According to Breuer-Petruccione, the SDE quantum trajectory evolution for heterodyne detection
$$d\psi=-iH\psi dt-\frac{\gamma}{2}\left(C^\dagger C-\langle C^\dagger \rangle_{\psi} C+\frac{1}{2}\langle C\rangle_{\psi} \langle C^\dagger\rangle_{\psi} \right)\psi dt +\sqrt{\gamma}(C-\langle C\rangle_\psi)\psi dW(t) +\frac{\sqrt{\gamma}}{2}(\langle C\rangle_\psi dW(t)-\langle C^\dagger\rangle_\psi dW^*(t))\psi $$
is equivalent to the SDE of quantum state diffusion (stochastic collapse)
$$d\psi=-iH\psi dt-\gamma\left(\frac{1}{2} C^\dagger C-\langle C^\dagger \rangle_{\psi} C+\frac{1}{2}\langle C\rangle_{\psi} \langle C^\dagger\rangle_{\psi} \right)\psi dt +\sqrt{\gamma}(C-\langle C\rangle_\psi)\psi dW(t) $$
as can be obtained trough a phase transformation $\psi(t)\rightarrow e^{i\phi(t)}\psi(t)$.
My question is: how deep can this correspondence been interpreted:
- Is the correspondence only on the level of the density matrix (i.e. separate unraveling) or can individual QSD samples 'physically' be interpreted as a time- evolution under heterodyne measurement?
- Does the global phase of the ensemble remain the same when time evolves?
