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

120 Macroscopic Dynamics
The appearance of this quantity in the Euler equation (9.18), confirms this statement: There,
the quantity ∂/∂nM (m/m∗ )nM(m/2)v2
z replaces the term ∂/∂nM nM(m/2)v2
z we would
obtain without lattice where nM = ns.
The hydrodynamic equations (9.17,9.18) can be further generalized to account for larger
condensate velocities. This is the topic of the subsequent section.
9.3 Hydrodynamic equations for large currents
In order to derive the hydrodynamic equations (9.17,9.18) for larger condensate velocities,
we need to generalize the energy functional (9.12). This is done by modifying the energy
contribution due to the current in lattice direction: The contribution (m/2)(m/m ∗ (nM))v 2 zM +
ε(nM) is replaced by the more general expression ε(vzM =¯hk/m; nM), the latter being the
energy of a Bloch state condensate (6.4) at average density nM, quasi-momentum ¯hk and
macroscopic superfluid velocity vMz =¯hk/m. Note that, for convenience, we include the
groundstate energy ε(nM) in the term ε(vzM =¯hk/m; nM). In this way, we obtain the energy
functional
E =
m
dr
2 nMv 2 Mx + m
2 nMv 2
My + nMε(vMz =¯hk/m; nM)+nMVext , (9.21)
which in terms of the phase defined in (9.10) can be rewritten in the form
m
E = dr
2 nM
2 ∂SM
+
∂x
m
2 nM
2
∂SM
+ nMε(∂zSM = k; nM)+nMVext .(9.22)
∂y
In writing this expression we have used the fact that according to (9.7,9.10)
∂zSM = k. (9.23)
Using this identity and the action principle employed above, we find the following hydrodynamic
equations
∂
∂t nM + ∂x(vMxnM)+∂y(vMynM)+∂z
m ∂
∂t vM
+ ∇
1
¯h ∂kε(k;
nM)nM
Vext + m
2 v2 Mx + m
2 v2 My + µopt(k; nM)
=0. (9.24)
=0. (9.25)
The second equation involves the chemical potential (6.5) of a Bloch states condensate with
quasi-momentum ¯hk at average density nM for Vext =0. When using the hydrodynamic
equations (9.24,9.25) it is important to keep in mind that k is a function of r and essentially
represents the z-component of the macroscopic velocity field vMz =¯hk/m.
The generalized hydrodynamic equations (9.24,9.25) have been reported in [138] for the
tight binding regime with the density-dependent effective mass and the equation of state
µopt =˜gnM and in [107] for a general equation of state µopt(nM) and general dispersion
ε(k; nM), hence for any lattice potential depth.