# Difference between revisions of "Documentation:Bosons in an optical lattice"

From ALPS

(→Implementation in Python) |
(→An example) |
||

Line 62: | Line 62: | ||

== Implementation in Python == | == Implementation in Python == | ||

− | === | + | === Example 1 === |

These can be easily evaluated in the following: | These can be easily evaluated in the following: |

## Revision as of 17:48, 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

These can be easily evaluated in the following:

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