Quantum fluctuations in atomic Josephson junctions: the role of dimensionality

We investigate the role of quantum fluctuations in the dynamics of a bosonic Josephson junction in $D$ spatial dimensions, by using beyond mean-field Gaussian corrections. We derive some key dynamical properties in a systematic way for $D=1, 2, 3$, namely, we compute the Josephson frequency in the low population imbalance limit, and we obtain the critical strength of the macroscopic quantum self-trapping. Our results show that the quantum corrections increase Josephson frequency in the $D=2, 3$ case, and a decrease in the $D=1$ case. Also, we show that the macroscopic quantum self-trapping critical strength is decreased in the $D=2, 3$ case, and increased in the $D=1$ case with respect to the mean-field calculations. We show that the difference between the cases of $D=2$ and $D=3$ on one side, and $D=1$ on the other, can be related to the qualitatively different dependence of the interaction strength on the scattering length in the different dimensions.