Quantum Trajectories (Overview)

Two views of the same physics

The Lindblad equation describes the average evolution of $\rho(t)$ over many runs. But a single experimental run — one atom watched by one photodetector — does not see a smooth density matrix. It sees a trajectory: smooth no-click evolution punctuated by sudden quantum jumps when the detector fires. Quantum trajectory theory (also called the Monte Carlo wavefunction or unraveling of the master equation) describes these individual runs, and recovers the Lindblad equation when averaged. It is both a conceptual picture and a powerful numerical method, since it propagates a state vector of dimension $d$ rather than a density matrix of dimension $d^2$ .

The unraveling: continuous evolution plus jumps

Replace the density matrix by a stochastically evolving pure state $|\psi(t)\rangle$ . Over a small step $dt$ , two things can happen.

No jump. With high probability the state evolves under the non-Hermitian effective Hamiltonian

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

followed by renormalization. The anti-Hermitian part gradually shrinks the norm; the lost norm is the probability that a jump occurred.

Jump. With probability $dp_k = \langle\psi|L_k^\dagger L_k|\psi\rangle\,dt$ , channel $k$ fires and the state jumps discontinuously,

|\psi\rangle \;\longrightarrow\; \frac{L_k|\psi\rangle}{\lVert L_k|\psi\rangle\rVert}.

For amplitude damping ( $L=\sqrt\gamma\,\sigma_-$ ) a jump is the detection of an emitted photon and collapses the qubit to its ground state; between jumps the no-jump evolution actually increases the ground-state amplitude, because not seeing a photon is itself information.

Recovering the master equation

Average the stochastic update over the jump/no-jump branches, weighting each by its probability. The jump term contributes $\sum_k L_k\rho L_k^\dagger$ (the recycling term), and the no-jump non-Hermitian evolution contributes the anticommutator $-\tfrac12\{L_k^\dagger L_k,\rho\}$ together with the unitary part. Identifying $\rho = \mathbb{E}[\,|\psi\rangle\langle\psi|\,]$ , the ensemble average exactly reproduces

\dot\rho = -\tfrac{i}{\hbar}[H,\rho] + \sum_k\!\left(L_k\rho L_k^\dagger - \tfrac12\{L_k^\dagger L_k,\rho\}\right).

So the Lindblad equation is the mean over all trajectories — but there are infinitely many unravelings (jump unraveling, quantum-state diffusion, homodyne) that all average to the same $\rho(t)$ while describing different conditional measurements of the bath.

Why it is useful

For a system of $n$ qubits, the density matrix has $4^n$ entries but a trajectory state vector has only $2^n$ . Simulating many trajectories and averaging often beats integrating the full master equation, especially for large systems — this is the basis of Monte Carlo wavefunction solvers in software like QuTiP. Conceptually, trajectories also connect open-system dynamics to continuous measurement and feedback control, where one conditions the state on the actual detector record rather than averaging it away.