Master equation is the fundamental differential equation governing open quantum systems. It describes how the density matrix $\rho(t)$ evolves when the system is coupled to an environment:
$$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H, \rho] + \mathcal{D}[\rho]$$
The first term is unitary (Hamiltonian evolution); the second is the Lindblad dissipator, modeling irreversible processes.
The dissipator is a sum over Lindblad operators $L_k$:
$$\mathcal{D}[\rho] = \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2} \{L_k^\dagger L_k, \rho\} \right)$$
Each $L_k$ represents a different decay channel: energy loss, dephasing, spontaneous emission, etc. The dissipator preserves positivity and trace of $\rho$—necessary for a valid density matrix.
QuTiP's mesolve integrates the master equation forward in time. Provide the Hamiltonian, collapse operators, initial state, and time points; get back expectation values of observables.
from qutip import * import numpy as np # Two-level system with decay H = 0.5 * sigmaz() gamma = 0.1 c_ops = [np.sqrt(gamma) * sigmam()] # Decay operator times = np.linspace(0, 10, 100) psi0 = basis(2, 0) # Start in ground state result = mesolve(H, psi0, times, c_ops, [sigmaz(), sigmam()]) # result.expect[0] is <σ_z>(t) # result.expect[1] is <σ_->(t)
The master equation is the standard framework for open quantum systems. Understanding it and Lindblad operators is essential for any QuTiP work.