Quantum Trajectory Theory

A quantum Trajectory Theory unravelling represents a Lindblad equation by an ensemble of stochastic pure states. In units $\hslash =1$, define the non-Hermitian effective HamiltonianFor a normalized state $|\psi ⟩$, channel $\alpha$ has instantaneous jump rate $r_{\alpha }=⟨\psi |L_{\alpha }^{†}{L}_{\alpha }|\psi ⟩$. During a small interval $dt$, that jump occurs with probability $r_{\alpha }dt$ and sends the state to $L_{\alpha }|\psi ⟩/\sqrt{r_{\alpha }}$. A zero rate means that channel cannot jump. Otherwise, with total no-jump probability $1-{\sum }_{\alpha }{r}_{\alpha }dt$, evolve with $I-i{H}_{\mathrm{eff}}dt$ and renormalize.

To verify the ensemble evolution, put $\rho =|\psi ⟩⟨\psi |$. The conditional normalizations cancel their probability weights, givingExpanding recovers the Lindblad equation. Different monitoring choices can give different trajectories with the same ensemble density matrix. For spontaneous emission, $L=\sqrt{\Gamma }|g⟩⟨e|$ gives jump rate $\Gamma |⟨e|\psi ⟩{|}^2$ and resets the atom to its ground state.^[ref]

A pure stationary state is dark when every jump operator acts on it as a scalar, after an allowed shift of the jump operators, and the corresponding effective Hamiltonian preserves its ray. Every trajectory that reaches such a state remains on that same ray.