Consistent theory of self-bound quantum droplets with bosonic pairing

We revisit the Bogoliubov theory of quantum droplets proposed by Petrov {[}Phys. Rev. Lett. $\mathbf{\text{115}}$, 155302 (2015){]} for an ultracold Bose-Bose mixture, where the mean-field collapse is stabilized by the Lee-Huang-Yang quantum fluctuations. We show that a loophole in Petrov’s theory, i.e., the ignorance of the softening complex Bogoliubov spectrum, can be naturally removed by the introduction of bosonic pairing. The pairing leads to weaker mean-field attractions, and also stronger Lee-Huang-Yang term in the case of unequal intraspecies interactions. As a result, the equilibrium density for the formation of self-bound droplets significantly decrease in the deep droplet regime, in agreement with a recent observation from diffusion Monte Carlo simulations. Our construction of a consistent Bogoliubov theory paves the way to understand the puzzling low critical number of small quantum droplets observed in the experiment {[}Science $\mathbf{\text{359}}$, 301 (2018){]}.