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

12.4 Tight binding regime 171
Quantum depletion and decoherence
Result (12.72,12.73) is linked to the coherence theory of 1D systems, where the off-diagonal
1-body density exhibits the power law decay
n (1) (|z − z ′
|) |z − z ′ −ν
|
→
(12.74)
n
ξ
at large distances. If the exponent ν is much smaller than 1, the coherence survives at large
distances
n (1) (|z − z ′
|) |z − z ′ −ν
|
|z − z ′
|
→
≈ 1 − ν ln
, (12.75)
n
ξ
ξ
and the application of Bogoliubov theory is justified
∆Ntot L
≈ ν ln ≪ 1 . (12.76)
Ntot ξ
For a superfluid, the value of ν in (12.74) is fixed by the hydrodynamic fluctuations of the
phase and is given, at T =0, by the expression (12.73) [1].
In terms of the Josephson parameters (10.34,10.49) we can also write
EC
ν = , (12.77)
Nν
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
8π 2 EJ
0
0 5 10 15 20 25 30
Figure 12.2: The quantity Nν (see Eq. (12.73) with N the number of particles per well) as a
function of the lattice depth s with m ∗ c as obtained for gn =0.5ER.
s