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

6.1 Bloch states and Bloch bands 71
φjk according to (6.15) corresponds to a Bloch state with quasi-momentum ¯h(k + A)! This
implies that
εj(k, A) =εj(k + A) , (6.16)
where εj(k) are the usual energy Bloch bands. Comparing with (6.14) we find finally that
Ij(k) =n ∂εj(k)
¯h∂k
, (6.17)
in analogy with the single particle case (see section 4.15). Hence, the group velocity ¯vj(k) is
given by
¯vj(k) = ∂εj(k)
. (6.18)
¯h∂k
The effective mass m∗ is defined by
1
m∗ := ∂2εj=1(k) ¯h 2 ∂k2
, (6.19)
k=0
and characterizes current and group velocity at small quasi-momenta in the lowest band
Ij=1(k) → n ¯hk
, (6.20)
m∗ ¯vj=1(k) → ¯hk
. (6.21)
m∗ The generalization of the definition of the effective mass (6.19) to any value of band index
and quasi-momentum reads the same as in the single particle case
1
m∗ j (k) := ∂2εj(k) ¯h 2 . (6.22)
∂k2 The fact that the expressions (6.17,6.18,6.19) are the same as for a single particle is not
trivial: It shows that it is the energy band spectrum which determines these quantities and not
the chemical potential band spectrum! Formally, we can of course define an effective mass
associated with the lowest chemical potential at small k
1
m∗ :=
µ
∂2 µj(k)
¯h 2 ∂k2
, (6.23)
k=0
in analogy to definition (6.19). In fact, in section 7 we will find that the k-dependence of
the chemical potential and in particular the effective mass (6.23) play a role in the description
of the spectrum of the Bogoliubov excitations of the condensate in a stationary state of the
Bloch-form (6.2). The energy and chemical potential effective mass are linked by the relation
1
m ∗ µ
:= ∂
n
∂n m∗
, (6.24)