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

36 Single particle in a periodic potential
In Fig. 4.2, we plot the quantity |ϕjk| 2 for different values of the quantum numbers j and k
in the case of the optical lattice potential V = sERsin 2 (πx/d) with s =5. In the lowest band
the particle tends to be concentrated close to the potential minima. The higher the band index
the more probable it becomes to find the particle in high potential regions. This is because
the particle is less affected by the presence of the lattice and its wavefunction resembles more
the free particle delocalized plane wave solution. Moreover, the flatter a band the less the
distribution changes when varying k.
d|ϕjk| 2
3
2
1
0
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
3
2
1
0
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
3
2
1
a)
b)
c)
0
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
x/d
Figure 4.2: Modulus squared of the Bloch function ϕjk (4.3) at k =0(solid line), ¯hk =0.5qB
(dashed line) and ¯hk = qB (dash-dotted line) with a) j =1,b)j =2and c) j =3of a
particle in the optical lattice potential V = sERsin 2 (πx/d) for s =5.
Momentum and Quasi-momentum
The Bloch functions (4.3), being characterized by a certain constant wave number k, have
a certain similarity with the plane wave states of a free particle of momentum p =¯hk. For
this reason, the quantity ¯hk is often called quasi-momentum. It is however important to point
out that due to the presence of the external potential, which has only a discrete translational
invariance, there is no conserved momentum. In fact, in a stationary state with a given quasimomentum
¯hk, the momentum can have values ¯h(k+l 2π/d) with l integer. The corresponding