Ph.D. Thesis - Physics

7.3 Magnetic gradient forces
Now that we have seen that in many cases quantum simulations may be performed in an
elliptical trap (or other trap with nonzero micromotion), we are ready to discuss the actual
source of the state-dependent forces F. Optical forces have been discussed at length in
Ref. [PC04b], and in the thesis of Ziliang Lin [Lin08]. We focus in this section, rather, on
fields of a magnetic origin. This has been treated in Ref. [CW08], but for the case of an array
of microtraps, similar to the trap design studied in Ch. 5. The weakness of interactions
between ions contained in different trapping regions motivates our work here on ions in the
same trap. Two questions drive this work:
1. What types of forces may be created using wires in the ground plane of the elliptical
trap?
2. How does the magnitude of this force and the magnitude of the J coupling rate scale
with the trap size?
A state-dependent force based on magnetic fields requires a field gradient in space, giving
rise to a force F = −∇(m · B), where m is the magnetic moment of the atom. For the
present work, we will consider the Zeeman-split sublevels of the ground S state in a 40 Ca +
or 88 Sr + -like ion. The absolute value of the magnetic moment is then m = gJµBmJ, where
µB is the Bohr magneton and mJ is the magnetic quantum number for the projection on the
ˆz axis of the total angular momentum J. This choice is made to facilitate straightforward
estimates for the types of ions discussed in this thesis, and indeed, coherence times of
several seconds have been observed for such qubits encoded in decoherence-free subspaces
[HSKH + 05]. However, hyperfine levels may prove to be a better choice because of their
excellent coherence times even without such encoding.
The state-dependent force thus depends on the alignment of the atom’s magnetic mo-
ment in space, which follows the orientation of the local magnetic field. This force, for
example along direction ˆy, is given by
Fˆy = gJµB
mˆx
∂Bˆx
∂y
∂Bˆy
+ mˆy + mˆz
∂y
7.3.1 Calculation of the gradients and interaction strengths
∂Bˆz
. (7.7)
∂y
The calculation of the fields and field gradients can be done by direct numerical integration
of the applied surface currents. Methods used are similar to those employed by Wang et al.
[WLG + 09] in their design of magnetic gradients for individual ion addressing. We assume
that the wires are infinitesimally narrow; this becomes less accurate as the trap scale is
decreased, and more sophisticated methods must be employed. Also, we limit the current
through a given wire to 1 A, comparable to the maximal currents employed in neutral atom
traps [HHHR01]. The fact that eventually, to reduce heating rates, these traps will need to
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