We study the dynamical phase diagram of a dilute Bose-Einstein condensate (BEC) trapped in a periodic potential. The dynamics is governed by a discrete nonlinear Schrödinger equation: intrinsically localized excitations, including discrete solitons and breathers, can be created even if the BEC's interatomic potential is repulsive. Furthermore, we analyze the Anderson-Kasevich experiment [Science 282, 1686 (1998)], pointing out that mean field effects lead to a coherent destruction of the interwell Bloch oscillations.

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.

In this review recent investigations are summarized of many-body quantum systems with long-range interactions, which are currently realized in Rydberg atom arrays, dipolar systems, trapped-ion setups, and cold atoms in cavities. In these experimental platforms parameters can be easily changed, and control of the range of the interaction has been achieved. The main aim of the review is to present and identify the common and mostly universal features induced by long-range interactions in the behavior of quantum many-body systems. Discussed are the case of strong nonlocal couplings, i.e., the nonadditive regime, and the one in which energy is extensive, but low-energy, long-wavelength properties are altered with respect to the short-range case. When possible, comparisons with the corresponding results for classical systems are presented. Finally, cases of competition with local effects are also reviewed.

We predict a dynamical classical superfluid-insulator transition in a Bose-Einstein condensate trapped in an optical and a magnetic potential. In the tight-binding limit, this system realizes an array of weakly coupled condensates driven by an external harmonic field. For small displacements of the parabolic trap about the equilibrium position, the condensates coherently oscillate in the array. For large displacements, the condensates remain localized on the side of the harmonic trap with a randomization of the relative phases. The superfluid-insulator transition is due to a discrete modulational instability, occurring when the condensate center of mass velocity is larger than a critical value.

We report the first experimental observation of nonlinear self-trapping of Bose-condensed ${}^{87}\mathrm{R}\mathrm{b}$ atoms in a one-dimensional waveguide with a superimposed deep periodic potential . The trapping effect is confirmed directly by imaging the atomic spatial distribution. Increasing the nonlinearity we move the system from the diffusive regime, characterized by an expansion of the condensate, to the nonlinearity dominated self-trapping regime, where the initial expansion stops and the width remains finite. The data are in quantitative agreement with the solutions of the corresponding discrete nonlinear equation. Our results reveal that the effect of nonlinear self-trapping is of local nature, and is closely related to the macroscopic self-trapping phenomenon already predicted for double-well systems.

Simulating electronic transport with atoms: Two superconductors connected by a bridge made out of nonsuperconducting material form a so-called Josephson junction (see the Perspective by Belzig). Valtolina et al. replaced the superconductors with two reservoirs of a superfluid Fermi gas and connected them by a weak link to allow atoms to move from one side to the other. Then they made one reservoir more populated than the other and studied the ensuing dynamics as a function of interaction strength between the atoms. In a related experiment, Husmann et al. kept the interaction strength at its maximum, but varied the temperature and the properties of the link. As temperature increased, the superfluid disappeared and thermal transport took over. Science , this issue p. 1498 , p. 1505 ; see also p. 1470