## On the Design and Simulation of Electromagnetic Traps and Guides for Ultra-Cold Matter

##### Abstract

The objective of this thesis is the design and simulation of new electromagnetic traps and guides for ultra-cold matter. The traps and guides are intended for future experiments with small amounts of alkali atoms to study the quantum-mechanical effects of condensation and coupling between trapped drops of cold matter.
The main results are with the development and simulation of new wire traps and guides based on the dressing effect realised in strong DC magnetic and RF fields of certain frequencies. Some designs are proposed using only trapping by the DC magnetic field.
The principal methodology used in the thesis is to first develop the necessary theory and design formulas to make an initial design, followed by analytical and numerical simulation of the effective trapping potential. This may be followed by optimization of the geometry and the DC driving currents to enhance the trapping performance of the structure.
A wire carrying both DC and RF currents is surrounded by a cylindrical minimum potential manifold and can be used as a guide for cold atoms. Bias rings are necessary around the wire to avoid a potential minimum of zero and to move the resulting circular potential minimum up and down along the wire. The minimum potential surfaces around two crossed or two parallel wires touch each other for certain critical values of the DC currents in the two wires. The DC currents must be in opposite directions in two parallel wires. Equations are derived in Chapter 2 for the distance to the circular minimum potential manifold for a single wire, for two crossed wires and for two parallel wires. It is then explained how prospective cold atom transfer between two crossed wires can be achieved by changing the magnitudes of the RF currents in the bias rings around the wires. Electrically controlled atom transfer between two parallel wires does not seem to be practical.
A four-wire cell trap made from two crossing pairs of parallel wires has been designed and optimized using a simple Matlab script. It can be used to trap both strong- and weak-field-seeking atoms and may possibly be used to study collision and entanglement between the two types of atoms. With only DC excitation the trap becomes a trap for weak-field-seeking atoms. It then unfortunately has a potential minimum of zero at its centre. A similar 3 x 3 wire dual-well trap has also been designed and optimized in Matlab. It is prospective for the study of entanglement of BEC matter placed in the two wells. A quite low potential barrier in the direction normal to the wireplanes when the two wells are merged could however entail that the trap is inadequate for this purpose or that additional bias fields are necessary.
Several multi-wire cell-grids that may find use as part of a quantum register are also described. The cell-grids can be stacked in threedimensions and can trap both strong- and weak-field-seeking atoms. The optimization, also here performed in Matlab, showed weaknesses due to a lack of complexity. A different and better optimization technique is most likely necessary to improve the optimization further.
Scaling to micrometre and nanometre size is demonstrated in Chapter 3. When scaling to micrometre size thermally induced spin-flip transitions should be considered. Scaling to nanometre size demands that both thermally induced spin-flips and the effect of the Casimir-Polder force must be taken into account. The effect of the Casimir-Polder force is minimized by the use of carbon nanotubes as conductors. The minimum feasible trapping distance is expected to be no less than 100 nm from the surface of a carbon nanotube.
A four micro-wire cell and a 3 x 3 micro-wire structure, both adapted for future realization on a micro-machined substrate, are given as examples of micrometre size structures. Several nanometre size structures are also demonstrated. It is shown that prospective atom transfer between two crossed nanotubes can be done essentially in the same way as for two crossed wires.
A four-nanotube cell and several nanotube cell-grids are also exemplified. The depth of the trapping potential is found to be proportional to the RF frequency. If the RF frequency is increased then the DC current level must also increase to maintain the same DC current to angular frequency ratio. The depth of the trap is accordingly also proportional to the DC current level in the conductors. The depth of the trap is thus ultimately limited by the maximum conductor current.
A quadrupolar trap similar to the well known Ioffe-Pritchard trap is studied in Chapter 4 with combined DC and RF current excitation of the bias rings. A non-uniform potential minimum is found around the local maximum at the centre of the trap, but this does not prevent the trap from being used to trap weak-field-seeking atoms. The potential maximum at the centre of the quadrupolar trap is more than sufficient for trapping strongfield- seeking atoms. The quadrupolar trap can therefore be used to trap both strong- and weak-field-seeking atoms if the DC bar currents are large enough. Simulations also indicate that the bias rings can be placed relatively closely together to compress clouds of cold atoms into successively smaller traps. As the gap distances become very small the B-field becomes very strong between the bias rings and there is a risk of dielectric breakdown.
A metallic cylinder atom guide consisting of a cylinder with a small hole and an external wire is described analytically in Chapter 5 and simulation results from Amperes are compared favourably with the results of calculations in Matlab. It is found that there can only be a B-field zero at the centre of the hole in the cylinder when there is a second field zero further inside the cylinder. The barrier between the two field zeros typically increases in width with increasing cylinder radius and in height with decreasing cylinder radius for a given cylinder current (DC). The smallest cylinder had the highest barrier between the field zeros, but also required the highest DC current in the external wire. Bias rings around the guide must be centred on the hole in the cylinder and the DC ring currents and the spacing between the bias rings must be scaled by the same factor as the ring radius to maintain the same shape and height of the trapping potential along the centre of the hole. The cylinder guide looks promising as a hermetic guide for cold matter. Bias rings are required both to pump atoms along the guide and to remove the zero in the B-field inside the hole.