial representation of this condition is shown in figure 1b. The precision of such a quantum enhanced measurement is ξ S / √ N , whereas the standard quantum limit set by shot-noise is 1/ √ N . In this Letter, we report on the observation of entangled squeezed states in a Bose-Einstein condensate of 87 Rb atoms. The particles are distributed over a small number of lattice sites (between 2 and 6) in a one dimensional optical lattice (see figure 1a). The occupation number per site ranges from 100 to 1100 atoms. The two modes supporting the squeezing are two states of the external atomic motion corresponding to the condensate meanfield wave-functions in two adjacent lattice sites. These modes are spatially well separated and thus represent an ideal starting condition for a spatially split interferometer. Labeling a † and b † the creation o...