In this article, we illustrate the scaling properties of a family of solutions for $N$ attractive bosonic atoms in the limit of large $N$. These solutions represent the quantized dynamics of solitonic degrees of freedom in atomic droplets. In dimensions lower than two, or $d=2-\epsilon$, we demonstrate that the number of isotropic droplet states scales as $N^{3/2}/{\epsilon }^{1/2}$, and for $\epsilon =0$, or $d=2$, scales as $N^2$. The ground-state energies scale as $N^{2/\epsilon +1}$ in $d=2-\epsilon$, and when $d=2$, scale as an exponential function of $N$. We obtain the universal energy spectra and the generalized Tjon relation; their scaling properties are uniquely determined by the asymptotic freedom of quantum bosonic fields at short distances, a distinct feature in low dimensions. We also investigate the effect of quantum loop corrections that arise from various virtual processes and show that the resultant lifetime for a wide range of excited states scales as $N^{\epsilon /2}{E}^{1-\epsilon /2}$.

• Received 10 June 2014

DOI:https://doi.org/10.1103/PhysRevA.90.063609