What is this funky plot?

This widget lets you play with graphs I stumbled upon during my Masters Project with the MBQD group in Cambridge. Geometrically the pattern is something called a Quasicrystal, i.e. a pattern which although it is very regular (e.g. has Octagonal symmetry) is not periodic, like an infinite chessboard would be. In the chessboard, every point would be the same as every other point, but that's not the case in a quasicrystal. Interestingly, Quasicrystals can be generated as projections of regular (periodic) patterns onto surfaces in fewer dimensions. That's how I made these plots: I started with a periodic "hypercubic" pattern in 4D and then projected it onto a 2D plane as you see above.

More physically, these plots represent the wavefunction of a particle which starts at one point on such a quasicrystalline lattice and is allowed to move around by hopping to neighbouring lattice points. The magnitude of the wavefunction at each point is represented by the size of the dot, and the complex phase by the colour.

Two key concepts in quantum mechanics are the Wavefunction, \(\ket{\psi}\) and the Hamiltonian \(\hat{H}\). The wave function is a special vector, it describes the probability of finding a particle, and the Hamiltonian is an operator (a bit like a matrix). It tells us how the wavefunction will change. Mathematically, we can find the new wavefunction at a given time, \(t\) as: \begin{equation}\label{Eqn: UnitaryEvo} \ket{\psi(t)} = e^{-i \hat H t/\hbar }\ket{\psi(0)}. \end{equation} This is a slightly strange equation because \(\hat{H}\) is an operator, but you can work it out by using the power series expansion of the exponential function. Or even easier, there's a Python library function especially for it! That's what I used to make these plots!

Play around with the sliders to see how \(\ket{\psi}\) spreads out as the particle hops around the Quasicrystal. Use the buttons to change the Hamiltonian. The "free particle" case corresponds to a particle which is free to hop in any direction, but the "trapped" case corresponds to a particle which feels an attraction to the centre of the 2D plane.

I used the code which makes these plots to do a feasibility study for a new experimental technique for the MBQD experiment. Their experiment consists of very cold atoms in a vacuum chamber, which can be put in an 8-fold symmetric optical lattice. Their system can be described by the "trapped" Hamiltonian in these plots, but the experimental technique we studied will, in theory, be able to give the "free particle" \(\hat{H}\). This is a simulation of a particle moving freely on a 4D lattice, and could be interesting for many reasons. (known best by Ulrich Schnieder!)