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We propose and analyse a pumped Bose-Hubbard dimer as a source of continuous-variable Einstein-Podolsky-Rosen (EPR) steering with non-Gaussian statistics. We use the truncated Wigner representation to calculate third and fourth order cumulants, finding clear signals of non-Gaussianity. We also calculate the products of inferred quadrature variances which indicate that states demonstrating the EPR paradox are present. Our proposed experimental configuration is extrapolated from current experimental techniques and adds another possibility to the current toolbox of quantum atom optics., 11 pages, 5 figures. arXiv admin note: text overlap with arXiv:1607.01911

We propose and analyse analogs of optical cavities for atoms using two-well Bose-Hubbard models with pumping and losses. With one well pumped, we find that both the mean-field dynamics and the quantum statistics show a quantitative dependence on the choice of damped well. Both the systems we analyse remain far from equilibrium, preserving good coherence between the wells in the steady-state. We find a small degree of quadrature squeezing and mode entanglement for some parameter regimes. Due to recent experimental advances, it should be possible to demonstrate the effects we investigate and predict., 15 pages, 6 figures

Labouvie \etal (\prl {\bf 115}, 050601, (20015)) recently demonstrated negative differential conductivity (NDC) in a multi-well Bose-Einstein condensate. They stated "we demonstrate that NDC originates from a nonlinear, atom number dependent tunneling coupling in combination with fast collisional decoherence." We show theoretically how the essential feature of NDC, a reduction in atomic current caused by an increase in chemical potential, is present in unitary dynamics through the well-known mechanism of macroscopic self-trapping (MST), and that the collisional decoherence merely serves as a quantitative modification of this., Comment: Submitted as a comment. A more extensive article is in preparation

We propose and analyse the use of a three-well Bose-Hubbard model for the creation of two spatially separated entangled atomic samples. Our three wells are in a linear configuration, with all atoms initially in the middle well, which gives some spatial separation of the two end wells. The evolution from the initial quantum state allows for the development of entanglement between the atomic modes in the two end wells. Using inseparability criteria developed by Hillery and Zubairy, we show how the detected entanglement is time dependent, oscillating with the well occupations. We suggest and analyse a method for preserving the entanglement by turning off the different interactions when it reaches its first maximum. We analyse the system with both Fock and coherent initial states, showing that the violations of the Hillery-Zubairy inequality exist only for initial Fock states and that the collisional nonlinearity degrades them. This system is an early step towards producing entangled atomic samples that can be spatially separated and thus close the locality loophole in tests of non-local quantum correlations., 12 pages, 4 figures. Rejection process has begun

We compare and contrast the entangling properties of a three-well Bose-Hubbard model and an optical beamsplitter. The coupling between the different modes is linear in both cases, and we may identify two output modes. Obvious differences are that our Bose-Hubbard model, with only the middle well initially occupied, does not have a vacuum input port, there is no equivalent of a collisional, $\chi^{(3)}$ nonlinearity with the beamsplitter, and the results of the Bose-Hubbard model show a time-dependence. In the non-interacting case, we obtain analytic solutions and show that, like a beamsplitter, the Bose-Hubbard system will not produce entanglement for classical initial states. We also show that whether inseparability or entanglement are detected depends sensitively on the criteria measured, with different criteria giving contradictory predictions., Comment: 16 pages, 6 figures, theory. arXiv admin note: text overlap with arXiv:1507.08451, arXiv:1506.03673

We consider how to generate and detect Einstein-Podolsky-Rosen (EPR) entanglement and the steering paradox between groups of atoms in two separated potential wells in a Bose-Einstein condensate (BEC). We present experimental criteria for this form of entanglement, and propose experimental strategies for detecting entanglement using two or four mode ground states. These approaches use spatial and/or internal modes. We also present higher order criteria that act as signatures to detect the multiparticle entanglement present in this system. We point out the difference between spatial entanglement using separated detectors, and other types of entanglement that do n}ot require spatial separation. The four-mode approach with two spatial and two internal modes results in an entanglement signature with spatially separated detectors, conceptually similar to the original EPR paradox., 15 pages, changes made

The SU(2) and SU(3) Lie algebras lend themselves naturally to studies of two- and three-well Bose-Einstein condensates, with the group operators being expressed in terms of bosonic annihilation and creation operators at each site. The success of these representations has led to the purities associated with these algebras to be promoted as a measure of entanglement between the atomic modes in each well. In this report, we show that these purities do not provide an unambiguous measure, but instead give results which depend on the quantum statistical states of each atomic mode. Using the example of totally uncoupled modes which have never interacted, we quantify these purities for different states and show that completely separable states can give values which have been claimed to indicate the presence of entanglement.

We introduce the study of dynamical quantum noise in Bose-Einstein condensates through numerical simulation of stochastic partial differential equations obtained using phase space representations. We derive evolution equations for a single trapped condensate in both the positive-$P$ and Wigner representations, and perform simulations to compare the predictions of the two methods. The positive-$P$ approach is found to be highly susceptible to the stability problems that have been observed in other strongly nonlinear, weakly damped systems. Using the Wigner representation, we examine the evolution of several quantities of interest using from a variety of choices of initial state for the condensate, and compare results to those for single-mode models., Comment: 8 figures, submitted to Phys. Rev. A