Mcsolve (Monte Carlo Trajectories)

Mcsolve runs stochastic trajectories of quantum evolution, applying collapse operators at random times. Each trajectory is a possible quantum history; averaging over many trajectories gives the ensemble result (matching mesolve).

Monte Carlo simulations are useful for understanding how noise manifests at the individual-event level and for systems where many collapse operators dominate.

from qutip import *
import numpy as np
 
# Simple decay
H = 0.5 * sigmaz()
c_ops = [0.1 * sigmam()]
times = np.linspace(0, 10, 50)
psi0 = basis(2, 1)
 
# Run 1000 trajectories
result = mcsolve(H, psi0, times, c_ops, [sigmaz()], ntraj=1000)
 
# result.expect: ensemble average (matches mesolve)
# result.trajectories: individual runs (if stored)

Each trajectory evolves via an effective non-Hermitian Hamiltonian:

$$H_{\text{eff}} = H - \frac{i\hbar}{2} \sum_k L_k^\dagger L_k$$

At each step, with probability $p_k(dt)$, collapse operator $L_k$ is applied (normalized jump). Otherwise, evolution continues. After $N$ trajectories, expectations average to the master equation solution.

Advantages:

• Natural interpretation of quantum jumps
• Efficient for many collapse operators
• Can track individual quantum events

Trade-offs:

• Slower than mesolve for small $N_{\text{traj}}$ (needs many runs for good statistics)
• More memory if storing trajectories
• Statistical noise in results

Use mcsolve when you need trajectory-level understanding or when many dissipation channels dominate.