# Bose-Einstein Condensation at higher critical temperature

The critical temperature $T_{c}$ of a Bose-Einstein Condensate is directly proportional to $n^\frac{2}{3}$, where $n$ is the density of the system which is to be condensed. The current $T_{c}$ for dilute atomic gases is in the nano-kelvin range. Has there been an attempt to achieve Bose-Einstein Condensate (BEC) by dropping the temperature till the micro-kelvin range and suddenly compressing it to increase the density?

• Ultracold atomic gases are closed, isolated systems to an excellent approximation. What will happen to the temperature if the gas is suddenly compressed? – Mark Mitchison Jul 23 '16 at 11:07
• You are right, the temperature of the gas would rise. However, would this rise in temperature be exactly such that the temperature of the gas would be pushed above the critical temperature at that density? – Abhijit Jul 23 '16 at 11:58
• Well, that is now a good question :) Have you tried to calculate this? It should be possible using undergraduate thermodynamics: the ideal gas approximation will be pretty good for dilute atomic gases above $T_c$. – Mark Mitchison Jul 23 '16 at 12:00
• I have not tried to calculate it and currently have no idea how to go about it. You are correct that the ideal gas approximation would be fine before the compression. However, while compressing, the inter-atomic interactions would play a huge role. Since now the particles are too close to each other, they might very well 'see' the interaction potential and the s-wave approximation would break down. I think this process would be key to calculate the rise in temperature. – Abhijit Jul 23 '16 at 12:06
• I don't think you are likely to see BEC if you compress the gas so much that the gas parameter $(na_s^3)^{1/3}$ is no longer small ($a_s$ is the $s$-wave scattering length). Probably you will just see runaway heating and lose all the atoms from your trap. If you want to guess the change in temperature you can neglect the interaction energy completely to a first approximation, which will at least estimate a lower bound on the temperature increase. Sorry I can't be of more help, I'm rather busy ATM. – Mark Mitchison Jul 23 '16 at 12:21