= Quantum Trajectory Theory
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= Quantum-trajectory unravelling
{synonym}
A <quantum Trajectory Theory> unravelling represents a <Lindblad equation> by an ensemble of stochastic <pure states>. In units $\hbar=1$, define the non-Hermitian effective <Hamiltonian>
$$
H_{\rm eff}=H-\frac i2\sum_\alpha L_\alpha^\dagger L_\alpha.
$$
For a normalized state $|\psi\rangle$, channel $\alpha$ has instantaneous jump rate $r_\alpha=\langle\psi|L_\alpha^\dagger L_\alpha|\psi\rangle$. During a small interval $dt$, that jump occurs with <probability> $r_\alpha dt$ and sends the state to $L_\alpha|\psi\rangle/\sqrt{r_\alpha}$. A zero rate means that channel cannot jump. Otherwise, with total no-jump <probability> $1-\sum_\alpha r_\alpha dt$, evolve with $I-iH_{\rm eff}dt$ and renormalize.
To verify the ensemble evolution, put $\rho=|\psi\rangle\langle\psi|$. The conditional <normalizations> cancel their <probability> weights, giving
$$
\mathbb E[\rho(t+dt)]=M_0\rho M_0^\dagger+dt\sum_\alpha L_\alpha\rho L_\alpha^\dagger+O(dt^2),\qquad M_0=I-iH_{\rm eff}dt.
$$
Expanding recovers the <Lindblad equation>. Different monitoring choices can give different trajectories with the same ensemble <density matrix>. For spontaneous emission, $L=\sqrt\Gamma|g\rangle\langle e|$ gives jump rate $\Gamma|\langle e|\psi\rangle|^2$ and resets the atom to its <ground state>.<A Wave Function approach to dissipative processes by Yvan Castin and others>[Sections 1–2, equations (4)–(16), jump probabilities, conditional normalization and master-equation equivalence]{ref}
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