Catastrophes in non-equilibrium many-particle wave functions: universality and critical scaling

As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-body wave functions after a quench, focusing on two-mode systems that include the two-site Bose Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space plus time certain characteristic structures generically appear that we identify as cusp caustics. In the vicinity of such a catastrophe the wave function takes on a universal form described by the Pearcey function and obeys scaling relations which depend on the total number of particles $N$. In the thermodynamic limit ($N \rightarrow \infty$) the cusp becomes singular, but at finite $N$ it is decorated by an interference pattern. This pattern contains an intricate network of vortex-antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench takes the form of a $\delta$-kick we show how to analytically map the wave function onto the Pearcey function and hence obtain the scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern.