We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The Boson Josephson Junction (BJJ) dynamics is described by the two-mode non-linear Gross-Pitaevskii equation, that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of new dynamical regimes for the phase difference across the junction and for the population imbalance, that are not accessible with Superconductor Josephson Junctions (SJJ). These include oscillations with either, or both of the following properties: 1) the time-averaged value of the phase is equal to π (π − phase oscillations); 2) the average population imbalance is nonzero, in states with "macroscopic quantum self-trapping" (MQST). The (non-sinusoidal) gener-alization of the SJJ 'ac' and 'plasma' oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and total number of condensate atoms) onto a single universal curve, for the inverse period of oscillations.Analogies with Josephson oscillations between two weakly coupled reservoirs of 3 He-B and the internal Josephson effect in 3 He-A are also discussed.

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By solving numerically the tunneling quantum dynamics between two trapped Bose-Einstein condensates ͑BEC's͒, we find two distinct time scales. On the short time scale, we recover semiclassical predictions such as ''macroscopic quantum self-trapping'' and the ''amplitude transition,'' first studied in the polaron context, and the '' states,'' previously discovered in the context of BEC's. On a much longer time scale, quantum dynamics shows that self-trapping is destroyed in contrast to semiclassical behavior. However, this time scale increases exponentially with the total number of condensate atoms, indicating that self-trapping and '' states'' would be relevant and experimentally observable in the tunneling of BEC's. ͓S1050-2947͑99͒51409-1͔ PACS number͑s͒: 03.75.Fi, 05.30.Jp, 32.80.Pj, 74.50.ϩr Bose-Einstein condensation in weakly interacting alkalimetal gases was detected initially by several groups ͓1͔. The precise manipulation of these Bose-Einstein condensates ͑BEC's͒ ͓2,3͔ has raised the possibility of tailoring these new BEC systems to a degree not possible with superfluid systems such as 4 He, 3 He, as well as with superconductors. The experimental demonstration of spatial coherence through the observation of interference fringes in two overlapping condensates ͓4͔ and the measurement of the relative phase of two condensates in different hyperfine spin states ͓5͔ naturally raise the question of measurement and exploitation of temporal phase coherence by means of a Josephson junction between two condensates. Aspects of the question have already been theoretically addressed in the context of BEC in the limit of noninteracting atoms ͓6͔ and for small-amplitude Josephson oscillations ͓7,8͔. The semiclassical mean-field dynamics using the Gross-Pitaevskii equation has been studied and interesting phenomena such as macroscopic quantum self-trapping discussed by Milburn et al. ͓9͔ and Smerzi et al. ͓10,11͔, and the existence of states and oscillations ͑dynamical states, wherein the time-averaged quantum phase difference across the junction equals ), have been predicted by Smerzi et al. ͓10,11͔ in a weakly coupled double BEC forming a boson Josephson junction. Similar studies have been conducted to investigate driven two-component BEC ͓12͔. Finite-temperature effects describing damping have also been studied ͓8,13͔. Quantum corrections have been included to describe collapse and revival sequences ͓9,14͔, phase decoherence and dephasing of Josephson oscillations ͓15-17͔, phase squeezing ͓18͔, and phase diffusion and renormalization of oscillation frequencies ͓14͔. The approaches to include quantum corrections that gave rise to collapses and revivals and departures from semiclassical dynamics ͓9,14͔ have been different and complementary.The Gross-Pitaevskii equation describing the mean-field dynamics of a BEC is formally identical to the nonlinear Schrödinger equation that has appeared earlier in other fields. Therefore, many of the results, methods, and insight can be fruitfully applied from those other fields t...

Bose-Einstein condensates in a double-well trap, as well 3 He-B baths connected by micropores, have been shown to exhibit Josephson-like tunneling phenomena. Unlike the superconductor Josephson junction of phase difference that maps onto a rigid pendulum of energy cos(), these systems map onto a momentumshortened pendulum of energy Ϫͱ1Ϫp 2 cos() and length ͱ1Ϫp 2 , where p is a population imbalance between the wells/baths. We study here the effect of damping on the four distinct modes of the nonrigid pendulum, characterized by distinct temporal mean values, ͗͘ and ͗p ͘. Damping is shown to produce different decay trajectories to the final equilibrium ϭ0ϭp state that are characteristic dynamic signatures of the initial oscillation modes. In particular, damping causes -state oscillations with ͗͘ϭ to increase in amplitude and pass through phase-slip states, before equilibrating. Similar behavior has been seen in 3 He-B experiments.