Lindblad Operators

Lindblad operators (also called jump operators or collapse operators) specify the decay channels in the master equation. Each $L_k$ models one physical process: energy loss, dephasing, spontaneous emission, etc.

A Lindblad operator is any operator $L_k$. Its effectiveness is controlled by a rate $\gamma_k$ (decay rate). For a two-level system, common operators are:

• Decay: $L = \sigma_- = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ (lowering operator)
• Dephasing: $L = \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
• Bit flip: $L = \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Pass Lindblad operators to the master equation solver as a list:

from qutip import *
import numpy as np
 
# Decay rate (T1)
T1 = 10.0  # microseconds
gamma1 = 1.0 / T1
 
# Dephasing rate (T2*)
T2 = 5.0
gamma2 = 1.0 / T2
 
# Collapse operators: decay and dephasing
c_ops = [
    np.sqrt(2 * gamma1) * sigmam(),     # Energy loss
    np.sqrt(gamma2) * sigmaz()          # Dephasing
]
 
H = 0.5 * sigmaz()
psi0 = basis(2, 0)
times = np.linspace(0, 20, 100)
result = mesolve(H, psi0, times, c_ops, [sigmaz()])

The rate scaling (e.g., $\sqrt{2\gamma}$ for decay) comes from the Lindblad master equation form. Different physical systems have different collapse operators—cavity QED, superconducting qubits, and trapped ions all use different sets.