We propose an experiment to measure the slow log(N ) convergence to mean-field theory (MFT) around a dynamical instability. Using a density matrix formalism, we derive equations of motion which go beyond MFT and provide accurate predictions for the quantum break-time. The leading quantum corrections appear as decoherence of the reduced single-particle quantum state.A Bose-Einstein condensate is described in mean field theory (MFT) by a c-number macroscopic wave function, obeying the Gross-Pitaevskii non-linear Schrödinger equation. MFT is closely analogous to the semiclassical approximation of single-particle quantum mechanics, with the inverse square root of the number N of particles in the condensate playing the role ofh as a perturbative parameter. Since in current experimental condensates N is indeed large, it is generally difficult to see qualitatively significant quantum corrections to MFT. In the vicinity of a dynamical instability in MFT, however, quantum corrections appear on timescales that grow only logarithmically with N . In this paper we propose an experiment to detect such quantum corrections, and present a simple theory to predict them. We show that, as the Gross-Pitaevskii classical limit of a condensate resembles single-particle quantum mechanics, so the leading quantum corrections appear in the single-particle picture as decoherence.We will consider a condensate in which particles can only effectively populate two second-quantized modes. This model can be realized with a condensate in a double well potential [1-6], or with an effectively two-component spinor condensate [7,8] whose internal state remains uniform in space. Such uniformity can be ensured, to a good approximation, by confining a very cold condensate within a size much smaller than [n| √ a 11 a 22 − a 12 |] −1/2 , where n is the mean total density and a ij is the s-wave scattering length between atoms in internal states i and j. The kinetic energy of spin non-uniformity then ensures that the spatial state of the condensate will adiabatically follow its internal state, which will evolve on slower time scales. Dynamical instabilities to phase separation are also frustrated in this regime, which since for available alkali gases all a ij differ only by a few percent, could be reached with small condensates (N ≤ 10 4 ) in weak, nearly spherical traps (ω ≤ 100 Hz). Stronger or less isotropic traps reach the two-mode regime at smaller N .In the double well realization, the nonlinear interaction may be taken to affect only atoms within the same well. In this case single-particle tunneling provides a linear coupling between the two modes, which can in principle be tuned over a wide range of strengths. Two internal states may be coupled by a near-resonant radiation field [9,10]. If collisions do not change spin states, there is also a simple nonlinear interaction in the internal realization.

Starting with a Gaussian variational ansatz, we predict anisotropic bright solitons in quasi-2D Bose-Einstein condensates consisting of atoms with dipole moments polarized perpendicular to the confinement direction. Unlike isotropic solitons predicted for the moments aligned with the confinement axis [Phys. Rev. Lett. 95, 200404 (2005)10.1103/PhysRevLett.95.200404], no sign reversal of the dipole-dipole interaction is necessary to support the solitons. Direct 3D simulations confirm their stability.

We study the dynamics of a two-mode Bose-Einstein condensate in the vicinity of a mean-field dynamical instability. Convergence to mean-field theory (MFT), with increasing total number of particles N , is shown to be logarithmically slow. Using a density matrix formalism rather than the conventional wavefunction methods, we derive an improved set of equations of motion for the mean-field plus the fluctuations, which goes beyond MFT and provides accurate predictions for the leading quantum corrections and the quantum break time. We show that the leading quantum corrections appear as decoherence of the reduced single-particle quantum state; we also compare this phenomenon to the effects of thermal noise. Using the rapid dephasing near an instability, we propose a method for the direct measurement of scattering lengths.

We study the system of coupled atomic and molecular condensates within the two-mode model and beyond mean-field theory (MFT). Large amplitude atom-molecule coherent oscillations are shown to be damped by the rapid growth of fluctuations near the dynamically unstable molecular mode. This result contradicts earlier predictions about the recovery of atom-molecule oscillations in the two-mode limit. The frequency of the damped oscillation is also shown to scale as √ N / log N with the total number of atoms N , rather than the expected pure √ N scaling. Using a linearized model, we obtain analytical expressions for the initial depletion of the molecular condensate in the vicinity of the instability, and show that the important effect neglected by mean field theory is an initially non-exponential 'spontaneous' dissociation into the atomic vacuum. Starting with a small population in the atomic mode, the initial dissociation rate is sensitive to the exact atomic amplitudes, with the fastest (super-exponential) rate observed for the entangled state, formed by spontaneous dissociation.Recent photoassociation [1] and Feshbach resonance [2,3] experiments suggest the possibility of producing molecular Bose-Einstein condensates (BEC) [4][5][6][7][8][9]. Large amplitude coherent oscillations between an atomic BEC and a molecular BEC have been theoretically predicted [4][5][6]. A common theme to these studies is the use of the Gross-Pitaevskii (GP) mean-field theory (MFT), reducing the full multi-body problem into a set of two coupled nonlinear Schrödinger equations. These are then solved numerically to obtain the Josephson-type dynamics of the coupled atomic and molecular fields.The simple GP dynamics is substantially affected by condensate depletion due to inelastic collisions [5,7,10], spontaneous emission, and the inclusion of noncondensate modes [10][11][12][13][14]. Consequently, the atommolecule oscillations are expected to be strongly damped under current experimental conditions. The proposed remedy for this detrimental effect involves a recovery of an effective two-mode dynamics [13], thereby preventing the buildup of thermal population.In this article we point out that even in the perfect twomode limit, MFT fails to provide long-term predictions due to strong interparticle entanglement near the dynamically unstable molecular mode. Quantum corrections to MFT appear in the vicinity of its dynamical instabilities, on time scales that grow only logarithmically with the number N of condensate particles [15][16][17]. Thus, even in the absence of a 'proper' thermal bath of noncondensate states, the mean-field equations are coupled to a reservoir of Bogoliubov fluctuations [16,18]. The rapid growth of these fluctuations near the instability is analogous to the rapid population of the thermal cloud, similarly inhibiting the mean-field atom-molecule oscillations. Our results, obtained using the numerical solution of exact quantum equations, go beyond the HartreeFock-Bogoliubov approach [12]. The leading quantum effect is identi...

We employ a semiclassical picture to study dynamics in a bosonic Josephson junction with various initial conditions. Phase-diffusion of coherent preparations in the Josephson regime is shown to depend on the initial relative phase between the two condensates. For initially incoherent condensates, we find a universal value for the buildup of coherence in the Josephson regime. In addition, we contrast two seemingly similar on-separatrix coherent preparations, finding striking differences in their convergence to classicality as the number of particles increases.Comment: 18 pages, 8 figures, improved version, pedagogical orientation, 3 extra appendices that are not included in the published versio