Entanglement in quantum catastrophes

We classify entanglement singularities for various two-mode bosonic systems in terms of catastrophe theory. Employing an abstract phase-space representation, we obtain exact results in limiting cases for the entropy in cusp, butterfly, and two-dimensional catastrophes. We furthermore use numerical results to extract the scaling of the entropy with the nonlinearity parameter, and discuss the role of mixing entropies in more complex systems.

Figure 1

Entanglement properties of the cusp catastrophe. The von Neumann entropy $S$ in the macroscopic limit $\mu \to \infty$ (thick line) shows a divergence at the critical value $A=0$ where the potential changes from a double- to a single-well structure (inset sketches). Numerical results for finite $\mu$ show a peak near this point. Inset (a) shows the scaling with $\mu$ of the parameter value $A^{*}$ at which the entanglement maximum occurs, and (b) shows the value of the entropy $S^{*}$ at this point.

Figure 2

The von Neumann entropy of the Butterfly catastrophe with $A_4=-4∕\sqrt{3}$ as the potential undergoes a double-triple-single well transition, both for $\mu \to \infty$ and finite $\mu$. The profile of the entanglement is very different to that of the cusp as the transition here is induced by a level crossing in the spectrum. Inset shows scaling of $A_2^{*}$ as a function of $\mu$.

Figure 3

The von Neumann entropy of the two-dimensional molar catastrophe with $A=-1$ as a function of $\gamma$. Plots of the potential for $\gamma <1$ and $\gamma >1$ are shown at the top of the figure. The origin of the potential is unstable and there are four stable potential wells satellite to this. Lower right inset shows scaling of ${\gamma }^{*}$ as a function of $\mu$.