Introduction

My reseach area was in ultracold atomic physics, specifically the dynamics of solitons in Bose-Einstein condensates (BECs) and their uses in interferometry. I carried out this work during my PhD at Durham university. I was a member of the Quantum Light and Matter (QLM) group and my supervisor was Professor Simon Gardiner.

Bose-Einstein Condensates

Bose-Einstein condensates (BECs) are a quantum state which can be realised at extremely low temperatures. The first experimental observation of a BEC occured in 1995 in a system of alkali atomic gases. Choosing the atomic species such that each individual atom behaves as a boson (this means that the total spin of the atom is an integer), lasers are used to cool the atoms and at these low temperatures most of the atoms occupy the same quantum state. The dynamics are then describable using a wave formulation where the collection of atoms behaves as a single quantum wave.

Solitons and Interferometry

Solitons are exact solutions to non-linear wave equations. In an optical trap, one can have different trap depths in each of the three dimensions. When these depths are much stronger in two of the dimensions, we enter a regime where the BEC becomes pseuso-1D. We refer to this as psuedo-1D because the quantum dynamics still occupy a 3D space, but the wave properties of the collection of atoms are frozen into the lowest mode of a harmonic oscillator (which is a Gaussian wave packet). In such a system, we can model the dynamics using the Gross-Pitaevskii equation (1DGPE) which gives bright soliton solutions in the absence of external potentials and where interactions between atoms are attractive. The application of solitons to interferometry arises from their inherent robustness, whereby they maintain their shape following collisions with objects and other solitons.

Interferometry is a process where wave interference is used for precise measurement. The idea is that waves interfere (interact) in a manner which is dependent on the phase difference between them. For soliton interferometry, we take two solitons, of equal size, which follow different paths, thereby obtaining a phase difference. The two solitons then collide on a barrier and the result is dependent on the phase difference. The measurement of the phase difference can then be used to calculate precise properties of the path difference. For example, a potential use with clear commercial applications is in gravimetry, where the interferometer is used to precisely survey geophysical properties.

Computational Methods

The GPE can be written (in dimensionless form) as $$ { \frac{i\partial \psi}{\partial t}-\frac{1}{2}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi - \vert \psi \vert^2 \psi . } $$ Being a non-linear equation, it is not necessarily solvable, except where $${V(x)}=0 ,$$ in which case we can find a bright soliton solution, $$ { \psi(x) = \frac{1}{2}\mathrm{sech}\bigg(\frac{x-(x_0+vt)}{2}\bigg) e^{ivx + (v^2/2 - 1/8 )t }, } $$ where the soliton is centered at x0 and moves with a velocity, v.

In the presence of a potential, we can model the soliton dynamics computationally. To do this, I employed the Fourier split step method. This uses two key techniques, namely advancing the kinetic energy term in momentum space and the potential term in position space, and setting up the terms so as to minimise the error arising from non-commutative kinetic and potential energy terms, given they are quantum operators. Up to third order in the timestep, this is written as, $$ { \psi(t + \delta t) = \mathcal{F}^{-1}(e^{-i\delta t\frac{k^2}{4}}\mathcal{F}(e^{-i\delta t(V(x)- \vert \psi \vert^2 )\psi}\mathcal{F}^{-1}(e^{-i\delta t\frac{k^2}{4}}\mathcal{F}( \psi(t))))). } $$


Research

The research I conducted examined the dynamics of solitons when considering multiple component BECs. This could be achieved by overlapping BECs of two different elements. However, a single atomic species can exist in multiple quantum states and these can also constitute the multiple components. In this case, transitions between states can occur either through inter-atomic collisions or by driving them using a laser with a frequency tuned close to the resonance of the transition.

This possibility is taken advantage of for one of the research topics, where a three component set-up with spatial variation is used to produce effective barriers for use in interferometry which are very narrow. This narrowness is desirable for carrying out the interferometry as it approaches an idealised system where the barrier only splits and recombines the solitons , resulting in no other unwanted secondary effects. This research was published in the paper:

C. L. Grimshaw, S. A. Gardiner & T. P. Billam, Soliton interferometry with very narrow barriers obtained from spatially dependent dressed states, Physical Review Letters 129, 040401 (2022) (arXiv, journal)

The other research topic looked at a system of two components and applied some theoretical as well as computational approaches to modelling their collisions on potential barriers under the variation of different parameters which arise in a two component system, as opposed to the single component GPE outlined above. This research was publised in the paper:

C.L. Grimshaw, S. A. Gardiner & B. A. Malomed, Splitting of two-component solitary waves from collisions with narrow potential barriers, Physical Review A 101(4): 043623 (2020) (arXiv, journal)