Intro

In the following the Gross-Pitaevskii equation, $$ i\partial_t \psi = \frac{-\hbar}{2m}\nabla^2 \psi + \alpha |\psi|^2 + G\left(n_R + \frac{\eta P}{\Gamma}\right) + \frac{i(Rn_R - \gamma_{LP})}{2} $$ where $$\partial_t n_R = -(\Gamma + R|\psi|^2)nR + P$$ is solved with the following parameters: $\alpha$ = 0.0004 µm² ps^-1, $\gamma_{LP}$ = 0.2 ps^-1, $\Gamma$ = 0.1 ps^-1, $G$ = 0.002 µm² ps^-1

The system is driven with Gaussian pumps of the form $P(\vec{r}) = p\exp(-(\frac{\vec{r} - \vec{r}_0}{\sigma})^2)$. In the following graphs $\sigma$= 1.2 µm, corresponding to a FWHM of 2.826 µm, and p is tuned so that they are just over the threshold power density.

All the code is available on Github

The Penrose crystal

Here the pumps are placed at the vertices of a Penrose tiling with diameter d.

r-space image of $\psi$, d=180 µm, p = 24 µm^-2ps^-1

k-space image of $\psi$, linear scale, same parameters as above

k-space image of $\psi$, log scale, same parameters as above

And then the dispersion relation: {{ graph(name="penrosedispersionr90d4p24n1024s1.2dt0.1.png", caption="Dispersion relation of E and k_x, linear scale) }}

We can compare to simulations with smaller or larger tilings with the same number of pumps.

r-space image of $\psi$, d=200 µm, p = 25 µm^-2ps^-1

Dispersion relation of E and k_x, linear scale

r-space image of $\psi$, d=160 µm, p = 23 µm^-2ps^-1

The Monotile Crystal

Here the pumps are placed at the vertices of a hat monotile tiling, using a method by Hastings Greer. A large tiling is generated and saved to a file. The tiliing is then cropped in the simulation so that a rectangular grid is obtained.