2D magnetic traps for ultra-cold atoms: a simple theory using complex numbers

The properties of two-dimensional magnetic traps for laser-cooled atoms are analysed using complex functions. The two components of the magnetic field from a series of parallel, infinitely long, current-carrying wires are represented by a single complex number. The regions of the field where paramagnetic atoms can be trapped occur where the magnetic field is zero. The locations of the zeroes of the field are obtained as the solution to a polynomial and the multiplicity m of the solution determines both the 2(m + 1)-pole nature of the trap and the field gradient through the centre. The zeroes of the field can be merged or split by varying the locations of the currents, their strengths or by applying a uniform magnetic field. The theory is applied to magnetic traps created from long thin wires or permanent magnets on a substrate. The properties of a number of magnetic trap configurations used for atom guides are discussed.