Ph.D. Thesis - Physics

Figure 5-17: Dependence of the trap depth D on the trap scale r0 for a constant secular
frequency ω. The values for r0 = 1 mm, the approximate scale of the lattice trap presented
in this chapter, are D = 0.3 eV, ω = 2π×200 kHz, and Ω = 10ω.
ω ∝ ecV
mr 2 0 Ω.
(5.10)
We note that in order to hold ω constant as r0 is varied, the following scaling law must
hold: r2 0 ∝ V/Ω. Whether one reduces V or increases Ω, or some combination thereof,
the trap depth is reduced in proportion to either V 2 or 1/Ω2 . Taking some typical baseline
experimental parameters for the lattice trap, we plot in Fig. 5-17 the trap depth as a function
of the trap scale. In this particular case, we varied the drive frequency Ω, but according to
the above argument, varying V would yield the same trap depth.
The result given in Fig. 5-17 completes our exposition of the scaling behavior of lattice
traps. For ion-ion distances for which an appreciable coupling rate might be achieved, as
in our above example for J = 1 kHz at d = 50 µm and ω = 2π×200 kHz, the trap depth
is reduced to the order of a mere 100 µeV. While there is no fundamental reason why a
laser-cooled ion may not be trapped at such a depth, such trapping has never been reported,
to our knowledge, in the literature. In fact, this value is below the Doppler limit for most
ions. We again note that even a hypothetical factor of 100 improvement in the ratio of
trap depth to secular frequency would result in a trap depth of only ≈ 10 meV, still a very
challenging figure.
5.7 Conclusions
In this chapter we have presented the design, testing, and evaluation of a macroscopic
lattice ion trap. Our main experimental achievement was verifying that, for two very
different systems (atomic and macroion), our theoretical model of the trap is accurate. The
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