Collapse Operators

Collapse operators are synonymous with Lindblad operators—they're the quantum jumps that model irreversible evolution. The name comes from the Monte Carlo trajectory interpretation: each collapse operator represents a possible “jump” (measurement outcome or dissipative event) the system can undergo.

In Monte Carlo simulation, collapse operators are applied stochastically. The system evolves unitarily until a random time, when a collapse operator “jumps” the state, then unitary evolution resumes. Running many trajectories gives the ensemble average, which matches the master equation result.

Collapse operators depend on the physical system:

Superconducting qubits:

• Energy decay: $\sigma_- = \frac{1}{\sqrt{2}}(s_x + is_y)$ (relaxation)
• Dephasing: $\sigma_z$ (pure dephasing)

Cavity QED:

• Photon loss: $\sqrt{\kappa} a$ (cavity decay, $a$ is annihilation)
• Atomic decay: $\sqrt{\gamma} \sigma_-$ (spontaneous emission)

Trapped ions:

• Spontaneous emission: $\sqrt{\gamma} \sigma_-$
• Heating: $\sqrt{\dot{n}} a^\dagger$ (vibrational mode heating)

Collapse operators are phenomenological—they capture environment effects without explicitly simulating the environment. Well-chosen operators match experimental measurements of decoherence rates (T1, T2, linewidths, etc.).