Optical Bloch equations for a driven, damped two-level atom

Units are SI, with Ω, δ and Γ angular frequencies in s⁻¹ (divide by 2π to compare with spectra in MHz); no metric signature enters, because the atom is non-relativistic and the laser is a classical field. Take a ground state |g⟩ and an excited state |e⟩ split by ħω₀, in the field E₀ε̂ cos(ω_L t). In the frame rotating at ω_L, with the terms at 2ω_L dropped, Quantum Mechanics II's rotating-wave approximation (quoted) leaves H = −(ħδ/2)σ_z + (ħΩ/2)σₓ, with δ = ω_L − ω₀ and the field's phase chosen so that Ω is real and positive. Topic 14.03 derived the vacuum's term Γ(σ₋ρσ₊ − ½σ₊σ₋ρ − ½ρσ₊σ₋) for an undriven atom, and it is kept unchanged. The frame change cannot alter it, because the rotating phases that σ₋ and σ₊ acquire are opposite and cancel in every product. Keeping it is an approximation with a size: the drive splits the dressed atom by about Ω, while the vacuum's coupling varies only on the scale ω₀. For rubidium's 780 nm line at Ω = Γ that ratio is 1.6 × 10⁻⁸.

For any operator A the master equation gives d⟨A⟩/dt = (i/ħ)⟨[H, A]⟩ + Γ⟨σ₊Aσ₋ − ½σ₊σ₋A − ½Aσ₊σ₋⟩. Apply it to the Pauli matrices, with u = ⟨σₓ⟩, v = ⟨σ_y⟩ and w = ⟨σ_z⟩. H is (ħ/2)(Ωσₓ − δσ_z), and [σᵢ, σⱼ] = 2iεᵢⱼₖσₖ turns its part into the precession dr/dt = a × r of a spin in a field, with r = (u, v, w) and a = (Ω, 0, −δ). In the dissipator, σ₋σ₋ = 0 and σ₋σ₊σ₋ = σ₋ give dσ₋/dt = −(Γ/2)σ₋, while σ₊σ_zσ₋ = −σ₊σ₋ gives dσ_z/dt = −Γ(1 + σ_z). Together, derived: du/dt = δv − Γu/2, dv/dt = −δu − Ωw − Γv/2, dw/dt = Ωv − Γ(w + 1). Each term is a rate times a dimensionless component, so the dimensions agree. Two limits check them. With Γ = 0 and δ = 0, starting from the ground state, w = −cos Ωt and v = sin Ωt: Quantum Mechanics II's lossless Rabi flopping. With Ω = 0 the population decays as e^(−Γt) and the coherence as e^(−Γt/2), Unit 12's Weisskopf-Wigner amplitudes.

The inversion relaxes at Γ and the transverse components at Γ/2, which defines T₁ = 1/Γ and T₂ = 2/Γ. The factor of two is derived, not fitted: the excited amplitude decays as e^(−Γt/2), the coherence is linear in it and the population quadratic. An environment that randomises the phase without causing transitions, a σ_z jump operator at rate γ_φ/2 as in topic 14.02's phase damping (quoted), adds γ_φ to the transverse rate only, so 1/T₂ = 1/(2T₁) + γ_φ and T₂ can only fall below 2T₁. With Γ/2 → 1/T₂ and Γ → 1/T₁ these are Felix Bloch's 1946 equations for nuclear magnetisation, quoted, with T₁ and T₂ as measured parameters; in NMR the reservoir is thermal, and topic 14.04's w → −1/(2n̄ + 1) replaces −1, an equilibrium polarisation of only 3.2 × 10⁻⁵ for protons at 9.4 T and 298 K. Measurements respect the bound: water protons have T₂ ≈ T₁ of a few seconds, tissue protons typically T₂ ≈ T₁/10, and an isolated rubidium atom's weak-probe line is Γ = 2π × 6.07 MHz wide, as its 26.23 ns lifetime predicts. A weak probe measures 2/T₂, so that is T₂ = 2T₁.

Set the derivatives to zero. The first equation gives u = 2δv/Γ, the third v = Γ(w + 1)/Ω, and the second then fixes w. With D = 4δ² + Γ² + 2Ω², derived: u = 4δΩ/D, v = 2ΓΩ/D, w = −(4δ² + Γ²)/D, and ρₑₑ = (1 + w)/2 = (Ω²/4)/(δ² + Γ²/4 + Ω²/2). Every term of D is a rate squared, so ρₑₑ is dimensionless. The third equation is also an energy balance: Ωv/2 = Γρₑₑ, photons taken from the laser per second equal photons scattered. For a weak drive ρₑₑ → (Ω²/4)/(δ² + Γ²/4), and R = Γρₑₑ becomes (πΩ²/2)L(δ), with L a normalised Lorentzian of full width Γ: Quantum Mechanics II's golden-rule rate, the weak-field limit. The steady state is mixed, |r|² = 1 − 4Ω⁴/D²; at Ω = Γ on resonance (u, v, w) = (0, 2/3, −1/3) and |r| = 0.745. It is reached through damped Rabi oscillations: on resonance the v-w block has eigenvalues −3Γ/4 ± i√(Ω² − Γ²/16), so once Ω exceeds Γ/4 the transient oscillates at √(Ω² − Γ²/16) and dies away at 3Γ/4.

Raise Ω at fixed δ and ρₑₑ climbs towards 1/2, never past it. The field that drives |g⟩ → |e⟩ drives |e⟩ → |g⟩ by stimulated emission at the same rate, so a strong drive can only equalise the populations, and the Bloch vector shrinks to the centre of the ball. The scattering rate R = Γρₑₑ is capped at Γ/2. For rubidium-87, whose measured 26.23 ns lifetime gives Γ = 3.812 × 10⁷ s⁻¹, that is 1.91 × 10⁷ photons per second, a radiation-pressure force of at most ħkΓ/2 = 1.62 × 10⁻²⁰ N and a deceleration of 1.12 × 10⁵ m s⁻². Write s₀ = 2Ω²/Γ²; then ρₑₑ = (s₀/2)/(1 + s₀ + 4δ²/Γ²), half the peak falls at 4δ² = Γ²(1 + s₀), and the full width is Γ√(1 + s₀), derived. With ħΩ = dE₀, I = ½cε₀E₀² and Unit 12's Γ = ω₀³d²/(3πε₀ħc³), quoted, s₀ = I/Iₛₐₜ with Iₛₐₜ = πhcΓ/(3λ³) = 16.7 W m⁻² for rubidium. At low intensity the same numbers give the resonant cross-section ħω₀Γ/(2Iₛₐₜ) = 3λ²/(2π) = 2.91 × 10⁻¹³ m², the one absorption imaging uses to count cold rubidium atoms.

Each input has a limit with a number on it. The rotating-wave approximation drops counter-rotating terms that shift the line by the Bloch-Siegert amount Ω²/(4ω₀), quoted: 4 × 10⁻⁹ Γ for rubidium at Ω = Γ, but many linewidths for a superconducting qubit driven at Ω/ω₀ of a per cent or more. The Markov step needs Ω and Γ far below the inverse of the vacuum's correlation time, which is of order ω₀. Two levels need the others far away, which is why experiments use a closed cycling transition and circular polarisation. A moving atom sees δ − k⋅v, so the Doppler spread of a room-temperature rubidium vapour, 0.51 GHz, is inhomogeneous broadening: the formula averaged over velocities, not a shorter T₂. And the field must be classical. In a cavity whose coupling g beats both its loss and Γ, the photon number of the mode matters, the vacuum Rabi splitting appears, and Unit 11's Jaynes-Cummings dynamics replaces these equations. The steady state also leaves something out: the spectrum and photon statistics of the scattered light need the quantum regression theorem, quoted in topic 14.07.