An open qubit coupled to an environment obeys the Lindblad (GKSL) master equation for its density matrix ρ:
dρ/dt = -i[H,ρ] + Σ_k γ_k ( L_k ρ L_k† - ½{L_k†L_k, ρ} )
Solving this directly gives the smooth, deterministic decay of coherence and purity you can read off above. But the Monte Carlo wavefunction method (quantum-jump / trajectory unraveling, Dalibard–Castin–Mølmer 1992) reaches the exact same ρ(t) a different way: evolve each pure state |ψ⟩ under a non-Hermitian effective Hamiltonian
H_eff = H - (i/2) Σ_k γ_k L_k†L_k
which continuously shrinks the state's norm. At every timestep the lost norm dp = 1-‖ψ‖² is the probability of a real quantum jump: with probability dp the state instead collapses discontinuously to L_k|ψ⟩/‖L_k|ψ⟩‖, chosen with weight γ_k⟨ψ|L_k†L_k|ψ⟩; otherwise it just renormalizes and continues. Each colored dot on the sphere is one such stochastic trajectory. Average the individual pure-state Bloch vectors over many trajectories and — as the count grows — you recover exactly the smooth ρ(t) of the master equation: this is the white arrow shrinking into the sphere. This is the same algorithm QuTiP's mcsolve uses, and it is how real photon-counting / continuous-measurement experiments on qubits are simulated, because each trajectory literally corresponds to one possible measurement record of the environment.
- Amplitude damping — L = √γ σ₋: relaxation to |0⟩, e.g. spontaneous photon emission (T₁ decay).
- Pure dephasing — L = √(γ/2) σ_z: random phase kicks that kill coherence without changing populations (T₂ decay).
- Depolarizing — equal-rate σ_x, σ_y, σ_z jumps: isotropic noise that shrinks the Bloch vector uniformly toward the maximally-mixed state.
- Ω (R_x drive) — a coherent Rabi rotation about x competing with the dissipation, so the trajectories spiral instead of just decaying.
- Coherence / Purity — |ρ₀₁| = ½√(r_x²+r_y²) and Tr(ρ²) = ½(1+|r|²) computed from the ensemble-averaged Bloch vector r; Tr(ρ²)=1 is pure, 0.5 is maximally mixed.