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Approximate theories for binary magnetic quantum droplets

We develop two approximate theories to describe the miscible and immiscible droplets that can occur in a binary mixture of highly magnetic bosonic atoms. In addition to allowing simpler calculations, the approximate theories provide insight into the role of quantum fluctuations in the two regimes. Results are validated by comparison to those from the extended Gross-Pitaevskii equation. As an application we solve for the ground state droplets crossing the miscible-immiscible transition as function of the short-ranged interspecies interaction parameter. We consider regimes where the transition occurs suddenly or as a smooth cross-over. Using dynamical calculations we show that the character of the transition is revealed in the number of domains produced when ramping the droplet into the immiscible regime.