Copyright by Kirsten Viering 2006 - Raizen Lab - The University of ...

and
c ′ g(t) = cg(t) (2.12)
c ′ e(t) = ce(t)e −iδt , (2.13)
where δ = ωl − ωa defines the detuning from resonance. By making this substitution
and making the Rotating Wave Approximation [9] one derives the following equations
[10]
and
i dc′ g(t)
dt = c′ e(t) Ω
2
(2.14)
i dc′ e(t)
dt = c′ g(t) Ω
2 − c′ eδ. (2.15)
It is now possible to diagonalize the matrix for the perturbative part of the
Hamiltonian to the following form
H ′ =
2
thus the shifted energy levels are given by
−2δ Ω
Ω 0
, (2.16)
Eg,e =
2 (−δ ± Ω′ ), (2.17)
where Ω ′ ≡ √ Ω 2 + δ 2 . If we assume Ω ≪ δ the energy levels are shifted by
respectively.
∆Eg,e = ± Ω2
4δ
Similarly, if we assume Ω ≫ |δ| the levels shift by
∆Eg,e = ±sgn(δ) Ω
2
A schematic for the two-level energy shift is shown in fig. 2.1.
6
(2.18)
. (2.19)