Trapped Bose-Einstein Condensates with Attractive s-wave Interaction

Within the framework of the Gaussian-state theory, we show that the quantum many-body ground state of a trapped condensate with weakly attractive interaction is a single-mode squeezed vacuum state, as oppose to the coherent state under repulsive interaction. The spatial mode of the squeezed-state condensates satisfies a Gross-Pitaevskii like equation in which the interaction strength is augmented by a factor $3$ due to the large particle fluctuation of the squeezed state. We also study the collective excitations of the condensates by the tangential space projection, which leads to new two-particle excitations and confirms the phase transition from coherent-state to squeezed-state condensates. Our investigation clarifies the quantum states of the attractive condensates and will shed new light on research of the droplet phases in dipolar and multicomponent condensates.