BEC represents order in momentum space, which is fictitious. It occurs at k = 0 (or, more precisely, at the lowest momentum state, depending on the observer's reference frame). This corresponds to the infinite- (or large-) r limit in configuration space. How exactly? How does no (the condenate fraction) appear in this space (which is related to momentum space through, of course, a Fourier transform)?
BEC in experiments with ultracold dilute gases of atoms are done inside atom traps. You can think of these as a harmonic trapping potential in real space. In this case, condensation occurs into the lowest energy eigenstate (neglecting atom-atom interaction for arguments sake), or in other words into the ground state of the harmonic trap. This state has a finite width both in real space and in momentum space.
"An Infinite range static(Coulomb field-like) configuration" ---What else can it be?
Regards,
Rajat
BEC in experiments with ultracold dilute gases of atoms are done inside atom traps. You can think of these as a harmonic trapping potential in real space. In this case, condensation occurs into the lowest energy eigenstate (neglecting atom-atom interaction for arguments sake), or in other words into the ground state of the harmonic trap. This state has a finite width both in real space and in momentum space.
Quite often spatial effects are represented in physics by correlation functions. If we are talking about a uniform system rather than a trapped gas, then to see the effects of Bose-Einstein condensation one can look at the off-diagonal terms of the one-body density matrix in configuration space. The largest eigenvalue of the OBDM is the occupation number, n0, of the zero momentum state. The OBDM goes to a constant at large separation r - r'; the value of this limit is also n0. We say we then have off-diagonal long-range order when there is a Bose-Einstein condensate.
For a superconductor one has to look at the correlations in the two-body density matrix because it is Cooper pairs that form the "condensate".
Dear Dr. Mullin,
Many thanks for such a clear answer; clarity in both thought and expression has always been a characteristic of your work. While I am generally aware of the beautiful ODLRO literature (I even assign problems about the subject to graduate students), I believe that a dramatic 'event' like Bose-Einstein condensation should equally dramatically manifest itself in real space (I mean experimentally). My attempt some time ago to 'see' no in the large-r limit of g(r) did not exactly succeed! Any further illuminating ideas about this subject?
I take this opportunity to thank all other respondents for their well-appreciated answers.
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