Analogue Hawking radiation as a logarithmic quantum catastrophe

Abstract

Caustics are regions created by the natural focusing of waves. Some examples include rainbows, spherical aberration, and sonic booms. The intensity of a caustic is singular in the classical ray theory, but can be smoothed out by taking into account the interference of waves. Caustics are generic in nature and are universally described by the mathematical theory known as catastrophe theory, which has successfully been applied to physically describe a wide variety of phenomena. Interestingly, caustics can exist in quantum mechanical systems in the form of phase singularities. Since phase is such a central concept in wave theory, this heralds the breakdown of the wave description of quantum mechanics and is in fact an example of a quantum catastrophe. Similarly to classical catastrophes, quantum catastrophes require some previously ignored property or degree of freedom to be taken into account in order to smooth the phase divergence. Different forms of spontaneous pair-production appear to suffer logarithmic phase singularities, specifically Hawking radiation from gravitational black holes. This is known as the trans-Planckian problem. We will investigate Hawking radiation formed in an analogue black hole consisting of a flowing ultra-cold Bose-Einstein condensate. By moving from an approximate hydrodynamical continuum description to a quantum mechanical discrete theory, the phase singularity is cured. We describe this process, and make connections to a new theory of logarithmic catastrophes. We show that our analogue Hawking radiation is mathematically described by a logarithmic Airy catastrophe, which further establishes the plausibility of pair-production being a quantum catastrophe