Bose$-$Einstein condensation
This will be a hand-waiving introduction to Bose-Einstein condensate theory. I’ll begin by introducing what is a Bose-Einstein condensate, followed by how we model it (hint: we use the Gross-Pitaevskii equation). For more detailed derivations, see Pethick and Smith [ISBN: 978-0521846516] or Pitaevskii and Stringari [ISBN: 978-0198507192].
Again, this is another work in progress, so expect me to continually dump my thoughts here and on subsequent posts.
Schrödinger equation
To understand how we simulate a Bose$-$Einstein condensate, first we must examine the behaviour of the Schrödinger equation. I will assume that you are already familiar with the Schrödinger equation, but if not, I will recommend the usual material:
- Principles of Quantum Mechanics, Shankar [ISBN: 978-0306447907]
- Introduction to Quantum Mechanics, Griffiths [ISBN: 978-9332542891]
Single particle
The single particle Schrödinger equation describes the behaviour of a non-relativistic (i.e. much slower than the speed of light, and/or massless) quantum particle. Firstly, we define our Hamiltonian as
$$ \hat{H}_0 = -\frac{\hbar^2}{2m} \hat{\nabla}^2 , $$
where $~\hat{H}_0$ corresponds with the momentum-space operator for a free particle. Assuming the particle is trapped by an external potential energy, we can give the following Hamiltonian instead,
$$ \hat{H}_1 = \hat{H}_0 + \hat{V}_{\textrm{ext}}(\mathbf{r},t). $$
Now we have a potential and kinetic term for our system. In cold atomic systems, we nearly always have trapped atoms, by use of an external trapping potential [usually with optical or magnetic fields]. To simulate the dynamics of a single particle in a trapping potential, we use the time dependent Schrödinger equation,
$$ i\hbar\frac{\partial}{\partial t}{\Psi}(\mathbf{r},t) = \hat{H}_1 {\Psi}(\mathbf{r},t). $$
Many particle
$$ H_N = \displaystyle\sum\limits_{i=1}^{N} \left(-\frac{\hbar^2}{2m_i} \nabla_i^2 + V_\textrm{ext}(\mathbf{r}_i,t) + \displaystyle\sum_{j\neq i}^{N-1}U(\mathbf{r}_i,\mathbf{r}_j) \right) $$ Solving this for the ground state is hard, even numerically. The memory requirements grow exponentially as we increase the particle count, as every particle must interact in some way with all the others nearby. Therefore, in order to find the lowest energy state we must simplify things.
First, let’s assume that particles don’t interact over long, or even short ranges. What if they can only interact when they occupy the same position, as though $|\mathbf{r}_i - \mathbf{r}_j| = 0$. Also, we can say that the strength of this interaction is dependent upon some parameter, $g$, which depends on properties of the particle. We can also assume that the more particles in a particular region, then the larger the interactions in that specific region. Thus, we can say that these interactions can be approximated by $g\rho(\mathbf{r},t)$, where $\rho$ is the particle density in the region.
$$ i\hbar\frac{\partial}{\partial t}\hat{\Psi}(\mathbf{r}_1,\dots\mathbf{r}_N,t) = \left[\hat{\Psi}(\mathbf{r}_1,\dots\mathbf{r}_N,t), H_N \right] $$
GP equation
$$
i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \left[ -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r},t) + \color{violet}{g\vert\Psi(\mathbf{r},t)\vert^2} -\color{teal}{\mathbf{\Omega}\cdot \mathbf{L} }\right]\Psi(\mathbf{r},t)
$$
Madelung transform
$$ \Psi(\mathbf{r},t) = \sqrt{\rho\left(\mathbf{r},t\right)}\exp\left[i\theta\left(\mathbf{r},t\right)\right], $$ where $$\rho(\mathbf{r},t) = |\Psi\left(\mathbf{r},t\right)|^2$$
Evolution
$$ \Psi(\mathbf{r},t+\delta t) = \exp\left(-\frac{i H}{\hbar}\delta t\right)\Psi(\mathbf{r},t) $$