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

6.2 Tight binding regime 77
Comparison of Eqs.(6.30,6.33) and Eqs.(6.31,6.34) shows that
µ0j − ε0j = gnd
2
δµ,j − δj = −4gnd
L/2
−L/2
L/2
−L/2
Using this result and the general relation µ = ∂(nε)/∂n, we identify
n ∂ε0
∂n
n ∂δj
∂n
= gnd
2
= −4gnd
L/2
−L/2
L/2
−L/2
f 4 j (z) , (6.35)
f 3 j (z)fj(z − d)dz . (6.36)
f 4 j (z) , (6.37)
f 3 j (z)fj(z − d)dz . (6.38)
We recall that in the single particle case the l-th Fourier component of the energy band
(coefficient of eikld in the Fourier expansion) is given by the matrix element of the Hamiltonian
between Wannier functions at distance ld. In the tight binding regime this immediatly
yields the cos-dependence on the quasi-momentum. The situation is different in the presence
of interactions. For instance, also next-neighbour overlap can lead to higher frequency
contributions to the energy band since the term dzf 2 l f 2 l±1 , neglected here, would yield a
cos(2kd)-dependence.
Group velocity, current and effective mass
Exactly as in the single particle case, the knowledge of the form of the considered energy band
(6.29) in the tight binding regime, immediately permits to write down explicit expressions for
the group velocity (6.18), the current density (6.17) and the effective mass (6.19).
The group velocity (6.18) takes the simple form
¯vj(k) = dδj
¯h
and correspondingly, the current density reads
sin(kd) , (6.39)
Ij(k) = dδj
sin(kd) . (6.40)
¯hL
Both quantities are proportional to the tunneling parameter δ.
The effective mass (6.19) turns out to be inversely proportional to the tunneling parameter
1
m∗ = d2δ 2 . (6.41)
¯h
Thus, the exponential increase of m ∗ as a function of s in an optical lattice (see Fig.6.8)
reflects the exponential decrease of the tunneling parameter. We can now discuss why the
effect of interaction on m ∗ can not be neglected, not even at very large s (see Fig. 6.9): This is