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

a laser field with frequency ω. This electric field induces an atomic dipole moment p,
where [36]
p = α(ω) E. (2.23)
α(ω) is the complex polarizability, which depends on the laser frequency ω. The induced
dipole gives rise to an energy shift given by
∆EAC = − 1
2 〈p· E〉 = − 1
Re(α)I, (2.24)
2ǫ0c
where the angular brackets indicate the time average. The intensity I is given by I =
2ǫ0c| E| 2 .
In a quantum-mechanical description the scalar polarizability α is given by
α = 1
2
k
|〈k|µ|j〉| 2
1
ωkj +ω +
1
ωkj −ω
. (2.25)
The energy shift of an energy level j in the presence of a laser beam with intensity I is
thus described by
∆EAC = − 1
4ǫ0c
k
|〈k|µ|j〉| 2
1
ωkg +ω +
1
ωkg −ω
I, (2.26)
where the sum is over all possible energy levels k. This formula is evaluated further in
appendix A.
In a semi-classical approximation the AC-Stark shift of the ground state, typically
called the dipole potential, can be expressed by
1
ωD2 +ω +
Udip(r) = − Γ2 D2
8Isat
1
ωD2 −ω
I(r), (2.27)
where ΓD2 is the D2-line transition linewidth and Isat is the saturation intensity. In cases
where the rotating wave approximation can be invoked, this equation simplifies to
Udip(r) = 3πc2
2ω 3
Γ
I(r), (2.28)
∆
where the expression of the saturation intensity is inserted [36], and ∆ describes the
detuning from resonance, ∆ ≡ ωlaser − ωD2. The dipole potential then scales as I
∆ .
Depending on the sign of ∆, a repulsive (for blue-detuned light) or an attractive (for
18