Open Quantum Systems
Lindblad master equation — decoherence and dissipation of a qubit
Environment & Drive
Initial state
|+x⟩ (superposition)
|+y⟩
|e⟩ (excited)
|g⟩ (ground)
Decay γ (T₁)
0.3
Dephasing γφ (T₂)
0.5
Drive freq Ω
1.0
Drive amp
0.5
Reset State
Purity Tr(ρ²)
1.000
Bloch |r|
1.000
⟨σz⟩
0.000
Coherence |ρ₀₁|
0.500
T =
0.0
Lindblad equation:
dρ/dt = -i[H,ρ] + Σₖ γₖ(LₖρLₖ† - ½{Lₖ†Lₖ,ρ})
Collapse operators:
• L₁ = √γ·σ₋ (amplitude decay / T₁)
• L₂ = √(γφ/2)·σz (pure dephasing / T₂)
The
Bloch sphere
shrinks as decoherence destroys quantum superpositions. Mixed states live inside the sphere (|r| < 1). Driving creates Rabi oscillations while the environment damps them.