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

6.1 Bloch states and Bloch bands 67
Chemical potential and energy band spectra
Given a solution ˜ϕjk of the GPE (6.3), the energy per particle εj(k) can be calculated using
the expression
d/2
εj(k) =
−d/2
˜ϕ ∗
1
jk(z)
2m (−i¯h∂z +¯hk) 2 + sER sin 2 (z)+ 1
2
2
gnd| ˜ϕjk(z)| ˜ϕjk(z)dz.(6.4)
and differs from the chemical potential µj(k)
d/2
µj(k) = ˜ϕ ∗
1
jk(z)
2m (−i¯h∂z +¯hk) 2 + sER sin 2 (z)+gnd| ˜ϕjk(z)| 2
˜ϕjk(z)dz.(6.5)
−d/2
by the term (gnd/2) d/2
−d/2 | ˜ϕjk(z)| 4 . The chemical potential coincides with the energy per
particle only in absence of the interaction term. In general, µj and εj are linked to each other
by the relation
µj(k) = ∂[nεj(k)]
. (6.6)
∂n
In the uniform interacting system, a condensate in a stationary state is characterized by an
energy per particle and by a chemical potential. These two quantities differ from each other
due to interaction. Stationary states are plane waves and the GPE yields
ε(k; s =0) = gn
2 + ¯h2 k2 , (6.7)
2m
µ(k; s =0) = gn + ¯h2 k2 , (6.8)
2m
showing that both energy and chemical potential have the same free-particle k-dependence.
Going from k =0to k = 0changes the wavefunction by just the phase factor exp(ikz),
which physically corresponds to imparting a constant velocity ¯hk/m to the condensate. The
“excitation” to a state with k = 0corresponds to a simple Galileo transformation which adds
energy ¯h 2 k2 /2m to each particle and thus to the chemical potential. Hence, in the absence
of a lattice the two spectra ε(k) and µ(k) differ from each other only by an off-set due to the
groundstate interaction energy and don’t exhibit a different k-dependence.
In the presence of a lattice, analogously to the uniform case one can associate to each
stationary state ϕjk an energy per particle and a chemical potential which form two different
band spectra. However, in the presence of a lattice, the situation is very different from the
uniform case: Going from k =0to k = 0does not just correspond to a simple change
of reference frame since the barriers of the potential remain fixed. The consequence is a
dependence of the Bloch wave ˜ϕjk on k which gives rise in general also to a difference between
the k-dependence of energy and chemical potential.
In Fig.6.4 we plot the band spectra (6.4,6.5) for gn =0.5ER at different lattice depth s.
For any j, k the value of the chemical potential is always larger than the energy per particle.
In analogy to the properties of the single particle Bloch band spectrum, increasing s has three
major effects: The energy per particle and the chemical potential are shifted to larger values.
The gaps between the bands become larger while each band becomes flatter.
In Figs.6.5, we plot the band spectra (6.4,6.5) at depth s =5for gn =0and gn =1ER.
Fig. 6.5 a) shows again the upward shift of the energy (6.4) and even more of the chemical