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

9.1 Macroscopic density and macroscopic superfluid velocity 117
In accordance with the basic idea introduced above, we assume nM to be approximately
constant on the length scale D in the z-direction at any given time t.
In analogy to the definition of the average density nl(r⊥), we introduce the average zcomponent
of the superfluid velocity field at site l
vzl(r⊥,t)= 1
ld+d/2
¯h
dz
d ld−d/2 m ∂zS(r⊥,z,t)
= ¯h 1
m d [S(r⊥,ld+ d/2,t) − S(r⊥,ld− d/2,t)] , (9.3)
where S(r⊥,z,t) is the phase of the order parameter and (¯h/m)∂zS(r⊥,z,t) is the zcomponent
of the superfluid velocity field obtained by solving the time-dependent GP-equation
in the presence of optical lattice and an additional slowly varying external potential as for example
a harmonic trap. The quantity (9.3) is required to be approximately constant over a
distance D. Hence, we can rewrite (9.3) in the form
vzl(r⊥,t)= ¯h 1
m D [S(r⊥,ld+ D/2,t) − S(r⊥,ld− D/2,t)] . (9.4)
Now, we go a step further and require the approximate stationary state in the window of size
D around the site l to be of the Bloch form ϕk = exp(iklz)˜ϕk (6.2) with quasi-momentum
kl(r⊥,t). Note that kl(r⊥,t) must be a slowly varying function of the index l and the variable
r⊥. The phase S in the window of size D around the site l is then approximately given by
S(r⊥,z,t)=kl(t)z + ˜ Sk(r⊥,z,t) , (9.5)
where the second contribution ˜ Sk is the phase of the Bloch wave ˜ϕk and thus ˜ Sk(r⊥,z,t)=
˜Sk(r⊥,z+ ld, t). Inserting (9.5) into (9.4) we find
vzl(r⊥,t)= ¯hkl(t)
m
¯h ˜Sjk(r⊥,ld+ D/2,t) −
+
m
˜ Sjk(r⊥,ld− D/2,t)
. (9.6)
D
Since ˜ Sjk is periodic in z with periodicity d, the second contribution becomes small for D>>d.
It can consequently be neglected for sufficiently large D and we are left with
vzl(r⊥,t)= ¯hkl(r⊥,t)
. (9.7)
m
We smooth (9.7) in the z-direction in the same way as the average density profile
l → z = ld ⇒ vzl(r⊥,t) → vMz(r⊥,z,t) , (9.8)
where the z-component of the macroscopic superfluid velocity field vM is given by
vMz(r⊥,z,t)= ¯hk(r⊥,z,t)
m
. (9.9)
The x, y-components vMx,vMy are given by the usual expressions (¯h/m)∂x,yS(r⊥,z,t).
It is important to note that vMz =¯hk/m does not coincide with the group velocity ¯vk
of a Bloch state condensate with quasi-momentum ¯hk. This is a very peculiar feature of a