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 $V(\vec{r}) = \sum_{x_\alpha = x,y,z} V_0^{x_\alpha} \sin^2 (\pi x_\alpha)$

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 $\left[\frac{1}{\pi^2} \left( -i \nabla + 2\pi \vec{k} \right)^2 + \sum_{x_\alpha = x,y,z} V_0^{x_\alpha} \sin^2 (\pi x_\alpha) \right] u_k (\vec{r}) = \epsilon_k u_k(\vec{r})$

which is clearly separable to say the x-component: $\left[\frac{1}{\pi^2} \left( -i \partial_x + 2\pi k_x \right)^2 + V_0^{x} \sin^2 (\pi x) \right] u_{k_x} (x) = \epsilon_{k_x} u_{k_x}(x) \,\,\,\,\, ,\,\,\,\,\, k_x = 0, \frac{1}{L_x} ,\cdots \frac{L_x-1}{L_x}$

In the plane wave basis, $u_{k_x} (x) = \frac{1}{\sqrt{L_x}} \sum_{m \in \mathbf{Z}} c_m^{(k_x)} e^{i2m\pi x}$

we arrive at a tridiagonal diagonalization problem: $\left[ 4(m + k_x)^2 + \frac{V_0^x}{2} \right] c_m^{(k_x)} - \frac{V_0^x}{4} c_{m-1}^{(k_x)} - \frac{V_0^x}{4} c_{m+1}^{(k_x)} = \epsilon_{k_x} c_m^{(k_x)}$

The wannier function is defined as: $w(x) = \frac{1}{\sqrt{L_x}} \sum_{k_x} u_{k_x} (x) e^{i 2\pi k_x x} = \frac{1}{L_x} \sum_{k_x} \sum_{m \in \mathbf{Z}} c_m^{(k_x)} e^{i 2\pi (m+k_x) x}$

and from there, one can calculate the onsite interaction: $U = g \int | w(x) |^4 dx = \frac{4 \pi a_s \hbar^2}{m} \int | w(x) |^4 dx$

After a little bit of algebra, we arrive at the hopping strength: $t = -\frac{1}{L_x} \sum_{k_x} \epsilon_{k_x} e^{-i2\pi k_x}$

Finally, the Fourier transform of the wannier function is: $\tilde{w}(q_x) = \frac{1}{\sqrt{L_x}} \int w(x) e^{-i2\pi q_x x} dx = \frac{1}{\sqrt{L_x}} \sum_{k_x} \sum_{m \in \mathbf{Z}} c_m^{(k_x)} \delta_{q_x, k_x+m}$

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 $\hat{H} = -t \sum_{\langle i,j \rangle} \hat{b}_i^+ \hat{b}_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) - \sum_i ( \mu - V_T ( \vec{r}_i) ) \hat{n}_i$

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. $V_T (\vec{r}_i) = K_x x_i^2 + K_y y_i^2 + K_z z_i^2$.

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 $Z = \mathrm{Tr} \, \exp \left(-\beta \hat{H} \right)$

and physical quantities such as the local density $\langle n_i \rangle = \frac{1}{Z} \, \mathrm{Tr} \, \hat{n}_i \exp \left(-\beta \hat{H} \right) = \frac{1}{Z} \sum_{\mathcal{C}} n_i (\mathcal{C}) Z(\mathcal{C})$

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.

© 2013 by Matthias Troyer, Ping Nang Ma