Application of projection techniques to the optical Bloch vector equation

Abstract

An exact formal solution to the problem of the two level atom under an arbitrarily time- dependent field was recently proposed by Muriel using the Method of Iterated Projections (MIP). The starting equation was the quantum Liouville equation for the density matrix operator:
iħ∂ρ/∂t = [H,ρ],
where
H = H₀ + H_int = ħ[(ω_g + ω_e)σ₀ − (ω_g − ω_e)σ_z]/2 − pE(t)σ_x.
By defining the following
s_x = ρ₁₂ + ρ₂₁,
s_y = i(ρ₁₂ − ρ₂₁),
s_z = ρ₁₁ − ρ₂₂,
one may arrive at the optical Bloch vector equation
ds/dt = Ω x s,
where s = (s_x, s_y, s_z) and the torque vector Ω = (Ω_x, Ω_y, Ω_z) with
Ω_x(t) = −2pE(t)/ħ,
Ω_y = 0,
Ω_z = ω = ω_e − ω_g.

In this paper, we show the consistency of the proposed solution by using the same method (MIP) to the Bloch equation.