Optical Bloch Saturation and Power Broadening

Worked Example

Consider a system with Drive Rabi frequency Ω = 1, Detuning Δ = 0, and Pure dephasing \gamma_φ = 0. The population decay rate is fixed at \Gamma_1 = 1, and the coherence decay rate is \Gamma_2 = 1/2 + \gamma_φ = 0.5.

1. The total decay parameter D = Δ² + \Gamma_2² = 0² + 0.5² = 0.25.
2. The saturation parameter is s = (Ω² \Gamma_2)/D = (1² × 0.5)/0.25 = 2.
3. The steady-state inversion w = -1/(1+s) = -1/(1+2) = -0.333333333333333.
4. The excited state population p_e = (1+w)/2 = 0.333333333333333.
5. The quadrature coherence v = -(Ω \Gamma_2 w)/D = -(1 × 0.5 × (-0.3333))/0.25 = 0.666666666666666.
6. The in-phase coherence u = 0 since detuning Δ = 0.
7. The population line HWHM is √(\Gamma_2² + Ω² \Gamma_2) = √(0.5² + 1² × 0.5) = √(0.75) ≈ 0.866025.

Experiment

Open the Optical Bloch Saturation interactive module. Observe how the parameters alter the steady state:

1. Set Drive Rabi frequency to 1, Detuning to 0, and Pure dephasing to 0. Note the saturation parameter is 2, and the excited population is roughly 1/3, as calculated above.
2. Increase Drive Rabi frequency to 4. Notice that the excited population approaches 0.5 (specifically ~0.485) and the saturation parameter becomes large (32). The line HWHM also increases dramatically to ~2.87, demonstrating power broadening.
3. Set Drive Rabi frequency to 0. The system falls purely into the ground state (inversion = -1, excited population = 0). The line HWHM reports "No driven line" because there is no coherent drive to produce a measurable spectrum.
4. Introduce Pure dephasing of 0.5 with Drive Rabi frequency 2 and Detuning 1. This increases the total coherence decay rate \Gamma_2 from 0.5 to 1.0, thereby affecting the saturation parameter and the line width.

Limitations

This model provides only the steady-state solutions, not the transient Rabi oscillations that occur immediately after the drive is turned on. It assumes a fixed population decay rate \Gamma_1 = 1, and does not account for thermal detailed balance (it assumes zero temperature where upward thermal transitions are negligible).

References

The optical Bloch equations and the phenomena of saturation and power broadening are derived in detail in the context of atomic physics in classical texts and theses, such as this diploma thesis (Huber).