Qubit Lindblad Relaxation and Dephasing

The Markovian Lindblad master equation describes the evolution of a quantum system interacting with its environment. For a qubit with energy splitting ω, we can model distinct physical processes: energy relaxation (amplitude damping) and pure dephasing.

The Hamiltonian is H = ω |e⟩⟨ e| (setting ℏ=1). The dissipation is modeled by jump operators: downward transitions |g⟩⟨ e| with rate \Gamma_(down), upward transitions |e⟩⟨ g| with rate \Gamma_(up), and pure dephasing modeled by √(\gamma_φ/2) \sigma_z.

worked-example

Let us calculate the state at t = 1 starting from the completely excited state (p_e(0) = 1, c(0) = 0). We assume \Gamma_(down) = 1, \Gamma_(up) = 0, and \gamma_φ = 0.

The equilibrium population is p_(eq) = \Gamma_(up) / (\Gamma_(down) + \Gamma_(up)) = 0.
The population decay rate is \Gamma_(down) + \Gamma_(up) = 1.
Therefore, the excited state population at t = 1 is p_e(1) = 0 + (1 - 0) e^(-1 × 1) = e⁻¹ ≈ 0.36787944117144233.
The coherence starts at 0 and remains 0.
The density matrix purity is p_e² + (1-p_e)² = (0.3678...)² + (1-0.3678...)² ≈ 0.5349116841303407.

experiment

You can interact with the quantum trajectory of the qubit in theInteractive Qubit Lindblad Relaxation and Dephasing module.

Set Initial state to PlusI to start with a coherent superposition.
Increase the Pure dephasing rate to 0.25 to see the off-diagonal elements shrink faster than the populations change.
Notice how the Purity drops below 1 as the environment extracts information from the qubit.
Observe the Minimum Eigenvalue; it remains non-negative at all times, ensuring the state remains a valid, physical density matrix.

limitations

The Lindblad equation assumes the Born-Markov approximation: the environment is large, memoryless, and unaffected by the qubit. It also ignores any non-secular terms that could become relevant if the decay rates are comparable to the qubit frequency ω. This phenomenological model defines independent rates which do not necessarily follow detailed balance for a bath at a specific positive temperature.

references

A comprehensive introduction to open quantum systems and the derivation of the Lindblad equation can be found in Preskill's lecture notes:John Preskill, Physics 219: Quantum Computation, Chapter 3.

Qubit Lindblad Relaxation and Dephasing