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

9.2 Hydrodynamic equations for small currents 119
Even though this expression is written in terms of macroscopic variables, the energy change
brought about by the lattice is implicitly accounted for by the effective mass m ∗ (nM) and
by the function ε(nM), which have been calculated microscopically solving the stationary
GP-equation in presence of the optical lattice.
Action principle
Our derivation of the equations of motion for the macroscopic density nM and the macroscopic
velocity field vM is based on the requirement that the action
t ′
∂
A = dt E − i¯h
(9.14)
0
∂t
is stationary and hence satisfies the condition
δA =0. (9.15)
The integrand in (9.14) contains the quantity
∂
= dr Ψ
∂t
∗ ∂
M
∂t ΨM
1 ∂
= dr
2 ∂t nM
∂
+ inM
∂t SM
. (9.16)
ThefirsttermdoesnotcontributetoδA since the variation at t =0and t = t ′ is zero by
assumption.
Imposing (9.15) on the variation of the action with respect to SM and using (9.10) for
the relation between SM and the macroscopic superfluid velocity, we find the equation of
continuity
∂
∂t nM
m
+ ∂x(vMxnM)+∂y(vMynM)+∂z vMznM =0. (9.17)
m∗ On the other hand, upon variation of the action with respect to nM and imposing (9.15), we
obtain an Euler equation for the macroscopic density
m ∂
∂t vM
+ ∇ Vext + µopt(nM)+ m
2 v2 Mx + m
2 v2 My + ∂
∂nM
where µopt(nM) =∂ [nMε(nM)] /∂nM denotes the groundstate chemical potential for Vext =
0 (see Eq.(5.9)).
It is interesting to note that in (9.17), the current in the lattice direction is modified by the
factor m/m∗ . Since the respective current component can be written as the product of the
superfluid velocity component and the superfluid density
the superfluid density results to be
m m
nM
m∗ 2 v2
Mz =0, (9.18)
I = nsvMz , (9.19)
ns = nM
m
. (9.20)
m∗