Bosons in an Optical Lattice

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 $V(\vec{r})$, where

in the units of recoil energy $E_r^\alpha = \frac{\hbar^2}{2m} \left( \frac{2\pi}{\lambda_\alpha} \right)^2$ and lattice spacing $\frac{\lambda_\alpha}{2}$.

The quantum mechanical behaviour of the single particle follows

which is clearly separable to say the $x$-component:

where $k_x = 0, \frac{1}{L_x} ,\cdots \frac{L_x-1}{L_x}$.

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 ($\vec{q}$) space, the (squared) wannier function $|\tilde{w}(\vec{q})|^2$ 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 $t$, onsite interaction strength $U$, and chemical potential $\mu$ at finite temperature $T$ via Quantum Monte Carlo implemented in the directed worm algorithm. Here, $\hat{b}$ ($\hat{b}^+$) is the annihilation (creation) operator, and $\hat{n}_i$ 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 $\mathcal{C}$ in the complete configuration space, with inverse temperature $\beta = 1/T$ . Here, the units will be cleverly normalized later on.

Contributors

• Ping Nang Ma
• Matthias Troyer