Difference between revisions of "Documentation:Bosons in an optical lattice"
From ALPS
(→An example) |
(→Example 1) |
||
Line 63: | Line 63: | ||
=== Example 1 === | === Example 1 === | ||
− | |||
− | |||
import numpy; | import numpy; |
Revision as of 17:51, 13 September 2013
Contents
Bandstructure of an homogeneous optical lattice
Theory
At this first moment, we shall look at the simplest case, i.e. a single particle of mass m which experiences a periodic potential , where
in the units of recoil energy and lattice spacing
.
The quantum mechanical behaviour of the single particle follows
which is clearly separable to say the x-component:
In the plane wave basis,
we arrive at a tridiagonal diagonalization problem:
The wannier function is defined as:
and from there, one can calculate the onsite interaction:
After a little bit of algebra, we arrive at the hopping strength:
Finally, the Fourier transform of the wannier function is:
Implementation in Python
Example 1
import numpy; import pyalps.dwa; V0 = numpy.array([8. , 8. , 8.]); # in recoil energies wlen = numpy.array([843., 843., 843.]); # in nanometer a = 114.8; # s-wave scattering length in bohr radius m = 86.99; # mass in atomic mass unit L = 200; # lattice size (along 1 direction) band = pyalps.dwa.bandstructure(V0, wlen, a, m, L);
>>> band Optical lattice: ================ V0 [Er] = 8 8 8 lamda [nm] = 843 843 843 Er2nK = 154.89 154.89 154.89 L = 200 g = 5.68473 Band structure: =============== t [nK] : 4.77051 4.77051 4.77051 U [nK] : 38.7018 U/t : 8.11272 8.11272 8.11272 wk2[0 ,0 ,0 ] : 5.81884e-08 wk2[pi,pi,pi] : 1.39558e-08