# Documentation:Bosons in an optical lattice

## 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

### An example

For instance:

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);

A first glance of the band structure:

>>> 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

Well, the values of t (nK), U (nK), and U/t can be obtained via:

>>> numpy.array(band.t()) array([ 4.77050984, 4.77050984, 4.77050984]) >>> >>> numpy.array(band.U()) array(38.7018197381118) >>> >>> numpy.array(band.Ut()) array([ 8.11272192, 8.11272192, 8.11272192])

In momentum () space, the (squared) wannier function can be obtained in the x-direction from:

>>> numpy.array(band.q(0)) array([-5. , -4.995, -4.99 , ..., 5.985, 5.99 , 5.995]) >>> >>> numpy.array(band.wk2(0)) array([ 7.57249518e-15, 7.88189086e-15, 8.20434507e-15, ..., 1.62988573e-18, 1.56057426e-18, 1.49429285e-18])

and the y- or z- direction by replacing the index 0 to 1 and 2 respectively.

# Bosons in an optical lattice trap

## Boson Hubbard model

### Hamiltonian

Bosons in an optical lattice trap can be effectively described by the single band boson Hubbard model

with hopping strength , onsite interaction strength , and chemical potential at finite temperature via Quantum Monte Carlo implemented in the directed worm algorithm. Here, () is the annihilation (creation) operator, and being the number operator at site i. Bosons in an optical lattice are confined, say in a 3D parabolic trapping potential, i.e.

.

due to the gaussian beam waists as well as other sources of trapping.

### Finite temperature

At finite temperature T, the physics is essentially captured by the partition function

and physical quantities such as the local density

for some configuration in the complete configuration space, with inverse temperature . Here, the units will be cleverly normalized later on.

© 2013 by Matthias Troyer, Ping Nang Ma