Entanglement Entropy of analogue Black Holes

Abstract

This thesis investigates the two seemingly unrelated topics of analogue black holes and entanglement entropy. After a review of the basic mechanisms behind earlier water-based analogues and analogue black holes in Bose-Einstein condensates, we retrace the ideas behind a recently proposed analogue black hole in a Bose-Einstein condensate of light, in which the authors demonstrate that the acoustic horizon emits phononic radiation. It is suggested that the creation of entangled phonon pairs at the horizon might be behind this phenomenon, which could potentially be confirmed by calculating the entanglement entropy of the acoustic radiation. As the phonons are governed by the equation of motion for free, massless scalar fields, we consider a treatment of entanglement entropy based on the discretization of scalar fields in the direction normal to the entangling surface. The regularized theory is mapped to a finite one-dimensional chain of harmonic oscillators, for which the reduced density matrix is known exactly. This result is used to numerically predict the entanglement entropy of oscillator-chains representing scalar fields in d ∈ {1, 2, 3} spatial dimensions on flat backgrounds, in preparation of the method’s future extension to curved spacetimes. We confirm that the analytical results for a (1+1)-dimensional scalar field are approximated by this numerical method, and proceed to verify the area-law for scalar fields in d = 3 spatial dimensions. The approach is extended to a method which requires only the position and momentum correlators restricted to subsystems of the full lattice, which allows us to obtain improved results for the studied cases. It is shown that this method can be efficiently used to study the entanglement entropy of scalar fields mapped to square lattices, which we demonstrate explicitly for a discrete circle as a function of its perimeter, and compare its predictions to those of the one-dimensional chain representation of scalar fields.