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Klaus Mølmer helped develop a wave-function description of dissipative quantum evolution with Yvan Castin and Jean Dalibard. Their 1992 manuscript, reproduced in 2008, represents individual conditioned evolutions through stochastic jumps and intervening nonunitary evolution.^[ref]

Averaging many trajectories recovers the corresponding Lindblad equation. This quantum-trajectory unravelling connects an ensemble density operator with possible individual measurement records; the trajectory depends on how the environment is monitored.

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]

Yvan Castin, with Jean Dalibard and Klaus Mølmer, formulated a stochastic wave-function approach to dissipative quantum systems. Their construction samples conditional jumps and no-jump evolution so that the ensemble reproduces a density-operator master equation.^[ref]

Using state vectors can reduce the cost of numerical simulations compared with directly storing a full density matrix, although many trajectories may be needed for precise averages. The 2008 posting reproduces a 1992 manuscript and explicitly acknowledges earlier related quantum-filtering work.