Two-level Bloch master equation in the rotating frame with drive Ω, detuning Δ, T₁ relaxation and T₂ transverse decay. The Bloch vector spirals inside the sphere — the geometric picture of decoherence with live populations P(|0⟩), purity Tr ρ², and time traces of u_x, u_y, u_z.
About this model
This page integrates the Bloch master equation for a driven two-level system in the rotating frame: coherent drive Ω, detuning Δ, longitudinal relaxation T₁, and transverse decay T₂. The Bloch vector u = (u_x, u_y, u_z) spirals inside the unit sphere as populations and coherences damp — the geometric picture of open-system decoherence. Live readouts show P(|0⟩), purity Tr(ρ²), and time traces of the three Bloch components. The model is a Markovian Lindblad / Bloch–Redfield caricature of a qubit coupled to a memoryless bath; it omits non-Markovian noise, multi-level leakage, and pulse distortions. You vary Ω, Δ, T₁, and T₂ to see Rabi nutation compete with relaxation and dephasing, including the usual constraint T₂ ≤ 2 T₁ for physical decay rates.
Who it's for: Quantum information, NMR, superconducting qubits, and advanced quantum optics courses on open two-level systems.
Key terms
Lindblad equation
Bloch vector
T1 relaxation
T2 dephasing
Decoherence
Purity
How it works
Two-level Lindblad / Bloch master equation: drive Ω, detuning Δ, T₁ relaxation and T₂ transverse decay shrink the Bloch vector inside the sphere — a hands-on visualization of qubit decoherence and Rabi flopping with dissipation.
Frequently asked questions
Why does the Bloch vector shrink inside the sphere?
Unit length |u| = 1 corresponds to a pure state (Tr ρ² = 1). Amplitude damping and pure dephasing drive the density matrix toward a mixed state, so |u| < 1 and purity falls. A common misconception is that decoherence only rotates u; dissipative channels also reduce its length.
How are T₁ and T₂ related?
Energy relaxation at rate 1/T₁ also destroys transverse coherence, so T₂ cannot exceed 2 T₁ in the standard Bloch equations. Additional pure dephasing shortens T₂ further: 1/T₂ = 1/(2T₁) + 1/T_φ. Setting T₂ > 2 T₁ is unphysical in this model.
What happens if Ω = 0 and Δ = 0?
Without drive or detuning the longitudinal component u_z relaxes exponentially toward the ground state on timescale T₁, while u_x and u_y decay on T₂. Populations approach thermal (here effectively |0⟩ at zero temperature in the usual pedagogical limit) and coherences vanish.