Thesis (pdf) - Swinburne University of Technology

Chapter 2: Theoretical Background
trapped in the local minimum of the magnetic field. These are called low or
weak field seeking states.
The simplest way to create a field with a minimum is a quadrupole field,
which can be created by two coils in an anti-Helmholtz configuration. This
field has a zero at its centre and from there a linear increase in the absolute
field strength. Different mF -states thus differ in their energy by an amount
proportional to the magnetic field:
∆E = gF µBB∆mF
(2.37)
The linearity of the field of this trap is a drawback though. Atoms moving
towards the centre have their magnetic moment counter-aligned with the di-
rection of the field vectors. When they pass through the field’s zero crossing,
the atom sees a reversed direction of the field and accordingly experiences a
force that expels it from the centre. This has been called a spin flip or a Ma-
jorana spin flip, although it is not the spin that changes, but the direction of
the magnetic field. In fact, if the spin and thus the magnetic moment is able
to change with the field then the atom will not be lost.
Another simple design creates a quadrupole trap in two dimensions. A
current I is running through an infinitely long wire, and a homogeneous bias
field B0 is applied perpendicular to the wire. There is one point r0 ∝ I/B0
where the field of the wire Bw ∝ I/r is just canceled by the bias field. This
field can be well approximated by a quadrupole field. If we now add a second
bias field along the wire, the overall field never crosses zero, although it has a
field minimum at the same position as before. Here the atomic spin can now
adiabatically follow the changing field direction and thus Majorana spin flips
are suppressed. Such a configuration is generally called a Ioffe-Pritchard (IP)
configuration [Pri83].
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