Ultra-cold atoms in shell and ring structures

This thesis looks at the three-dimensional dynamics of wave packets within the context of atomic trapping. We study three-dimensional dynamics from a theoretical standpoint, a field of high interest due to the emergence of quantum atom-based technologies, including quantum atomic metrology. In particular, we look at the dynamics of freely expanding wave packets at the point of release from an atom trapping scheme consisting of different geometries. We derive a new methodology for generating analytic expressions for the free expansion of these wave functions that utilise the infinite summation of individually expanding and interacting Gaussian distributions. We can then demonstrate that the resulting expressions show high fidelity to results obtained through numerical simulation of the freely-expanding Gaussian and hollow shell wave packets, validating the methodology used. The advantages of these analytic expressions are three-fold. Firstly, the method does not rely on intermediary time steps allowing for a quick system evaluation. Secondly, having an analytic expression allows for a greater understanding of the system’s behaviour. In particular, we can now analyse the system’s interference fringes and more easily predict the impact various properties have on the overall system. Finally, the analytical model is not subject to the pitfalls of numerical simulation such as numerical drift, high dependence on the choice of time step and considerations such as cross-boundary and boundary interactions. In developing these expressions, we observe the emergence of interference fringes, particularly a region of high wavefunction density that developed in the centre of the hollow shell and toroidal wave packets. Following on from this, we look at how an initial asymmetry in the starting parameters of the system affects the resulting free expansion. In particular, we look to simulate the effects of microgravity on the hollow shell and how tilting affects the toroidal wave packet.

Additionally, this thesis studies atom dynamics in the hollow shell. We use a Kapitza-style, driven inverted pendulum, technique to generate a stable region above the equatorial plane in a classical atomic-scaled system. We demonstrated that it iii is possible to achieve a stable inverted position on an atomic scale, even when the system has a degree of elasticity. We demonstrate that the capture angle for such a system is quite broad and can even be just above the equatorial plane in some cases. We demonstrate that achieving a stable inverted position for a wide range of starting variables is possible. Additionally, we demonstrate that in-situ cooling of the system may be possible by linearly reducing the driving frequency of the system. This phenomenon is experimentally interesting since it may lead to a new technique for systemic cooling, allowing for prolonged lifetimes and a reduction in evaporation rate. We also demonstrate that it is possible to localise a pendulum through multi-directional driving, allowing for more refined control and manipulation of the system. After demonstrating the viability of an inverted pendulum on the scale of an atom trapping scheme, we then took steps to improve the model to reflect an atomic system better to see whether such a technique could be applicable in atomic trapping systems. In testing this idea, we found that when modulating an atomic quadrupole trap in the manner outlined for the mechanical system, the system did not readily result in a stable region in the inverted position. Instead, we found that throughout all the parameters tested, the trap operated in a linear regime when vertically driven, meaning that for these parameters, the Kapitza-style driving was insufficient to stabilise the north pole in this system. Instead, we were able to find an alternative approach utilising a time-averaged adiabatic potential at the north pole of the trapping potential, which led to stable systems and might be utilised instead of the vertical Kapitza-style driving.