Stabilization of Bose-Bose mixtures on a spherical surface induced by the Lee-Huang-Yang correction

Motivated by recent experimental advances in bubble traps which confine atoms on a thin shell, we investigate the stabilization of droplet states of a Bose-Bose mixture on a spherical surface due to quantum fluctuations. We analytically calculate the energy of a weakly interacting Bose-Bose mixture on a spherical surface with attractive interspecies and repulsive intraspecies interactions using the Bogoliubov approximation. Our findings show that the energy per particle has a minimum at a finite density in the presence of the Lee-Huang-Yang (LHY) term, which indicates the presence of liquidlike droplet state stabilized by quantum fluctuations. We also numerically solve the extended Gross-Pitaevskii equation, incorporating both bubble traps and an additional repulsive LHY correction derived from the Bogoliubov spectrum of a three-dimensional homogeneous mixture. Our calculations show that, due to the size limitations of the sphere, droplet states can only exist within specific parameter ranges. We also verify the transition between droplet states and uniform condensates in spherical confinement potentials, as well as the coexistence of the two. Our results pave the way for future exploration of quantum droplets in curved geometries with nontrivial real-space topologies.