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

transitions for the excited state polarizability (data taken from [137]). At wavelengths
where the 2s polarizability matches the 1s polarizability, a magic wavelength exists. In
the range from 400 nm to 1 µm and above three distinct magic wavelengths exist. The
broadest of these is located around 512.4 nm.
polarizabilitya.u.
1000
500
0
500
4.10 7
1000
5.10 7
6.10 7
7.10 7
wavelengthm
8.10 7
9.10 7
1.10 6
Figure A.1: Scalar polarizability of the 1s (red) and 2s (blue) states of atomic hydrogen,
calculated using equation A.1.
The method outlined above is limited in precision by the number of states that
are included in the sum. Fortunately, there are other approaches to calculating the
scalar polarizability of hydrogenic atoms [138, 139]. A closed analytical expression for
the scalar polarizability (in atomic units) is given by [138]
α1s(ω) = − 1
2 512ν
−
ω2 m=1
9 1m
(ν2 1m −1) 2 (ν1m +1) 8 (ν1m −2)
2
ν1m −1
·2F1 4,2−ν1m,3−ν1m;
ν1m +1
α2s(ω) = − 1
−
ω2
·2F1
2
m=1
2 18 ν 9 2m
(ν 2 2m +2) 7 (ν2m −4) 3
4,2−ν2m,3−ν2m;
176
2
ν2m −2
,
ν2m +2
(A.3)
(A.4)