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

4.2 Tight binding regime 43
describing the capability of a particle to tunnel through the barriers of the potential. As the
potential is made deeper, the smaller becomes the tunneling parameter δj and hence the flatter
becomes the band (4.29).
For exactly zero tunneling (δj =0), i.e. when the Wannier function does not extend over
neighbouring wells, all Bloch states of a band are degenerate and the Wannier functions are
also eigenstates of the Schrödinger equation. In the case of an electron in a crystal, they
coincide with the atomic orbitals in this limit.
When tunneling is possible, the degeneracy between the Bloch states is lifted and the
particle can gain energy δj by spreading over the wells of the potential. This is analogous to
the case of a particle in a double-well: When tunneling contact is established the two, initially
degenerate, energy levels split by 2δ. The new groundstate corresponds to the symmetric
combination of the single-well solutions and its energy is lowered by δ.
The value of δj can be calculated using the semi-classical form of the single-well wavefunction
in the region of the barrier (see [99], §50). The exponential decay of the semi-classical
wavefunction as a function of the height of the potential barrier gives rise to an exponential
decay of δj in s (see also section 4.2 below).
Group velocity, current and effective mass
The knowledge of the form of the energy band (4.29) considered in the tight binding regime,
immediately permits to write down explicit expressions for the group velocity (4.15), the current
density (4.17) and the effective mass (4.18).
The group velocity (4.15) takes the simple form
¯vj(k) = dδj
¯h
and correspondingly, the current density reads
sin(kd) , (4.31)
Ij(k) = dδj
sin(kd) . (4.32)
¯hL
The two quantities (4.31) and (4.32) are proportional to the tunneling parameter δj. Hence
they decrease in the same manner as δj when the potential depth is increased.
The effective mass (4.18) turns out to be inversely proportional to the tunneling parameter
of the lowest band
1
m∗ = d2δ 2 . (4.33)
¯h
Thus, the exponential increase of m∗ as a function of s in an optical lattice (see Fig.4.6)
reflects the exponential decrease of the tunneling parameter.
The generalized k-dependent effective mass (4.21) takes the form
1
m ∗ j
= d2δj 2 cos(kd) . (4.34)
¯h