We model an atomic Bose-Einstein condensate (BEC) near an instability, looking for universal features. Instabilities are often associated with bifurcations where the classical field theory provided here by the Gross-Pitaevskii equation predicts that two or more solutions appear or disappear. Simple examples of such a situation can be realized in a BEC in a double well potential or in a BEC rotating in a ring trap. We analyze this problem using both Bogoliubov theory and exact diagonalization. The former describes elementary excitations which display complex frequencies near the bifurcation. We make connections to the description of bifurcations using catastrophe theory but modified to include field quantization.

Quantum bifurcations in a Bose-Einstein condensate

The creation and annihilation of particles is a fundamental feature of relativistic quantum fields. A famous example of this is provided by Schwinger’s 1951 prediction that the vacuum is unstable to particle-antiparticle production if a static electric field is applied to it. In this talk we examine the classical field limit of Schwinger pair production by mapping the Klein-Gordon equation with an electric field onto the non-relativistic Schrödinger equation with a 1/r^2 potential. This latter problem has two regimes depending on the depth of the potential: a sub-critical regime where PT symmetry is preserved and a supercritical regime where PT symmetry is broken and one finds ``fall-to-the-centre’’ where probability is absorbed at the origin. Schwinger pair production occurs in the latter regime and can be described in terms of non-Hermitian quantum mechanics.

Schwinger pair production as a non-Hermitian problem

In optics, caustics are bright, sharp lines and shapes created by the natural focusing of light. Some examples include rainbows, the wavy lines on the bottom of swimming pools and the patterns produced by gravitational lensing. The intensity at a caustic diverges in the classical ray theory, but can be smoothed by taking into account the wave nature of light. In this work we consider a new type of caustic that occurs in quantum systems due to phase singularities; because phase is such a central concept in wave theory, this heralds the breakdown of the wave description and is an example of a quantum caustic. In particular, we consider analogue black holes which can be formed in a flowing Bose-Einstein condensate (BEC) gas. Waves flowing near the event horizon (describing analogue Hawking radiation) appear to suffer a logarithmic phase divergence, which is known as the trans-Planckian problem for gravitational black holes. We describe the regularization procedure required to cure this quantum caustic in our BEC system, and make connections to catastrophe theory.

Hawking radiation as a quantum caustic

M1-1  Degenerate Quantum Gases and cold Atoms and Molecules (DAMOPC) / Gaz quantiques dégénérés et atomes et molécules froids (DPAMPC)

technical paper

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Duncan O'Dell

3

SHORT BIO

Presentations

Quantum bifurcations in a Bose-Einstein condensate

Schwinger pair production as a non-Hermitian problem

Hawking radiation as a quantum caustic

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