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I would like to understand what quantum fluctuation really means. I think it's the particles that are not in the ground state. Am I right? But then, what are the differences between quantum fluctuations and thermal fluctuations?

Another question. How do these quantum fluctuations stabilize the BEC against the mean-field collapse? I've read something about soft and hard excitations, but I do not know how these stabilize the BEC.

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    $\begingroup$ Quantum fluctuations are not from a system being out of the ground state. "Quantum fluctuations" refers to the fact that, for example, $\langle 0 | \hat{x}^2 | 0 \rangle \neq 0$, i.e. the mean square value of an operator in the ground state may not be zero. $\endgroup$ – DanielSank Oct 19 '18 at 19:48
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Dinesh, this subject is also that I do not understand in detail and would like to understand better (especially the LHY correction), but I can say this much: BEC with attractive interaction are unstable in general. Atoms will rush to clump on top of another and decay via three-body loss. However, if there is another, repulsive interaction, which is effected by quantum fluctuation (to which the LHY correction is associated) , it will cancel out the attractive interaction at some length scale. Hence the BEC stabilizes into small droplets as per the paper you cite.

Note that the dipolar interaction (which is studied in your reference) is not crucial. If I recall correctly, its role is to provide enough repulsive interaction to reduce the large attractive s-wave interaction, so that the net interaction (without quantum fluctuation) is a small attractive interaction. LHY correction is quite tiny, so you wouldn't observe its effect unless the attractive term you are trying to balance is already small.

Here is an example of stable BEC droplet without dipolar interaction, by researchers from ICFO: https://arxiv.org/abs/1708.07806

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