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

Abstract

Abstract. As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-particle wave functions following 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 certain characteristic shapes, that we identify as cusp catastrophes, appear under generic conditions. In the vicinity of a cusp the wave function takes on a universal structure described by the Pearcey function and obeys scaling relations which depend on the total number of particles N . In the thermodynamic limit (N → ∞) 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 is a δ-kick the problem can be solved analytically and we obtain 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. Finally, we use these scalings to describe the wave function in the critical regime of a Z 2 symmetrybreaking dynamical phase transition.

Quantum caustics and the hierarchy of light cones in quenched spin chains

“…The underlying common cause of this similarity is the presence of singularities, or more precisely nonanalyticity, in both cases. We emphasize that in the application to light cones we study here, this universality occurs out of equilibrium, and thus we have an example of universality in quantum dynamics [79,85,87].…”

“…A fourth scale appears in quantum fields due to discretization of excitations leading to "quantum catastrophes" [75][76][77][78][79][80] (rippling mirrors give analogous effects [81]). Going to the continuum (classical field) limit returns us to a wave catastrophe.…”

Caustics in quantum many-body dynamics

“…Similar "dynamical phase transitions" have been seen in other systems [59][60][61][62][63] and, like their equilibrium versions, they obey scaling laws in the critical regime thereby providing examples of universality in dynamics. The many-body caustics we will describe in this paper are another example of universality in dynamics characterized by a hierarchy of universal shapes in Fock space that have self-similar scaling [64][65][66].…”

Emulating a many-body topological transition with an aberrated optical cavity

“…The operator H BH is related to the Lipkin-Meshkov-Glick model [20] in nuclear physics and to Ising models with long-range interactions [21][22][23][24]. It is able to describe processes not captured by the mean-field approximation, such as squeezing of the number difference [25][26][27][28], Bloch oscillations and Bragg resonances [29,30], quantum revivals [31], and caustics in Fock space [32][33][34]. The first term in H BH accounts for on-site particle-particle interactions characterized by the "charging energy" E c , while the second describes hopping between sites at frequency J/ .…”