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

10.2 Lowest Bogoliubov band 137
µl
∆Sl
˙ = −
δµ
+
¯h ¯h
1 ∂µl
−
¯h ∂nl
nl=n
∆nl +
nl=n
l ′ =l+1,l−1
δ +2n ∂δ
∂n
4n¯h
∂δ δ − 2n ∂n
∆nl ′ −
4n¯h ∆nl
. (10.20)
To obtain (10.19) we have taken the value of δl,l′ at equilibrium since the dependence on ∆Sl
is of first order. Instead, to get Eq.(10.20) one has to expand δ l,l′
µ to first order in the density
fluctuations ∆nl
δ l,l′
µ ≈ δµ + ∂δl,l′
µ
∆nl +
∂nl
nl=n
∂δl,l′ µ
∂nl ′
∆nl
nl=n
′
= δµ + 3 ∂δ
2 ∂n ∆nl + 1 ∂δ
∆nl ′ , (10.21)
2 ∂n
where in the last step we have neglected terms involving ∂f/∂n, as done previously in sections
6.2 and 7.3. Taking the derivative of (10.20) with respect to time and and inserting (10.19)
we find
∆Sl
¨
1 ∂µl nδ
= −
¯h ∂nl nl=n ¯h (2∆Sl − ∆Sl+1 − ∆Sl−1)
+ nδ
¯h 2
∂δ δ +2n∂n [(2∆Sl+1 − ∆Sl+2 − ∆Sl)+(2∆Sl−1−∆Sl − ∆Sl−2)]
4n
∂δ δ − 2n ∂n (2∆Sl − ∆Sl+1 − ∆Sl−1) . (10.22)
4n
−2 nδ
¯h 2
This equation is solved by
with
¯hω(q) = 2δ sin2
qd
2
2
∆Sl ∝ e i(lqd−ω(q)t) , (10.23)
δ +2n ∂δ
∂n
sin 2
qd
+2n
2
∂µl
∂n
∂δ
− 4n . (10.24)
∂n
Recalling the tight binding expression for the inverse compressibility (see Eq.(6.43)) we can
rewrite (10.24) in the form
¯hω(q) =
2δ sin 2
qd
2
2
δ +2n ∂δ
∂n
sin 2
qd
+2κ
2
−1
, (10.25)
in agreement with the result (7.38) obtained from the solution of the Bogoliubov equations
(7.14,7.15).
It is interesting to note that if we initially set √ nlnl ′ = nl in Eq.(10.11) we obtain the
result
¯hω(q) =2 √ δκ−1
sin
qd
2
(10.26)
for the dispersion of small groundstate perturbations. This proves that the first term in
the brackets of Eq.(10.25) has its physical origin in the fluctuations of the site occupations
occurring on a few-sites length scale which are excluded when setting √ nlnl ′ = nl. Note that