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.

We demonstrate how the freedom in the definition of the positive-P representation may be used to reduce stochastic sampling errors, using the single-mode anharmonic oscillator as an example. In the modeling of this system, which is well known for being problematic, we achieve an improvement of more than 20 orders of magnitude in the distribution of the trajectories.

Asymmetric steering is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. This phenomenon is possible because of an asymmetry in the definition of the original Einstein-Podolsky-Rosen paradox and in this article we show theoretically that it may be demonstrated, at least in the case where Alice and Bob can only make Gaussian measurements, using the intracavity nonlinear coupler.

Recent experimental measurements of atomic intensity correlations through atom shot noise suggest that atomic quadrature phase correlations may soon be measured with a similar precision. We propose a test of local realism with mesoscopic numbers of massive particles based on such measurements. Using dissociation of a Bose-Einstein condensate of diatomic molecules into bosonic atoms, we demonstrate that strongly entangled atomic beams may be produced which possess Einstein-Podolsky-Rosen (EPR) correlations in field quadratures in direct analogy to the position and momentum correlations originally considered by EPR. DOI: 10.1103/PhysRevLett.95.150405 PACS numbers: 03.65.Ud, 03.75.Gg, 03.75.Pp, 42.50.Xa The recent demonstrations of atomic correlation measurements at the shot noise level [1,2] are a significant step towards true quantum-atom optics. Quantum optics, which began with photon correlation measurements, has allowed for many fundamental tests of quantum mechanics. Importantly, the availability of lasers allowed the development of techniques to perform quadrature phase measurements. In quantum-atom optics, Bose-Einstein condensates (BEC) play the role of the laser. However, homodyne noise correlation measurements of atomic field quadratures have not been available.In this Letter, we suggest one route to achieve this, and outline a scheme which would allow for fundamental tests of quantum mechanics with massive particles. We base our proposal on the Einstein-Podolsky-Rosen (EPR) paradox [3]. The EPR paper introduced two particles with perfect correlations (entanglement) in momenta and positions, these persisting with spatial separation. Depending on which property of one particle we choose to measure, we can predict with certainty the same observable of the other particle. EPR concluded that local realism was inconsistent with the completeness of the quantum mechanical description of nature. As suggested by Reid in 1989 [4], products of variances of inferred optical quadratures can demonstrate the paradox by seeming to violate the Heisenberg uncertainty relation, although this is impossible for directly observed quadratures. This is applicable to realistic correlations, and was demonstrated experimentally by Ou et al.[5] in 1992.We show here that dissociation [6 -9] of a molecular BEC can also exhibit EPR correlations in atomic quadratures. Such tests of quantum mechanics [see also Ref. [10] ] are a step toward understanding the properties of mesoscopic superpositions of massive particles, since they introduce couplings to gravitational fields, not previously tested in quantum measurement experiments.There has been much experimental progress [11,12] and intense theoretical interest [13][14][15][16][17][18][19][20][21][22][23] in the production of molecular dimers from Bose condensed atoms. We will assume that this can be created from bosonic constituents and propose a realization of the EPR paradox via dissociation, which can automatically yield two counterpropagating beams through momentum conservation...