Decoherence is the loss of quantum coherence due to environmental interactions. Dissipation is the loss of energy. Both are modeled via collapse operators in the master equation.

Energy relaxation (T1 decay): Spontaneous emission of energy to the environment.

• Operator: $L = \sqrt{\gamma_1} \sigma_-$
• Timescale: $T_1$ (energy relaxation time)
• Result: system decays to ground state

Dephasing (T2* decay): Loss of phase coherence without energy loss.

• Operator: $L = \sqrt{\gamma_{\phi}} \sigma_z$
• Timescale: $T_2^* < T_1$ (pure dephasing time)
• Result: superposition $(\vert 0 \rangle + |1\rangle)/\sqrt{2}$ becomes mixture

Coherence time: $T_2 = 2 T_1$ (best case, limited by T1); usually $T_2 < 2T_1$ due to dephasing.

Real qubits have both T1 and T2:

from qutip import *
import numpy as np
 
# Superconducting qubit with measured decoherence
T1 = 10.0     # microseconds
T2 = 5.0      # microseconds
 
H = 0.5 * sigmaz()
c_ops = [
    np.sqrt(1/T1) * sigmam(),        # T1 decay
    np.sqrt(2/T2 - 1/T1) * sigmaz()  # T2 dephasing
]
 
times = np.linspace(0, 20, 100)
psi0 = (basis(2, 0) + basis(2, 1)).unit()
result = mesolve(H, psi0, times, c_ops, [sigmaz()])

Decoherence limits quantum computation. Understanding and measuring T1 and T2 are essential for quantum engineering.