Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

168 Condensate fraction
depletion (12.62) can be rewritten in the form ∆Ntot/Ntot = a/ √ 2πσ in coincidence with the
expression for the depletion in a disc (12.33). This means that we are not dealing with an array
of separated two-dimensional condensates, but with one condensate only being distributed over
several sites. Equivalently, we can refer to the system as a coherent array of two-dimensional
condensates, each of them containing a non-condensed fraction
∆N
N
giving rise to a non-condensed fraction of the whole system
∆Ntot
Ntot
= 1
∆N
Nw N
l
= ã
, (12.63)
d
= ã
, (12.64)
d
where the fact that we can simply added up the number of non-condensed atoms at each site
is a non-trivial step presupposing the coherence of the system as a whole.
It is interesting to note that the depletion (12.62) is obtained by letting the tunneling
parametergotozero(δ → 0). Still, the resulting depletion takes values much smaller than
one provided that the confinement within each well does not become infinitely large giving rise
to a diverging ˜g and hence ã 1 . The reason for this is that before letting δ → 0 we have taken
the limit Ntot, L→∞,n= const.. This forces the system as a whole to be coherent since
the ratio between the Josephson tunneling energy EJ and the Josephson charging energy EC
(see Eqs.(10.34,10.49)) diverges
l
l
EJ/EC = Nκδ/2 →∞ (12.65)
for fixed κ, δ as N →∞, while the average density n is kept constant. This indicates that
coherence is maintained across the whole sample (see discussion in section 10.1).
In conclusion, we can say that the result (12.62) does not tell us what happens to the
condensate fraction when the system size and the total number of particles are kept fixed
while the lattice is made very deep. Surely, we expect the true result to differ from (12.62)
in this case: In connection with the occurence of number squeezing and the approach to the
superfluid-insulator transition the condensed fraction should become small as a signature of
the decoherence of the system breaking up into disconnected parts each one of them occupying
one lattice site.
1D thermodynamic limit
We have seen that in the thermodynamic limit, the quantum depletion is upper-bounded by
the usually very small quantity ã/d. If the continuum approximation (12.58) in the radial
1 If δ → 0 is achieved by letting s →∞,also˜g →∞and thus ã →∞(see Eq.(6.54) with σ given by
(6.50)). Yet, there are other ways of taking the limit δ → 0 which do not affect ˜g: For example, the tunneling
rate can be made zero by increasing d or by raising the barriers of the periodic potential while keeping the
potential unchanged close to its minima.