Experiments to Control Atom Number and Phase-Space Density in ...

Appendix A
Magic Wavelength of Hydrogen
The concept of a magic wavelength is important for precision frequency mea-
surements of optically trapped atoms. In general, an optical potential will introduce an
AC-Stark shift of the energy levels, leading to a spatial dependence of the transition
frequency within the trapped cloud, and limiting the precision achievable. Many new
technologies rely on precision time; the Global Positioning System (GPS) is just one
important example.
This section introduces the idea of a magic wavelength trap, and how this provides
a unique environment for precision spectroscopy. Subsequently, the magic wavelength
for hydrogen is determined theoretically.
A.1 Magic Wavelength
An optical clock using trapped neutral atoms is desirable for two main reasons:
long interaction times and high short-term stability [127, 128]. Because trapped neutral
atoms exhibit both qualities, they can potentiall provide a frequency standard superior
to single-ion and atomic fountain clocks. This would be an important step towards new
tests of fundamental physics and improved measurements of fundamental constants.
Chapter 2.6.2 showed that the shift of atomic energy levels of an atom in the
presence of an external light field due to the AC-Stark effect is given by (compare
equation 2.26)
∆EAC = − 1
Re(α)I (A.1)
2ǫ0c
= − 1
|〈k|µ|j〉|
4ǫ0c
2
1
ωkg +ω +
1
I. (A.2)
ωkg −ω
k
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