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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.

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The first thing to realise is that, although we often talk about BEC as a low-temperature equilibrium state, in dilute alkali gases a BEC is a highly excited, non-equilibrium, metastable state of the system. While this may sound strange at first, it becomes rather obvious after a moment's thought: the ground state of a collection of lithium atoms is not a gas, it's a block of metal! However, in order for a BEC to form, we need the kinetic energy of the atoms to be small, so that the thermal de Broglie wavelength is larger than the mean interatomic distance. It is in this sense that we talk about the temperature being "small". Such a metastable gaseous state at low temperature is only possible at low density, so that the probability of a collision in which two atoms bind together to form a molecule is negligible.

In ultracold atomic gases this is usually achieved by evaporative cooling. Here, the depth of the atoms' confining potential is decreased extremely slowly, allowing only the most energetic atoms to escape. This eventually leads to a state of very low density but also low kinetic energy, such that the condition for BEC is obtained.

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