On the expansion of turbulent Bose-Einstein Condensates

In 1924, Indian physicist Satyendra Bose worked with Albert Einstein to develop Bose-Einstein statistics [ Bose, Einstein 1925]; the statistics that govern all half-spin particles (called Bosons). Bosons are different to their counterpart particles - Fermions - in that multiple can occupy the same energy state. Such a system naturally hence predicts that at absolute zero (the lowest temperature possible, -273.15C), when there is zero energy in the system, all particles drop to their lowest energy level and hence a new state of matter emerges. The theory of such a state was studied extensively throughout the 20th Century by some of the greatest physicists and mathematicians [London, 1938; Feynman, 1955] of their time, however was not experimentally observed until 1995 [Cornell et al. 1995].

Since their discovery the area of BECs has produced a plethora of experimental and theoretical results. We will, specifically, be studying exactly how BECs expand. A lot [for example, the many papers of Bagnato et. al.] of experimental setups involve the process of releasing the condensate from its trapping potential to optically view the state of the system. The problem here is that the experimentalists cannot have any way of telling exactly what properties the condensate processes before expansion or whether the state of the condensate after expansion is due to the turbulence applied or the actual act of expanding.

The research over these next three years takes the form of studying exactly how BECs expand from both a harmonic (a function taking the form of $\frac{1}{2}\omega^2x^2$) potential and a box potential. The first can be realised by simple dynamics which can be explained semi-analytically in the Thomas-Fermi limit (which can occur when the frequency of the trap $\omega \ll 1$) The second (box) potential takes a more complicated analytical form. Although well investigated experimentally [see papers by Nivon et. al., e.g. Nivon et. al. 2018], the theoretical aspects of such a problem is relativity untouched, apart from a smattering of recent papers [Ivanov and Kamchatnov 2019], which even then, are studying the problem in a one-dimensional limit. To break down the problem at hand, we will start from a one-dimension system, and then move to a two and three-dimensional system during these three years. Each dimension has its own problems to investigate - current research in the one-dimensional system covers the probing of turbulence which, in 1D, consists of sound waves (a sort of structureless random-ish noise travelling) and solitons (or, more accurately - solitary waves). Solitons are defined as a ``self-reinforcing solitary wave packet'', which essentially means it's a wave which does not break up as it travels through the medium and keeps its shape through propagation, through a delicate balance between the dispersion and the nonlinearity of the wave. In the case of BECs, which are governed by the Gross-Pitaevskii equation, dark solitons are supported. This means that the solitons formed are "negative waves", i.e., they form a dip in the condensate density instead of a peak. In order to help in the finding of a nice analytical result to the expansion with soliton-ic turbulence, the PhD student is looking at how similar research has taken place in nonlinear optics and general soliton research. This is allowed with the Gross-Pitaevskii equation being a nonlinear Schrodinger equation - and hence is similar to the Nonlinear Schrodinger Equation that is used ubiquitously in nonlinear optics and solitonic theory.

In the two and three-dimensional systems, the turbulence takes the form of vortices. Unlike in the classical world where (for example in the flow of water or air) turbulence can take the form of beautiful and complicated patterns, the turbulence in BECs is quantised; the vortices looking like long, thin cylindrical tubes with a width on the scale of the BECs healing length (the smallest length-scale of the system).