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I was just wondering about the role and effects of the trap in formation of BE condensate. In this respect if we have two potential, one small harmonic potential of energy E/10 and length L/100 inside a larger harmonic potential of energy E and Length L. Where is the BEC likely to be formed out of these two potentials ? This is just a hypothetical problem that came to mind. Can anyone share their insights regarding this ? Thanks in advance.

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A BEC is formed when the de Broglie wavelength of the atoms $\lambda_{\text{dB}}$ are on the order of the interparticle separation ($\sim 1/n$, where $n$ is the density, so particles/volume). This is expressed in a dimensionally correct form as: $$ D = n\lambda_{\text{dB}}^3,$$ known as the "degeneracy parameter", which quantifies phase-space density (position * momentum). BEC occurs when $D \sim 1$.

$\lambda_{\text{dB}} \propto 1/\sqrt{T}$, so going in temperature makes $D$ up. But going denser also make $D$ go up.

The BEC phase transition depends on both temperature and atom nmber density. So the trap is used not only to make sure your atoms don't fly away (that it, the trap depth is higher than the highest atomic energy), but also to make the peak density enough to allow for BEC. This is why in experiments sometimes people use a "dimple" trap, where they add something that makes a localised region of space have higher density, to trigger BEC around that point.

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  • $\begingroup$ Thank you for your explanation. So you mean to say that the trap with higher density will have greater chances of BEC formation, given both the potential have trap depth higher than the highest atomic energy ? $\endgroup$
    – AdShil00
    Sep 26, 2022 at 7:16

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