The GPE in 5 Steps¶

Here we describe how to numerically solve time-dependent Schrödinger-like equations:

\[\begin{gather*} \I\hbar \pdiff{}{t}\psi(\vect{r}, t) = \left( \frac{-\hbar^2\vect{\nabla}^2}{2m} + V(\vect{r}, t; \psi) \right)\psi(\vect{r}, t) \end{gather*}\]

Here we describe the superfluid with a “condensate wavefunction” \(\psi(\vect{r}, t)\) which has the following hydrodynamic interpretation:

\[\begin{gather*} \psi(\vect{r}, t) = \sqrt{n(\vect{r}, t)} e^{\I \phi(\vect{r}, t)}. \end{gather*}\]

Here \(n(\vect{r}, t)\) is the number density of particles at position \(\vect{r}\) and time \(t\). The gradient of the phase of the condensate wavefunction is related to the group-velocity of the superfluid:

\[\begin{gather*} \vect{v}_{\text{group}} = \frac{\hbar}{m}\vect{\nabla}\phi(\vect{r}, t). \end{gather*}\]

Expressing the condensate wavefunction in this way, we effect a [Madelung transformation] and obtain the following equivalent hydrodynamic description:

\[\begin{gather*} \pdiff{}{t}n(\vect{r}, t) + \vect{\nabla}\cdot\vect{j}(\vect{r}, t),\\ m D_t \vect{v}(\vect{r}, t) = -\vect{\nabla}\left( \frac{-\hbar^2}{2m}\frac{\vect{\nabla}\sqrt{n(\vect{r}, t)}}{\sqrt{n(\vect{r}, t)}} + V(\vect{r}, t) \right) \end{gather*}\]

where \(\vect{j}(\vect{r}, t)\) is the particle number current:

\[\begin{gather*} \vect{j}(\vect{r}, t) = n(\vect{r}, t) \vect{v}(\vect{r}, t), \end{gather*}\]

and \(D_t\) is the “material derivative” or “covariant derivative” which simply accounts for the fact that the fluid is moving:

\[\begin{gather*} D_t f(\vect{r}, t) = \pdiff{}{t}f(\vect{r}, t) + \vect{v}(\vect{r}, t)\cdot\vect{\nabla}f(\vect{r}, t). \end{gather*}\]

The first equation in the hydrodynamic formulation is simply the statement of conservation: particles are neither created nor destroyed. The second is simply the hydrodynamic form of Newton’s law \(m\vect{a} = \vect{F} = - \vect{\nabla}V\). Since we work at \(T=0\), the effective potential \(V(\vect{r}, t)\) includes both any external potentials, as well as the \(gn(\vect{r}, t) \equiv \mu(\vect{r}, t) = \mathcal{E}'(n)\), which is the thermodynamic chemical potential – the derivative of the energy density \(\mathcal{E}(n) = gn^2/2\). At finite temperature, this term should be replaced by the appropriate enthalpy.

All of the quantum effects are contained in the remaining term

\[\begin{gather*} Q = \frac{-\hbar^2}{2m}\frac{\vect{\nabla}\sqrt{n(\vect{r}, t)}}{\sqrt{n(\vect{r}, t)}} \end{gather*}\]

which is sometimes called the quantum pressure. For example, as we shall see in many of the demonstrations, superfluid vortices have quantized circulation where the wavefunction phase winds by integer multiples of \(2\pi\). This property is manifest in the wavefunction formulation where \(\psi(\vect{r}, t)\) must be single-valued and smooth. A vortex at the origin would thus have the general form

\[\begin{gather*} \psi(\vect{r}) \propto f(r) e^{\I n \theta} \end{gather*}\]

where \(n\) is the integer circulation, and \(f(r) \propto r\) vanishes at the core, otherwise the wavefunction could not smoothly be single-valued.

This quantization seems to be missing from the hydrodynamic formulation, but is present in the fact that the quantum pressure \(Q \propto 1/r\) diverges in the vortex core. These divergences must exactly cancel similar divergences in the velocity \(v \propto 1/r\): this matching and cancellation of divergences enforces vortex quantization, but is tricky to manage in the hydrodynamic formulation. It follows naturally from the wavefunction evolution.

Super_Hydro

The GPE in 5 Steps¶