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

160 Condensate fraction
12.1 Quantum depletion within Bogoliubov theory
Let us consider the problem in a 3D box of size L in x, y-direction and an optical lattice
with Nw sites oriented along z Note that the quantum depletion of the condensate has been
calculated in [123, 124] for different geometries. The quantum numbers of the elementary
excitations are the band index j and the quasi-momentum q along the z direction and the
momenta px and py in the transverse directions.
Within the framework of Bogoliubov theory, the quantum depletion of the condensate is
given by
∆Ntot
=
Ntot
1
dz dx dy |vj,q,px,py(r)|
Ntot j q,px,py,|p|=0
2 , (12.1)
where Ntot denotes the total number of atoms, ∆Ntot is the number of non-condensed parti-
cles, |p| =
p 2 x + p 2 y +¯h 2 q 2 and vj,q,px,py(r) are the Bogliubov v-amplitudes of the elementary
excitations. The sum runs over all bands j, over the quasi-momenta q in the first Brillouin
zone and the momenta of elementary excitations in the transverse directions px,py allowed by
the periodic boundary conditions
q = 2π
ν,
Nwd
ν=0, ±1, ±2, ..., ±Nw ,
2
(12.2)
px,py =¯h 2π
ν,
L
ν=0, ±1, ±2, ... . (12.3)
The Bogoliubov amplitudes vj,q,px,py(r) in Eq. (12.1) solve the Bogoliubov equations
− ¯h2
2m ∇2 + sER sin 2
− ¯h2
2m ∇2 + sER sin 2
πz
d
πz
d
+2dgn| ˜ϕ(z)| 2
− µ ujqpxpy(r)+gnd˜ϕ 2 vjqpxpy(r) =¯hωj(q)ujqpxpy (12.4)
+2dgn| ˜ϕ(z)| 2 − µ
vjqpxpy(r)+gnd ˜ϕ ∗2 ujqpxpy(r) =−¯hωj(q)vjqpxpy .(12.5)
They can be obtained from the three-dimensional time-dependent GPE
i¯h ∂Ψ(r,t)
= −
∂t
¯h2
2m ∇2 + sER sin 2
πz
+ g |Ψ(r,t)|
d
2
Ψ(r,t) , (12.6)
where the order parameter Ψ fulfills the normalization condition
dr |Ψ(r,t)| 2 = Ntot , (12.7)
by considering small time-dependent perturbations δΨ(r,t) of the groundstate Ψ0(r)
where
Ψ(r,t)=e −iµt/¯h [Ψ0(r)+δΨ(r,t)] , (12.8)
δΨ(r,t)=uσ(r)e −iωσt + v ∗ σ(r)e iωσt
as exemplified in section 7 for the case px = py =0.
(12.9)