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

118 Macroscopic Dynamics
condensate in a lattice. Note that group velocity and superfluid velocity are related to the
superfluid current I by the relations I = n¯v and I = nsvM respectively, where n is the total
density and ns is the superfluid density. The decrease of the superfluid current I due to the
lattice brings about a decrease of the group velocity and of the superfluid density rather than
of the superfluid velocity. It is this effect which underlies the difference between the expression
(9.7) and the group velocity.
We can also define a macroscopic phase SM through the relation
vM(r⊥,z)= ¯h
m ∇SM(r⊥,z) (9.10)
This fixes the quantity SM up to an irrelevant constant.
The introduction of the quantities nM and SM allows us to speak of an effective macroscopic
order parameter
whose evolution in time we are interested in.
ΨM = √ nM e iSM , (9.11)
9.2 Hydrodynamic equations for small currents
Using the notions of macroscopic density and macroscopic superfluid velocity we devise a
hydrodynamic formalism which is valid to the dynamics involving only small currents. This
restriction signifies that only small quasimomenta ¯hk are involved in the dynamics which are
associated with currents nM¯hk/2m ∗ . The generalization to large currents is done in the
subsequent section.
Macroscopic energy functional
The energy change per particle due to the presence of a small constant current in z-direction
is ¯h 2 k2 /2m∗ = m2v2 Mz /2m∗ , where we have used the expression (9.9) for the macroscopic
superfluid velocity. Using this fact, the total energy of the system can be written in terms of
the macroscopic density nM and superfluid velocity field vM
E =
m
dr
2 nMv 2 Mx + m
2 nMv 2 My + m
2
m
m∗ (nM) nMv 2
Mz + nMε(nM)+nMVext , (9.12)
for the total energy of the system. Here, Vext is an external potential supposed to vary on
length scales much larger than D and ε(nM) and m ∗ (nM) are, respectively, the groundstate
energy and the effective mass calculated at the average density nM in absence of the external
potential (see sections 5.1 and 6.1). We can rewrite the integrand of (9.12) in terms of the
density nM and the phase SM defined in (9.10). This yields
E = dr
nM
¯h 2 2 ∂SM
+ nM
2m ∂x
¯h 2 2 ∂SM
2m ∂y
+ m
nM
m∗ ¯h 2
2
∂SM
+ nMε(nM)+nMVext .
2m ∂z
(9.13)