# 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 = H_{0} + H_{int} = ħ[(ω_{g }+ ω_{e})σ_{0} − (ω_{g} − ω_{e})σ_{z}]/2 − pE(t)σ_{x}.

By defining the following

s_{x} = ρ_{12} + ρ_{21},

s_{y} = i(ρ_{12} − ρ_{21}),

s_{z} = ρ_{11 }− ρ_{22},

one may arrive at the optical Bloch vector equation

d**s**/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.

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## How to Cite

*Proceedings of the Samahang Pisika ng Pilipinas*

**13**, SPP-1995-TP-01 (1995). URL: https://proceedings.spp-online.org/article/view/SPP-1995-TP-01.