# Why is the density of a BEC so low?

I've just begun reading C. Pethick and H. Smith's textbook "Bose-Einstein condensation in dilute gases" (Cam. Uni. Press). In the Introduction, they contrast the density of atoms at the centre of a BEC cloud to other phases of matter. To quote from the text (pg 1, 2nd ed.):

The particle density at the centre of a Bose-Einstein condensed atomic cloud is typically 10^13–10^15 cm^−3. By contrast, the density of molecules in air at room temperature and atmospheric pressure is about 10^19 cm−3.

This has puzzled me, as instinctively I think the density of the BEC should be higher because the atoms are macroscopically occupying the ground state and would have a smaller separation between atoms than a 'warm' classical gas.

Why is the density of a BEC lower than normal phases, despite the large degree of occupation?

Answers on this thread state it is because of the large positional uncertainty of the atoms in the BEC: https://www.quora.com/Why-is-the-density-of-a-BEC-said-to-be-low-while-the-distance-between-atoms-is-low. But as far as I can see, this will only ensure the density of the BEC is uniform.

As a beginner to the subject, I would greatly appreciate if someone can help me to resolve my conceptual difficulty.