Thesis (pdf) - Swinburne University of Technology

system is expanded into
Ψ = �
i=1,2
Chapter 2: Theoretical Background
ciφi
where the coefficients ci are complex and normalise the state.
(2.14)
A pure state is defined by the density matrix ρ = |Ψ〉〈Ψ|. The time evolution of
the density matrix for a Hamiltonian H is given by the von Neumann equation:
i¯h dρ
dt
= [H, ρ] (2.15)
As we restrict ourselves to an atom with two levels only, with |φ1〉 = |g〉
the ground state and |φ2〉 = |e〉 the excited state, the density matrix can then
easily be written down
ρ =
⎛
⎝ ρee ρeg
ρge ρgg
⎞
⎠ =
⎛
⎝ |ce| 2 cec ∗ g
cgc ∗ e |cg| 2
⎞
⎠ (2.16)
Normalisation requires |cg| 2 +|ce| 2 = ρgg +ρee = 1. We can now insert equation
(2.16) into equation (2.15). To include spontaneous emission of photons by the
atoms, we include a finite lifetime Γ −1 of the excited state. The time evolution
of the two level system can now be written down as [All75]:
dρgg
dt = +Γρee + i
2 (Ω∗R ˜ρeg − ΩR ˜ρge)
dρee
dt = −Γρee + i
2 (Ω∗R ˜ρge − ΩR ˜ρeg)
d˜ρge
dt
d˜ρeg
dt
� �
Γ
= − + i∆
2
� �
Γ
= − + i∆
2
˜ρge + i
2 Ω∗ R(ρee − ρgg)
˜ρeg + i
2 ΩR(ρgg − ρee) (2.17)
Here ˜ρij = ρije −i∆t , with ∆ being the detuning. The Rabi frequency ΩR is
defined by equation (2.6).
Equations (2.17) are called the optical Bloch equations. Using the inversion
w = ρgg − ρee, the normalisation and the fact that ρeg = ρ ∗ ge, we can simplify
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